Humans sometimes make errors. Perhaps you have been sloppy with a calculation on a mathematics examination. You made an error. While many see machines and communications systems as more reliable than humans when it comes to making errors, machines and communications systems do make errors too. One can think of this as the consequence of there being noise in any physical communication system, which by chance alters the information involved. Advances in sending information electronically, including streaming sound and video, have led mathematicians to take an interest in increasing the reliability of data which is transferred electronically. Digital technologies work so well that we often now have lost track of the mathematical triumphs that make these technologies as reliable as they are. An earlier column looked at some aspects of using codes to correct errors in digital technology systems. Here I will look at a kind of error that communications systems are subject to, errors which arise due to a symbol in a code word being dropped (or inserted) rather than changed. Deletion errors have recently gotten renewed attention and are much less understood than other kinds of transmission errors. First, let me provide some background related to using mathematics to correct errors.
Mathematics is often viewed as studying properties of patterns involving numbers or shapes, the investigations that give rise to the areas of mathematics called algebra and geometry (topology). We are familiar with numbers and live with the fact that the number seventeen in English or dix-sept in French can be represented in different ways, depending on the system we choose for representing it. Thus, seventeen is written as 17 using the symbol alphabet 0, 1, 2 , ....,9, XVII using the Roman Numeral System of representation (where the alphabet is I, V, X, L, C, D, and M) and 10001 in binary (alphabet 0 and 1). But a recent insight is that a number written in decimal, say 130542, can be thought of as a string without thinking of its properties as a number written in decimal (base 10). Thus, 111011 is a string using the alphabet 0,1. GCTTAG is a short string (length 6) representing a small piece of DNA, where the alphabet consists of A,C,G,T. The letters used are the first letters of the nucleotides that make up DNA, adenine, cytosine, guanine and thymine. The letters in DNA representation are always used in groups of 3 symbols.
One leader in this approach was the Russian mathematician Vladimir Iosifovich Levenshtein (there are variant spellings) (1935-2017). It was he, together with other scholars both before and after him (notably Richard Hamming) (1915-1998), who studied in detail ways to compute the distance (how far apart) two strings are, as well as using strings to represent or code information for a wide variety of purposes.
Very briefly, the idea is to see how starting with one string, using a minimum number of insertions, deletions or substitutions it can be transformed into another string. From this point of view the string WORD is distance 2 from the string SWARD by using one substitution, O to A, and the insertion of an S. This approach to distance is one way to construct spell checking software for a word processor on a computer. If a word is typed that is not in the dictionary of words stored on the computer, perhaps the string which was typed is not really a word, so one tries to find a word in the dictionary that the data enterer might have meant by locating one or more words that are close or nearby to the typed word, using some way to measure the distance between strings.
Here is a very small example to give you the flavor of what can be done. To send an image, the image would be divided into a certain number of cells, often called pixels. In the simplest setting, each pixel might be black or white and one might send an image of the letter H represented in a 3x3 grid as shown below, using the code 0 for white and 1 for black. The image would be sent as 101111101 (one binary digit for each pixel). One may use the 0's and 1's to reconstruct the image, top to bottom and left to right.
In recent times, a ubiquitous use of black and white cells in digital images is the QR (for Quick Response) code, a sample of which is displayed below.
When codes are used to represent information, it is traditional to use codes where the number of symbols in each code word is the same. This makes it possible to work in an environment where some kind of delimiter, such as a comma or a space, is not needed to separate code words. Thus, if 00 represents white and 11 represents black then one might send the message 00111100 rather than 00, 11, 11, 00.
It is worth remembering the way that using mathematics in the world and doing mathematics for its own sake reinforce each other. Mathematicians have been fascinated by cubes in many cultures and since ancient times. Their interest stemmed from the fact that they were regular polyhedra. Initially, cubes showed up in Euclid's Elements as one of the 5 convex solids in 3-space whose vertices are all alike and whose faces are (convex) regular polygons. Euclid does not mention the issue of convexity, a concept developed later by other mathematicians. These five polyhedra are now known as the Platonic Solids. But later mathematicians looked at the question of what regular convex solids existed in higher dimensions than 3. Intriguingly there are 6 regular convex solids in 4-dimensional space but in dimensions 5 or more there are only 3 regular convex polyhedra. But who might have realized that when the mathematics community studied the properties of cubes, they might be used in error correction technology?
Ancient scholars (and artists) interested in polyhedra did not have the modern tool of coordinates to use as a tool. However, eventually it was observed that the vertices of an $n$-dimensional cube could be labeled with all of the possible binary strings of length $n$. One can get between the labeled version of a $n$-cube and a ($n+1$)-cube in a delightfully appealing way that is illustrated below. What is shown is one way to represent a labeled 3-dimensional cube with its 8 vertices labeled with the 8 binary strings of length 3. Note that the bottom square in the diagram has all of its third coordinates 0 while the top square has all of its bottom coordinates 1. The 4 vertical edges join corresponding vertices in the two copies of the square (also known as a 2-cube) together. More generally, to get an $(n+1)$-cube from two copies of a labeled $n$-cube, in one copy of the $n$-cube one uses the binary sequences of length $n$ with a 0 added in the $n+1$ position while in the other copy of the $n$-cube one adds a 1 in the last position.
The 3-cube has 8 vertices and 12 edges. You can verify for yourself that the first diagram below can be thought of as a 4-dimensional cube with 16 vertices and 32 edges (as well as 24 square, two-dimensional faces). Cubes often give rise to attractive and symmetrical drawings. The first diagram below is appealing, but the fact that it represents a 4-dimensional cube is not so clear. The 5-cube and the 4-cubes which make it up can also be seen below. Based on the constructions shown, you can convince yourself that an $n$-cube has $2^n$ vertices and $n 2^{n-1}$ edges.
Given a binary string of length $n$, we can think of this string as a label for a vertex of an $n$-dimensional cube. The string we send conveys the information, for instance the gray level of a pixel in an image. If a string that is sent is not one of the possible strings in the dictionary whose entries consist of the binary digits in the code word, we know an error has been made, but typically this does not allow us to correct the string that arrived to the string that was sent. So the way we will conceptualize a code word is that some of its digits are information digits and that other digits are added with the goal of correcting errors.
If we want to code black or white pixels, we might use 0 to represent white and 1 to represent black, but if a 0 is sent a 1 might arrive and if a 1 is sent a 0 might arrive. A first idea might be that we could send the code word repeatedly, and though we are wasting space and time with sending a longer string than we need, perhaps this longer string will allow correction. Thus with the code 0, 1 we can sent 00 for 0 and 11 for 1. However, note that if 00 is sent and a single error occurs so that 10 arrives, we cannot be sure if this is because 00 was sent with an error or that 11 was sent and 10 arrived. In the first case an error occurred in position one and in the second case an error occurs in the second position. Thus, a single repetition is not good enough. We need something longer—000 or 111 would work, because now a single error allows us to use the notation of Hamming distance to tell which was the more likely sent string. For two strings of the same length, the Hamming distance between them is the number of positions in which the two strings differ. Thus, the strings 000 and 111 and the strings 101 and 110 have Hamming distance 3 and 2, respectively.
Major contributions to the theory of error correcting codes have arisen from practitioners of many backgrounds. I have already mentioned Richard Hamming but also remarkable are the contributions of Vera Pless(1931-2020) who made important contributions to coding theory as a researcher, teacher, and in applied settings.
Suppose one wants to code that the pixels in an image are black or white. Thus, one is trying to code two states or pieces of information. One would prefer to code this with short strings rather than longer ones. Using one bit will not work because if one codes black with 1 and white with zero, and a single bit is deleted what one receives is the empty message, and one cannot tell if a 0 or 1 was sent. If one uses 10 for black and 01 for white, then the received messages with one deletion are either 1 or 0. However if 1 arrives this can be due to the fact that 01 was sent and a 0 was deleted or if 10 was sent and a 0 was deleted. If one uses 00 and 11 as the code words, now if 0 arrives then 00 was sent while if 1 arrives 11 was sent and one can recover the original information.
Think about this situation. Consider the following collection of binary code words of size 5:
$$00000, 10001, 01010, 11011, 11100, 00111$$
For each code word above, let us write down the result of one digit in the code word being deleted:
Notice that the number of possibilities can vary from one code word to another code word. In this example, since one deletion results in exactly 16 (1 + 3 + 5 + 3 + 2 + 2 = 16) possible binary sequences, this set of code words covers all of the possibilities and we can uniquely decide which string must have been sent for any choice of string that arrives.
When a set of code words is sent using hardware, it is possible that elements of the coded message may incur errors. Here are some ways that a communication channel might alter a sent code word:
To design ways to digitally work with texts and images sent via a communications system, the notion of a mathematical channel was pioneered. The idea is that one transmits a sequence of data in the form of binary digits but what one receives at the other end of the channel may be different from what is transmitted in specific ways. Thus, in one model a 1 might change to a 0 with certain probability, or a zero might be changed to a 1 with the same probability p. This channel is probably the most studied and is known as the binary symmetric channel. Other kinds of communications channels can be described that relate to the kinds of errors that were briefly looked at above.
The specific case that interests me is that of a communications channel where binary strings of the same length are used to encode information, and where, when a string is sent, it may be corrupted by a single deletion of one of the sent binary bits. Note that if one is expecting a code word, say, with 5 bits and only 4 bits arrive, one knows that a deletion error has occurred. It turns out that there is a strong relationship between the theory of codes involving deletion errors and insertion errors but only deletion error codes will be addressed here. In addition to Levenshtein, other pioneers of error deletion codes were Grigory Tenengolts and Rom Rubenovich Varshamov (1927-1929) a Soviet/Armenian mathematician).
So the general question to be studied is: If there are $k$ symbols, characteristics, etc. to be encoded, what is the largest number of binary code words of a fixed length $n$ that can used to capture the information one wants to transmit, when a code word might be subjected to the deletion of one binary bit? Above we saw that we could construct a binary code of length 5 with six code words which could correct one deletion error. You can verify that for strings of length 3, a code with two code words exists: 000, 101. Can you find a binary collection of code words of length 4 that will correct one error deletion? For strings of length 6 it is known that a code with ten code words exists, but it is not so easy to find such a code.
We have seen that the code 000 and 101 can be used to correct one deletion. So can the code 111 and 010, which arises from the first code by interchanging 1 and 0. You can check your understanding by considering the following question. If the set of binary strings $B$ of length $n$ allows one to use these strings to correct one deletion, will the set $B^*$ arising by interchanging the 0's and 1's in the strings of $B$ also give rise to a code which will enable one to correct one deletion error?
In using the error deletion codes discussed above, we have in essence imagined that we have a dictionary of the erroneous strings that arise and we correct the erroneous string using this dictionary. Study the idea of how one could correct the erroneous string by doing calculations on the erroneous string and based on these calculations, reconstruct the string that was sent.
Compared with other kinds of error correction codes, surprisingly little is known about error deletion codes. The discussion above can be extended to have the possibility of more than one deletion error (codes for when 2 error deletions might occur) or where the alphabet involves more than the two symbols 0 and 1. (For DNA, the alphabet uses 4 symbols.)
Enjoy thinking about and learning about using mathematics to study digital communications systems!
Abramson, N., Information and Coding, McGraw Hill, New York, 1963.
Berlekamp, E., Algebraic Coding Theory, McGraw-Hill, New York, 1968.
Berlekamp, E., (ed.), Key Papers in the Development of Coding Theory, IEEE Press, New York, 1974.
Blahut, R., Theory and Practice of Error Control Codes, Addison-Wesley, Reading, l983.
Blake, I., Algebraic Coding Theory: History and Development, Dowden, Hutchinson and Ross, Stroudsburg, 1973.
Hamming, R., Coding and Information Theory, Prentice-Hall, Englewood Cliffs, 1980.
Hill, R., A First Course in Coding Theory, Oxford U. Press, Oxford, l986.
Hoffman, D. et al, Algebraic Coding Theory, Charles Babbage Research Centre, Winnipeg, 1987.
Lin, S., An Introduction to Error-Correcting Codes, Prentice-Hall, Englewood Cliffs, 1970.
MacWilliams, Florence Jessie, and Neil James Alexander Sloane. The Theory of Error-correcting Codes. North-Holland, Amsterdam, 1977.
McEliece, R., The Theory of Information and Coding, Addison-Wesley, Reading, l977.
Peterson, W. and E. Weldon, Error-correcting Codes, 2nd. ed., MIT Press, Cambridge, l972.
Pless, V., Introduction to the Theory of Error-Correcting Codes, Wiley, New York, 1982.
Shannon, C. and W. Weaver, The Mathematical Theory of Communication, U. of Illinois Press, Urbana, 1949.
Sloane, Neil, On single-deletion-correcting codes, Codes and designs, 10 (2000) 273-291.
Tatwawadi, Kedar, and Shubham Chandak, Tutorial on algebraic deletion correction codes, arXiv preprint arXiv:1906.07887 (2019).
Thompson, T., From Error-Correcting Codes Through Sphere Packing to Simple Groups, Mathematical Association of America, Washington, 1983.
Viterbi, A. and J. Omura, Principles of Digital Communication and Coding, Mc-Graw Hill, New York, 1979.
Welsh, D., Codes and Cryptography, Oxford U. Press, Oxford, 1988.
]]>The nuts and bolts of what I do when thinking about math are very similar to the nuts and bolts of thinking about a policy problem…
Courtney Gibbons
Hamilton College
Thanks to the American Association for the Advancement of Science and their Science and Technology Policy Fellowships, I’ve spent the last year working for Senator Gary C. Peters with the majority staff of the Senate Committee on Homeland Security and Governmental Affairs (HSGAC, which we pronounce “HISS-gack”). Broadly speaking, my portfolio has lived in the “Governmental Affairs” realm. I’ve had a chance to work on things related to federal grants and cooperative agreements, federal data policies, artificial intelligence—and math! Especially the mathematics and statistics that power different kinds of AI systems (and why the math is relevant to the policy).
I’ve had the opportunity to serve in the Senate at the same time as Duncan Wright, the American Mathematical Society Congressional Fellow, worked in Senator Young’s office. Duncan tells me he’ll be writing about his experiences soon, too!
One of the most thrilling aspects of working for the United States Senate has been using my training as a mathematical problem-solver to work on public policy problems (what can I say, I’m easy to thrill), in a very different way than following a traditional path like being an NSF rotator or working for the DOD in some capacity.
In my “normal” life, I work on problems in commutative and homological algebra—not exactly the most sought-after technical knowledge in Congress. But the nuts and bolts of what I do when thinking about math are very similar to the nuts and bolts of thinking about a policy problem or solution.
For example, when I say that I think about “rings and modules” to another mathematician, we have to achieve some clarity to keep communicating. To me, “ring” means unital, commutative, probably Noetherian, and almost certainly local or graded with a unique homogeneous maximal ideal. When I say “module,” I mean finitely generated. Change any of those properties, and the tools I use—like Nakayama’s Lemma—are off the table.
It’s the same in policy. When I started, I jumped on one of the office priorities: simplifying and coordinating the federal grant application process for recipients (and applicants, and potential applicants, and…). The first thing my mentor had me do was find the 1970s legislation that defines what financial assistance from the government means. And just like the word “ring” comes with many flavors of adjectives now, the word “grant” does, too. Is it a competitive grant? A formula grant? Is it for basic research? Disaster relief? I very quickly needed experts to help me understand the nuances.
Luckily for me, Congress has many experts: analysts at the Congressional Research Service and auditors at the Government Accountability Office investigate topics at the request of Congress, and many, many, many people have asked for reports and analyses of grants policies. Senator Peters held a hearing on grants to learn about the issue from people with additional hard-won insights they’ve collected after years (in some cases, careers!) of navigating the systems and processes required to get, use, and report on a grant.
Eventually, my teammates and I started looking for existing solutions to problems that might work in this situation, and, just like algebraists borrowed Betti numbers from topologists to study invariants of rings and modules, we started borrowing from other policy areas to put together some options.
Like math papers, potential legislation goes through a kind of peer review called “technical assistance” where people provide feedback on the bill text. And, like peer review, navigating different (sometimes conflicting) suggestions makes it interesting to figure out how to move forward. (The first idea I had got the equivalent of a bright red REJECT stamp from some external parties, but with lots of helpful feedback that informed my next attempt. Helpful review is truly a wonderful gift!)
The next steps for legislation include finding cosponsors, introducing the bill, shepherding it through the markup process where the committee with jurisdiction over the legislation has a chance to debate and change it, and eventually get it to the floor of the Senate; then it goes to the House; then, hopefully, it gets signed into law. I’ve only been here since October, so I will only see the bills I worked on through part of their journey. Like my math research work, it’s unclear which ideas, if any, will make it to the end of the process (or if I’ll recognize them when they do). But it’s been a gratifying experience to see up close how one Senate office (from the staff to the boss himself!) approaches the work of making the country a better place. The people I’ve worked with have renewed my hope and confidence in this country’s strange and byzantine processes. And I’ll keep my eye on S. 2286, the Streamlining Federal Grants Act, with the same tenderness I feel for my best and most fun mathematical collaborations.
Stepping from math professor life into Senate staffer life (at the same time as becoming a mom!) has been a strange but rewarding change-up. Take it from me: it’s never too late (or too early) to start finding your way to policymaking. Thanks to this fellowship and all the different policy areas I had a chance to learn about and work on, every night I went home thinking about policy in ways that made my neurons tingle just like when I’m in math mode.
And, reader, aside from your tenacity in thinking about problems, you probably also have practitioner expertise as a working mathematician. Congress has a lot of opinions about what kinds of scientific activities (including research!) matter. Think math is apolitical? Think again! In 2020, a major piece of legislation (Pub. L. 116-283) required NSF to collaborate with other agencies and industry partners to create AI Research Institutes, determine the feasibility of a National Artificial Intelligence Research Resource, study the artificial intelligence workforce, and more. The new TIP directorate at NSF? You guessed it: Congress! The CHIPS and Science Act (Pub. L. 117-167) has an entire title devoted to telling the NSF how to prioritize its work. Someone, somewhere, needs you to speak up on behalf of math and its value.
I’ll vouch for Congress as a productive place to be that voice. (Did I mention they have trains?)
]]>Sara Stoudt
Bucknell University
Are you taking a break from mathematical thoughts and cozying up with a good beach read this summer? Understandable! But sometimes a reading “break” isn’t a complete break from the kinds of mathematical thinking you might do during work hours. Thanks to Sarah Hart, we can all become more aware of the connections between mathematics and literature these days. If you haven’t picked up a copy of “Once Upon a Prime” yet, I highly recommend it. She walks through mathematical themes, mathematical structures, and even mathematical characters in her new book that aims to show us that mathematics shares the creativity of literature, and literature thrives off of that pursuit of mystery that can inspire mathematical discovery. There isn’t a split between “numbers” people and “words” people. (Want more of this or can’t wait for your copy of the book to reach you at the library or in your mailbox? Marian Christie has more poetry and math resources to enjoy online.)
One of the topics Hart dives into is constraints on the writing itself that are imposed by mathematical concepts. For example, a sestina is a poem with permutation structure. A poet chooses six words, and those six words must appear at the end of certain lines in a six-line stanza. Then each word’s line number changes in each consecutive stanza according to a permutation. (For a self-aware sestina check out this one that Larry Lesser shared with me.)
I wasn’t always familiar with this type of poem. In the fall, I worked with a poetry professor here at Bucknell University, Katie Hays, to do a crossover class between her Intro to Poetry course and my first-year seminar on Storytelling with Data. Katie suggested that we work with the sestina when I had mentioned that we were learning about order mattering in the presentation of plots and tables. The more we brainstormed, the more we thought that by playing with the order of words in a story-like poem, we could drive that point home. Katie and my students collaborated on a sestina, but I never approached writing one myself. Until now!
However, as a statistician, I had to put a little extra wrinkle in. How might I add an element of chance to the endeavor of writing a sestina? Enter the set.seed idea. I could use a random number generator to tell me which order the words in the sestina should start in. I ended up with:
Shower
Read
Screen
Fan
Couch
Trip
Then the sestina permutation takes over, governing which words fall in which order from there on. And with that, here goes nothing!
(The first reprieve)
In academia, when it rains, it pours. A deluge of papers to grade, lectures to give - not a
sprinkling, not even a mere shower.
But then April’s more than showers bring May, full stop, and hobbies poke out of the ground to
bloom: I can go for a walk, I can brunch, I can read.
An unencumbered breath of outside air with a view of trees subs in for mask-filtered air and the
computer screen.
What to make of this new found quiet? No Slack clickety-clack, no inbox whoosh or ding.
Routineless? I’m a big fan!
It’s time to reacquaint myself with life’s simple pleasures: my non-dress pants, my power
playlist, my couch.
Time is infinite. I am limitless… maybe I should take a trip.
(The ambitious time)
Wouldn’t it be great if I could just get away? Turn off every notification and galavant off on the
perfect getaway trip.
I can already feel the stress evaporate just picturing it. No storm clouds in sight, just the
occasional sunshower.
Picture me taking in the sights, eating food not out of a Tupperware container, not ever thinking
about a desperate need for even the briefest of respites on the couch.
Think of all of that energy and inspiration just mine for the taking, making me ready to write just
for myself, for fun, not for progress. And in the same way, read.
With all of this cool new work I’ll produce, maybe I’ll amass more than one reader that counts
themselves as my academic fan.
I can dream, right? Me shining so bright it’s others who will need the sunscreen.
(The changing plan)
But then the friction kicks in - all that pre-planning, the jet-lag, the inevitable TSA screen.
I want to relax, not play travel agent to myself. I may not be optimizing my classroom time, but
instead, that pesky trip.
There is so much pressure to make the most of this “free time”. Of that, I am never a fan.
Not to mention that travel requires getting used to a new bed, and a new shampoo that
accompanies that new shower.
So I go ahead and leave those e-mails from my former ambitious self about that dream vacation
on read.
As June flies by and my friends ask about that vacation I was definitely going to take, I start to
couch.
(The veg out)
Sure a dream vacation would be nice, but there are so many beautiful things you can do from
the comfort of your couch.
On a stay-cation I can give myself over to rom-coms, reality shows, my guiltiest of pleasures.
Yes, I’m still watching. Thanks for checking, TV screen.
I can make an adult version of a blanket fort and timelessly read.
I can finally go through the registry and pick that perfect gift for my friend’s baby shower.
I can bask in complete silence and just stare up at my ceiling fan.
(The turning point)
But at some point, my surroundings and the silence start to lose their charm, even that hypnotic
ceiling fan.
Is that an urge to venture beyond the couch?
Do my sweatpants suddenly feel too comfortable? Am I perhaps up for a pre-noon shower?
I’m not ready to go all in, mind you, just a next semester pre-screen.
But… if I am going to eventually conjure some bold, new ideas, I’ll need Post-It Notes and new
colored pens (glitter preferably) to keep track of them. Time for a Staples trip!
There were all of those interesting pedagogy papers I e-mailed myself over the course of the
last year. Maybe it’s time to give them a quick, no pressure read.
A passive read turns active with highlighters, notes, mind maps, the occasional mutterings to
myself, and before I know it I’m a butterfly emerged recharged with strong wings like an
intricate fan.
You didn’t even need a fancy trip to get to this moment. That time was unwasted on the couch.
Without dread I return to my computer screen.
I even catch myself daydreaming about fun stats in the wild examples in the shower.
(The welcome back)
Welcome back students! Did any of you take a trip? What brought you joy… anything you read?
In this class you will be both a statistics shower and (story) teller. And yes, we’ll learn the details
too, like what residual plot shapes are dreaded. Oh no, not the fan!
I know the temptation of the couch, believe me, but I hope this class provides some activation
energy, helping you realize that you have power beyond your screen.
Now that you’ve read my sestina experiment, I’ll get a little meta and tell you how I approached writing it. As I’ve tried to lean into my creative side and write poetry, I’ve been helped along by the power of constraints to help me face the blank page. Write a poem? Daunting! Write a poem with specific rules about each line? Doable! I was searching for advice for writing a sestina and came across this source that framed a sestina as a story and then proceeded to give a sample sestina about a cousin’s job. That made me think of doing one about my job, and specifically my job now that the semester is over. (Don’t teachers just frolic all summer?)
With a premise in place I just had to pick my words and get going. Half of the words were repurposed from another poetry project I was stuck on, and the other half came from trying to round out a summer story (including thinking of words that could be used in multiple ways). Before I started to write, I listed phrases or ideas that could end with each of the terms. Then I tried to find sub-themes in these that would help me write each stanza. I decided on a narrative arc that tracks me through the summer: initial relief at getting a break, an ambitious start for trying to use all my free time wisely, a mini burn-out that requires real rest, and finally a return of energy and hopes for the future school year. I decided to explicitly label each stanza (see the parenthetical labels) for a bit of extra clarity, a roadmap through time if you will. Another thing you might have noted as you read was that my lines can get fairly long. I’m more of a prose-poem kind of person.
Enough about me, it’s your turn! Set the seed to something else and make your own academic summer sestina using my words. Or come up with your own words and go through the same process with a different theme. Be free! Does this still seem too daunting? Try out a titrina, the “square root of the sestina,” instead. Now randomness will play two roles: one to sample 3 out of 6 initial words and one to provide an initial ordering. Happy writing! Feel free to share your poems with me at sas072@bucknell.edu, on Twitter (@sastoudt), or in the comments here.
]]>Ursula Whitcher
Mathematical Reviews (AMS)
You've heard of periods at the ends of sentences and periods of sine waves. The word period also has a special meaning in number theory. These periods are surprisingly useful for solving problems in particle physics. In this month's column, I'll tell you more about what periods are, where the physics comes in, and how all of this relates to the geometry of doughnuts.
Maybe you've heard the joke that a topologist can't tell the difference between a coffee cup and a doughnut. (If it's new to you, check out a beautiful illustration of the transformation by Keenan Crane and Henry Segerman.) Geometers are able to distinguish coffee cups from doughnuts. We can even tell the difference between types of doughnuts. For example, here's a fat, cakey doughnut:
Here's a skinny, crunchy doughnut:
But the geometry of doughnuts is so fascinating that, once you begin examining it, it's hard to think about anything else!
Let's describe the difference between our two doughnuts more formally. An idealized mathematical doughnut surface is called a torus. We can characterize the shape of a torus using two circles, one that goes around the outside and one that goes through the hole in the center. On a fat, cakey torus, these two circles are roughly the same size.
On a skinny, crunchy torus, the outer circle is much larger than the inner circle.
In these examples, the circles are easy to measure. But sometimes tori appear in more complicated ways. For example, suppose $x$ and $y$ are complex variables and $t$ is a complex parameter. Consider the solutions to the equation
\[y^2 = x(x-1)(x-t).\]
This is the famous (for number theorists) Legendre family of elliptic curves. If we throw in a solution "at infinity," then, topologically speaking, it is a family of tori. It's hard to graph the solution to an equation in two complex variables, but we can graph the real values. Here's what it looks like when the parameter $t$ is set to be equal to 3:
You can think of graphing the real points as slicing through the doughnut at an angle. In this graph, you see a skewed version of one of the circles and part of a second circle.
Measuring the lengths of these two circles is tricky. There is a general mathematical strategy from calculus class that we can try: set up an integral to measure the arclength. In this case, the appropriate integral turns out to be:
\[\int_\gamma \frac{dx}{y} = \int_\gamma \frac{dx}{\sqrt{x(x-1)(x-t)}} \]
Here, the integral is over an appropriate simple closed curve $\gamma$ in the torus/elliptic curve.
But there's a problem! I'll let a cartoon of an easily confused orange cat explain it.
The cat isn't lying: this integral is really hard. Standard techniques from calculus class do not work. In fact, this integral has no closed-form algebraic solution.
The integral $\int_\gamma \frac{dx}{\sqrt{x(x-1)(x-t)}}$ is an example of a period. For a number theorist, a period is a number that you get by finding the integral of an algebraic expression over an appropriate subspace. (Technically speaking, we should be able to describe the regions we are integrating over using inequalities and systems of algebraic equations whose coefficients are rational numbers.)
Many interesting constants, such as $\pi$ and $\log 2$, can be written as periods. There are huge and interesting questions about periods: for example, how can we characterize which numbers arise as periods? Using operations on integrals, one can show that adding or multiplying periods produces a new period. This gives periods the structure of a ring. Another big open question is describing all the relations that the ring of periods satisfies.
Let's get back to trying to understand our specific period, $\int_\gamma \frac{dx}{\sqrt{x(x-1)(x-t)}}$. We know that the result of the integral is a number that depends on the parameter $t$, so let's think of the integral as a function $P(t)$. We can take derivatives of $P(t)$:
$$\frac{d}{dt} \int_\gamma \frac{dx}{\sqrt{x(x-1)(x-t)}} = \int_\gamma \frac{d}{dt} \frac{dx}{\sqrt{x(x-1)(x-t)}}.$$
As we take derivatives, the expression under the integral sign becomes more complicated, but it keeps the same general shape. By finding a common denominator, we can identify a relationship between $P(t)$, $P'(t)$, and $P''(t)$:
\[ t(t-1) P''(t) + (2t-1) P'(t) + \frac{1}{4} P(t) = 0.\]
This is a differential equation! (It's called the Picard-Fuchs equation, after the French mathematician Émile Picard and the German Jewish mathematician Lazarus Fuchs.) As a second-order differential equation, this Picard-Fuchs equation has two independent solutions. These solutions correspond to the two different circles on the torus.
A standard method for solving differential equations is to use an infinite series. In this case, one of the solutions to the differential equation for our period can be written in terms of the following series:
\[\sum_{n=1}^{\infty} \frac{((\frac{1}{2})(\frac{1}{2} + 1)\cdots (\frac{1}{2} +n-1 ))^2}{(n!)^2}t^n.\]
The numerator involves an expression, $(\frac{1}{2})(\frac{1}{2} + 1)\cdots (\frac{1}{2} +n-1 )$, that looks rather like a rising factorial shifted by $\frac{1}{2}$. If we replace this expression by the shorthand $(\textstyle{\frac{1}{2}})_n$, we get a more compact notation for our series:
\[\sum_{n=1}^{\infty} \frac{(\textstyle{\frac{1}{2}})_n^2}{(n!)^2}t^n.\]
This is a famous series known as the hypergeometric series, with numerator parameters $\frac{1}{2}, \frac{1}{2}$ and denominator parameter 1 (since there's only a single factorial in the denominator). The whole series is sometimes expressed by the even more compact notation ${}_2F_1\left(\textstyle{\frac{1}{2}, \frac{1}{2}}; 1 \,|\, t \right)$.
For more details about the solution process, including a description of the second independent period, see Don Zagier's in-depth essay The arithmetic and topology of differential equations. I'd like to show you a more complicated period that shows up in theoretical physics.
In particle physics, describing the interactions between fundamental particles such as electrons and photons involves doing difficult integrals. (Even worse, from a mathematician's standpoint, these integrals may not always be well-defined!) Physicists organize these computations using diagrams called Feynman diagrams of increasing complexity. There are specific rules for creating and manipulating Feynman diagrams, but at a first approximation, one can imagine they tell stories about particles that meet, interact and perhaps undergo a transformation, then go their separate ways.
One of the fundamental forces in the universe is the weak force. The weak force is involved in holding atoms together or breaking them apart. It's the force that controls the process of radioactive decay and makes carbon-14 dating possible.
To do computations involving the weak force, one must work with Feynman diagrams that contain loops. Here's a Feynman diagram with two loops that is sometimes called the sunset diagram.
The American mathematician Spencer Bloch and the French physicist Pierre Vanhove teamed up to study the sunset diagram. To simplify the problem, they worked with a model where there are only 2 space-time dimensions. (Imagine particles moving back and forth along a line as time passes.) They assumed that all the particles generated during the interaction have equal mass $m$, that there's a fixed external momentum $K$, and they threw in a constant $\mu$ to balance the units. The result is the following sunset integral:
\[\mathcal{I}_\circleddash = \frac{\pi^2 \mu^2}{m^2} \int_0^\infty \int_0^\infty
\frac{dx\,dy}{(1+x+y)(x+y+xy) - xy \frac{K^2}{m^2}} \]
This integral is really, really hard!
One of the key problems is that the denominator, $(1+x+y)(x+y+xy) - xy \frac{K^2}{m^2}$, might be 0. To understand more about where the denominator vanishes, we can set $t=\frac{K^2}{m^2}$. The result is a family of curves that depends on the parameter $t$:
$$(1+x+y)(x+y+xy) - t xy =0.$$
Here's the resulting graph for $t=11$.
The features of this graph might look familiar. We've got a skewed circle and part of another circle—the doughnut slices are back! In other words, $(1+x+y)(x+y+xy) - t xy =0$ is a parametrized family of elliptic curves.
Bloch and Vanhove pursued a strategy that might seem familiar. They set $\mathcal{J}_\circleddash = \frac{m^2}{\pi^2 \mu^2} \mathcal{I}_\circleddash$ to simplify the units, then looked for a differential equation involving $\mathcal{J}$:
\[\frac{d}{dt} \left(t(t - 1)(t - 9) \frac{d}{dt} \mathcal{J}_\circleddash \right) + (t-3) \mathcal{J}_\circleddash = -6.\]
Because the right-hand side of this differential equation is not zero, solving it is more complicated than solving the differential equation we saw earlier. Standard differential equation methods approach this kind of problem in two steps. First, solve the homogeneous equation where we pretend the right-hand side is zero. Then, find a solution to our inhomogeneous equation where the right-hand side is a nonzero constant.
Bloch and Vanhove showed that the homogeneous solutions to the Picard-Fuchs differential equation for $\mathcal{J}_\circleddash$ can be written in terms of the classical hypergeometric series ${}_2F_1\left(\textstyle{\frac{1}{12}, \frac{5}{12}}; 1 \,|\, - \right)$. This series places rising factorials involving $\frac{1}{12}$ and $\frac{5}{12}$ in place of the $\frac{1}{2}$ we saw earlier. I've used $-$ to indicate that a more complicated expression is plugged in for the series variable.
To solve the full inhomogeneous equation, we need another special constant, $\mathrm{Li}_2(z)$, known as the dilogarithm. The dilogarithm can be written as an infinite series. When $|z|<1$,
\[\mathrm{Li}_2(z) = \sum_{k=1}^\infty \frac{z^k}{k^2}.\]
The dilogarithm is also a period! We can write it using a double integral.
\[\mathrm{Li}_2(z) = \iint_{0 \leq u \leq v \leq z} \frac{du\,dv}{(1-u)v}.\]
Thus, periods give us a precise way to describe the solutions to the sunset diagram integral—as well as a reason to eat doughnuts!
I thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, England for support and hospitality during the K-theory, algebraic cycles and motivic homotopy theory program, where I presented a version of this material during the institute's 30th birthday celebrations. This
work was supported by EPSRC grant no EP/R014604/1.
Bill Casselman
University of British Columbia
“But however we analyze the difference between the regular and the irregular, we must ultimately be able to account for the basic fact of aesthetic experience, the fact that delight lies somewhere between boredom and confusion.”
E. H. Gombrich, in The sense of order (at the top of page 9) |
Breaking News. Just as this column goes to posting, the authors of the original
construction described below have announced a new aperiodic tiling that does not use
reflected tiles. See Kaplan’s new web page or the new arxiv submission for more information.
A tiling of the plane by a set of two-dimensional shapes is a partition of the entire plane by congruent copies of the shapes, without overlaps or gaps. One common example is the tiling by regular hexagons:
Not all shapes can make a tiling. For example, the only regular polygons that tile the plane are triangles, squares, and hexagons. Regular octagons require squares to fill in, making another common pattern.
These examples are periodic—they are invariant under a full discrete set of translations. More interesting from a mathematical point of view are aperiodic tilings, which are not invariant under any translations at all. Best known among these are Penrose tilings, in which pentagonal shapes play a role:
Penrose tilings use two shapes, a thick and a thin rhombus.
Up until recently, all known aperiodic tilings used a minimum of two shapes, and it has long been a major problem to find a tiling that uses just one. Quite recently this problem has been solved, although only with a qualification:
The single shape that makes this tiling resembles a hat. It is in some sense a rather elementary shape, since it fits nicely into the standard hexagonal tiling of the plane:
This construction is due to the four authors David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. They have posted a long article about it on the arXiv, and it will presumably be eventually published in a professional journal. When it first appeared, it got a lot of media attention in places such as the Guardian and the New York Times. One thing that might have caused this attention was that the tiling was nicknamed einstein. This has nothing to do with the famous physicist. The word “ein” is German for “one”, and “Stein” is German for “stone”, incorporated for example into the word “Spielstein”, a gaming token.
However, as I have said, there is a qualification to the assertion that it uses just a single tile. That is true, but both sides of that tile are used. In the following image, the mirrored tiles are coloured:
In fact, the mirrored tiles play an important role in understanding the structure of this tiling. So although it is a remarkable and valuable construction, it still leaves open the problem of finding an aperiodic tiling by a single shape.
What I want to do in the rest of this essay is just to give some idea of the peculiar features of the new hat tiling. As opposed to answering questions such as: How was this construction found? Are there variants? Why, exactly, is it aperiodic? These are all quite difficult, and at least partially answered in the arxiv paper. But even carrying out my limited intention is not a simple task. In doing this, I shall be basically expanding slightly on the exposition of Smith et al.
Let’s look more carefully at the tiling to see if we can discern some kind of pattern. One thing that becomes quickly evident is that the neighbourhoods of the mirrored hats are all similar. The most important observation of this kind is that three unmirrored hats in this neighbourhood always form congruent triads:
I’ll call these groups of four hats H-clusters.
There are now a couple of things that come to mind. One is the frequent occurrence of singleton hats inside triplets of H-clusters. Another is that near corners of the H-clusters are groups of three hats that are invariant under $120^\circ$ rotation:
These extend to larger groups of six hats, still invariant under $120^\circ$ rotation:
What now remains unmarked is fairly simple. The unmarked tiles are either isolated hats or a couple of hats sandwiched between two H-clusters. The remarkable conclusion is that the hats in the tiling can be grouped into one of four types of clusters, and we therefore arrive at a new tiling by these four types:
Each of these clusters can be associated to a simpler figure, called a metatile. They are named, apparently arbitrarily, T, H, P, and F.
Three of these shapes are invariant under some rotation, while the underlying groups of hats are not. So their orientations must be kept track of, here by arrows.
The tiling by clusters gives rise to a tiling by metatiles!
This transition from hats to metatiles is the main step to the explicit construction of a tiling of the plane, although it is not at all obvious how this goes.
The metatiles can themselves be grouped into supertiles …
… which can themselves be clustered into larger supertiles …
… and so on to infinity.
The point is that if one is given just one really large supertile one can construct all of the supertiles it contains (its descendants), and then repeat this process until one reaches the bottom layer of metatiles, from which one can lay down the hats they are associated to. In this way one can construct arbitrarily large patches of hats in a consistent manner, and some very general reasoning assures you that a covering of the whole plane exists, even though of course you cannot construct it. At first it might seem extraordinary that Smith et al. came up with this scheme, but in fact it is a variation on a well known technique in the construction of aperiodic tilings—for example the Penrose tilings. Stepping up from one layer of hat/meta/super tiles to one dominating it is called inflation, and stepping down to one it dominates is called deflation. The version implemented here has one entirely new and noteworthy feature, though—the tiles in different layers are not similar shapes, but deformed a bit from descendants and parents.
So now we ask: Given a tiling by metatiles, how do you construct the tiling by supertiles that dominates it? More generally, how to step from one tiling by supertiles to the one dominating it? To the one it dominates?
All of the supertilings that occur in this business have the same general appearance—what Smith et al. call a fixed combinatorial configuration. They are characterized by a small number of parameters: lengths $a$, $x$, $y$, a vector $z_{0}$ (along with rotations), and an angle $e$.
All directions except the $z_{\bullet}$ are aligned with the edges of a hexagon tilted at the angle $e$. Certain nodes are vertices in both a configuration and its descendants, and the configuration is determined by these together with the vectors $z_{*}$. The supertiles of a configuration and its descendants are not similar, as the analogues for the Penrose tiling are, but Smith et al. say that asymptotically the ratio of basic lengths is $\tau = (1 + \sqrt{5})/2$. The existence of these layers of configurations, and the fact that Smith et al. discovered them, seems to me a kind of miracle.
How configurations of different levels interact, and in particular how to construct any finite sequence of configurations, ought to be roughly clear from the following diagram:
With this diagram, one can use the nodes common to both layers to locate points in each, by paths in the configurations. For example, suppose we are given the configuration marked in gray, and want to find the parameters of the higher one (red). That means we are given, for example, the node $A$.
In the figure below, a few generations of F-tiles are drawn. The shapes don’t vary much and as I have said, in the limit the figure in one generation is that in the previous generation scaled by a constant related to the golden ratio. The degree of tilt starts off at $0^{\circ}$ and converges rapidly to a fixed angle, very roughly $-7^{\circ}$.
Here is a different version, with the tiles rescaled so as to be comparable:
Convergence is evident. It is curious, and perhaps a bit confusing, that the difference between the initial (meta)tile and the subsequent (super)tiles is so great.
The basic problem in drawing any aperiodic set of tiles is that construction is not local. That is to say, there is no way to start laying down tiles that does not involve some computation at a place far away. Furthermore, one can only plot tiles in a bounded region that must be fixed in advance. For hats, the process goes like this:
This page offers a collection of graphics involving tiles, with an unusual lack of restrictions on their use. (This contrasts with how Roger Penrose imposed a strong copyright on his tiles.)
This web page contains, for example, files to feed to a 3D printer to make hat tiles. (The tiles in the photograph at the top of this column were produced from one of these by Mladen Bumbulovic, Techical Director of the Physics Machine Shop at the University of British Columbia.)
ArXiv preprint of the paper in which the tiling by hats is constructed and analyzed. Most of this impressive paper is devoted to detailed proofs of various claims, which are quite complicated. Many involve computer assistance for difficult case-by-case elimination.
This shows that the X-ray spectrum associated to the hat tiling is discrete, as are those associated to numerous other aperiodic tilings. These results link aperiodic tilings to the mainstream of mathematics.
Maria Fox
Oklahoma State University
Dedicated to my Dad, Dr. Barry R. Fox.
The idea of distance is central to so much of the mathematics we do and teach that it's easy not to give it a second thought. But what do we mean when we say two rational numbers are "close" or "far apart"? For example, how do we decide if $\frac{7718}{23029}$ or $\frac{6056}{18143}$ is closer to $\frac{1}{3}$?
A common approach is to compare the decimal expansion of $\frac{1}{3} = 0.33333...$ with the expansions of these two numbers:
$$\frac{7718}{23029} = 0.33514264... \quad \quad \frac{6056}{18143} = 0.33379264...$$
From these expansions, both rational numbers seem quite close to $\frac{1}{3}$. Indeed, if we truncated all three numbers at the tenths place, they would be indistinguishable: all equal to 0.3. Even at the hundredths place, they still agree. It's only when we consider the thousandths place that we can tell them apart: truncating to 0.333, the rational number $\frac{6056}{18143}$ continues to agree with $\frac{1}{3}$, while $\frac{7718}{23029}$ truncates to 0.335 and does not.
This is an excellent way to understand our usual distance function: as we compute further and further decimal expansions of two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$, the longer they agree, the smaller the distance between them. To separate this idea from alternative notions of distance, we'll refer to this usual type of distance as the Archimedean distance, defined as $d_\mathrm{Arch}(\frac{a}{b}, \frac{c}{d} ) = \sqrt{ (\frac{a}{b} - \frac{c}{d})^2 }$.
Under the Archimedean distance, $\frac{6056}{18143}$ is closer to $\frac{1}{3}$ than $\frac{7718}{23029}$ is.
But there are other ways to compare rational numbers, especially if one happens to enjoy number theory. We can consider their reductions modulo a prime, for example $5$. Whenever $5$ does not divide the denominator of a rational number $\frac{a}{b}$ (which we'll assume for now), then $b$ has a multiplicative inverse modulo $5$, and we can use this inverse to make sense of $\frac{a}{b}$ modulo 5.
In particular, since $3 \cdot 2 \equiv 1 \pmod 5$, 2 is a multiplicative inverse of 3 modulo 5. Using this, we can define $\frac{1}{3}$ modulo 5 as $1 \cdot 3^{-1} \equiv 2 \pmod 5$. Similarly, we can view both $\frac{6056}{18143}$ and $\frac{7718}{23029}$ modulo 5:
\begin{align*}
\frac{7718}{23029} &\equiv 7718 \cdot 23029^{-1} &\pmod 5 \\
& \equiv 7718 \cdot 4 &\pmod 5 \\
& \equiv 30872 &\pmod 5 \\
&\equiv 2 &\pmod 5 \\
\end{align*}
\begin{align*}
\frac{6056}{18143} &\equiv 6056 \cdot 18143^{-1} &\pmod 5 \\
& \equiv 6056 \cdot 2 &\pmod 5 \\
& \equiv 12112 &\pmod 5 \\
&\equiv 2 &\pmod 5 \\
\end{align*}
Interestingly, both $\frac{6056}{18143}$ and $\frac{7718}{23029}$ agree with $\frac{1}{3}$ modulo 5. That is, if we use "reduction modulo 5" as a first glance at these competing rational numbers, they are both indistinguishable from $\frac{1}{3}$.
But we can take this a step further and compare these rational numbers modulo $5^2 = 25$. Since 5 does not divide the denominators, it is possible to compute the necessary multiplicative inverses and reduce these rational numbers modulo 25:
\begin{align*}
\frac{7718}{23029} &\equiv 7718 \cdot 23029^{-1} &\pmod{25} \\
&\equiv 7718 \cdot 19 &\pmod{25} \\
& \equiv 17 &\pmod{25} \\
\end{align*}
\begin{align*}
\frac{6056}{18143} & \equiv 6065 \cdot 18143^{-1} &\pmod{25} \\
&\equiv 6056 \cdot 7 &\pmod{25} \\
& \equiv 17 &\pmod{25}. \\
\end{align*}
Again, they both agree with $\frac{1}{3} \equiv 17 \pmod{25}$.
Moving on to the next power of 5, we can finally see a difference when comparing $\frac{1}{3} \equiv 42 \pmod{125}$ and:
$$\frac{7718}{23029} \equiv 42 \pmod{125} \quad \quad \frac{6056}{18143} \equiv 17 \pmod{125}.$$
Viewed modulo 125, the rational number $\frac{7718}{23029}$ continues to agree with $\frac{1}{3}$, while $\frac{6056}{18143}$ does not. In other words, using our reduction-modulo-powers-of-5 technique, it takes longer to distinguish $\frac{7718}{23029}$ from $\frac{1}{3}$ than it takes to distinguish $\frac{6056}{18143}$ from $\frac{1}{3}$. In this sense, $\frac{7718}{23029}$ is closer to $\frac{1}{3}$ than $\frac{6056}{18143}$ is.
This is the central idea behind the 5-adic distance. Assuming that $5$ does not divide the denominators of $\frac{a}{b}$ or $\frac{c}{d}$, we define:
$$d_5 \left( \frac{a}{b}, \frac{c}{d} \right) = \left( \frac{1}{5} \right)^{\nu( \frac{a}{b} - \frac{c}{d} )},$$
where $\nu( \frac{a}{b} - \frac{c}{d} )$ is the number of times 5 appears in the prime factorization of the numerator of $\frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd}$, or equivalently the largest integer $n$ for which $\frac{a}{b}$ and $\frac{c}{d}$ agree modulo $5^n$. This exactly captures our approach above: as we compare $\frac{a}{b}$ and $\frac{c}{d}$ modulo $5, \ 5^2, \ 5^3, \ 5^4,$ etc., the longer these rational numbers agree, the smaller the 5-adic distance $d_5( \frac{a}{b}, \frac{c}{d} )$ between them.
In particular, under the 5-adic distance, $\frac{7718}{23029}$ is closer to $\frac{1}{3}$ than $\frac{6056}{18143}$ is. (Notice that this is the opposite of our conclusion under the Archimedean distance!)
The 5-adic distance function behaves very differently than the Archimedean distance. For example the integers $5, 25, 125, 625, ...$ grow closer and closer to 0 under the 5-adic distance, but farther and farther from 0 under the Archimedean distance. If we extend the 5-adic distance in a natural way to allow 5 to appear in the denominators of rational numbers (in this case, we define $\nu(\frac{a}{b}-\frac{c}{d})$ to be $-n$, where $n$ is the number of times 5 appears in the prime factorization of the denominator of $\frac{a}{b} - \frac{c}{d} = \frac{ab-bc}{bd}$), then $\frac{1}{5}, \frac{1}{25}, \frac{1}{125}, \frac{1}{625}$, ... will grow farther and farther from 0 in the 5-adic distance, but closer and closer to 0 under the Archimedean distance.
Now that we have a new type of distance, what is the effect on some familiar geometric objects? As a first example, we'll consider how familiar properties of triangles suddenly change under the 5-adic distance.
First, we need to adjust our idea of triangles. Triangles are most commonly considered in 2-dimensional space, as in the image below.
But we can just as easily consider triangles in 1-dimensional space! We'll define a triangle inside the rational numbers to simply be a choice of its three vertices: that is, a choice of three rational numbers $\frac{a}{b}$, $\frac{c}{d}$, and $\frac{e}{f}$. The side lengths are then the distances between the three possible pairs of vertices. For example, below is an illustration of the triangle with vertices $\frac{1}{2}$, $\frac{7}{3}$, and $\frac{21}{4}$.
Triangles enjoy the familiar Triangle Inequality under the Archimedean distance: each side length is less than or equal to the sum of the other two side lengths:
$$d_\mathrm{Arch}\left(\frac{a}{b}, \frac{c}{d} \right) \leq d_\mathrm{Arch} \left(\frac{a}{b}, \frac{e}{f} \right) + d_\mathrm{Arch} \left(\frac{c}{d}, \frac{e}{f} \right).$$
This is true regardless of the way the vertices are labelled. After computing the side lengths of this triangle under the Archimedean distance,
our example above meets three Triangle Inequalities (depending on the labelling of the vertices):
\begin{align*}
d_\mathrm{Arch} \left(\frac{1}{2}, \frac{7}{3} \right) &\leq d_\mathrm{Arch} \left(\frac{1}{2}, \frac{21}{4} \right) + d_\mathrm{Arch} \left(\frac{7}{3}, \frac{21}{4} \right) &1.8\overline{3} < 4.75 + 2.91\overline{6} \\
d_\mathrm{Arch} \left(\frac{1}{2}, \frac{21}{4} \right) &\leq d_\mathrm{Arch} \left(\frac{1}{2}, \frac{7}{3} \right) + d_\mathrm{Arch} \left(\frac{21}{4}, \frac{7}{3} \right) &4.75 = 1.8\overline{3} + 2.91\overline{6} \\
d_\mathrm{Arch} \left(\frac{21}{4}, \frac{7}{3} \right) &\leq d_\mathrm{Arch} \left(\frac{21}{4}, \frac{1}{2} \right) + d_\mathrm{Arch} \left(\frac{7}{3}, \frac{1}{2} \right) &2.91\overline{6} < 4.75 + 1.8\overline{3}.
\end{align*}
But what happens under the 5-adic distance? Now our triangle has 5-adic side lengths as labeled below:
In fact, triangles in the 5-adic distance enjoy the even stronger Non-Archimedean Triangle Inequality:
$$d_5 \left(\frac{a}{b}, \frac{c}{d} \right) \leq \mathrm{max} \left\{ d_5 \left(\frac{a}{b}, \frac{e}{f} \right), \ d_5 \left(\frac{c}{d}, \frac{e}{f} \right) \right\},$$
regardless of the way the vertices are labelled. So, we have the three Non-Archimedean Triangle Inequalities from this example:
\begin{align*}
d_5 \left(\frac{1}{2}, \frac{7}{3} \right) &\leq \mathrm{max} \left\{ d_5 \left(\frac{1}{2}, \frac{21}{4} \right), \ d_5 \left(\frac{7}{3}, \frac{21}{4} \right) \right\} &1 = \mathrm{max} \Bigl\{1, \frac{1}{5} \Bigr\} \\
d_5 \left(\frac{1}{2}, \frac{21}{4} \right) &\leq \mathrm{max} \left\{ d_5 \left(\frac{1}{2}, \frac{7}{3} \right), \ d_5 \left(\frac{21}{4}, \frac{7}{3} \right) \right\} &1 = \mathrm{max} \Bigl\{ 1, \frac{1}{5} \Bigr\} \\
d_5 \left(\frac{21}{4}, \frac{7}{3} \right) &\leq \mathrm{max} \left\{ d_5 \left(\frac{21}{4}, \frac{1}{2} \right), \ d_5 \left(\frac{7}{3}, \frac{1}{2} \right) \right\} &\frac{1}{5} < \mathrm{max} \Bigl\{ 1, 1 \Bigr\}.
\end{align*}
Unlike under the Archimedean metric, this triangle is "isosceles" under the 5-adic metric: two of its side lengths are equal. This is not an accident! Just as we saw in this example, whenever $ d_5(\frac{a}{b}, \frac{e}{f}) \neq d_5(\frac{c}{d}, \frac{e}{f})$, the third length $d_5(\frac{a}{b}, \frac{c}{d})$ is exactly equal to their maximum. As a result,
Theorem.
All triangles are isosceles under the 5-adic distance.
Given any three rational numbers $\frac{a}{b}$, $\frac{c}{d}$, $\frac{e}{f}$, at least two of the three distances $d_5(\frac{a}{b}, \frac{c}{d})$, $ d_5(\frac{a}{b}, \frac{e}{f})$, and $d_5(\frac{c}{d}, \frac{e}{f})$ will be equal.
This is certainly different from triangles under the Archimedean distance!
Circles also behave quite differently under the 5-adic distance. As before, even though it is most common to consider circles in 2-dimensional or 3-dimensional space—as illustrated below—
we will consider circles in 1-dimensional space. A circle in the rational numbers is determined by a center $\frac{a}{b}$ and a radius $r$. The circle $C(\frac{a}{b}, r)$ of radius $r$ about the point $\frac{a}{b}$ is defined to be all those rational numbers whose distance from $\frac{a}{b}$ is strictly less than $r$. (This is an "open circle," though we could have just as easily studied "closed circles" by allowing the distance to also be equal to $r$.)
For example, let $C_{\mathrm{Arch}}(\frac{7}{3}, 2)$ be the circle of radius 2 about $\frac{7}{3}$, under the Archimedean distance:
$$C_\mathrm{Arch}\left( \frac{7}{3}, 2 \right) = \left\{ \frac{a}{b} \ \middle| \ d_\mathrm{Arch}\left( \frac{7}{3}, \frac{a}{b} \right) < 2 \right\}.$$
From the definition of Archimedean distance, we can identify $C_\mathrm{Arch}( \frac{7}{3}, 2) $ with the open interval between $\frac{1}{3}$ and $\frac{13}{3}$:
Unfortunately, it's much more difficult to draw a circle under the 5-adic distance! For example, consider the circle $C_{5}(\frac{7}{3}, 2)$ of radius 2 about $\frac{7}{3}$, under the 5-adic distance:
$$C_5 \left( \frac{7}{3}, 2 \right) = \left\{ \frac{a}{b} \ \middle| \ d_5 \left( \frac{7}{3}, \frac{a}{b} \right) < 2 \right\}.$$
The first thing to notice is that the 5-adic distance between any two rational numbers will always be a power of 5 (including negative powers, like $\frac{1}{5}$). So requiring the 5-adic distance to be less than 2 is the same as requiring the 5-adic distance to be less than or equal to 1.
Whenever 5 does not divide $b$, the distance $d_5(\frac{7}{3}, \frac{a}{b})$ between $\frac{7}{3}$ and $\frac{a}{b}$ will always be at most 1, because the exponent $\nu(\frac{7}{3} - \frac{a}{b})$ will be positive or zero. So, for example, the rational numbers $-\frac{7}{8}$, $\frac{2}{3}$, $2$, and $\frac{9}{2}$ are all contained in this circle:
But so are infinitely more rational numbers. In fact, every single integer will be contained in this circle:
It's also illustrative to consider some points that do not lie in this circle. Both $-\frac{1}{5}$ and $\frac{8}{5}$ are 5-adic distance 5 from $\frac{7}{3}$, the rational number $\frac{14}{25}$ is distance 25 from $\frac{7}{3}$, and $\frac{401}{125}$ is distance $125$ from $\frac{7}{3}$. So, none of these points are contained in the circle $C_5( \frac{7}{3}, 2)$:
Not only are 5-adic circles challenging to draw, they also behave very differently from Archimedean circles. For example, using the fact that all 5-adic triangles are isosceles, one can show the following:
Theorem.
Any point in a 5-adic circle can be used as the center of the circle.
Whenever $\frac{c}{d} \in C_5( \frac{a}{b}, r)$, then the circles of radius $r$ about $\frac{a}{b}$ and $\frac{c}{d}$ are equal: $ C_5( \frac{a}{b}, r) = C_5( \frac{c}{d}, r).$
As an example, since $-\frac{7}{8}$, $\frac{2}{3}$, $2$, and $\frac{9}{2}$ are all contained in $C_5( \frac{7}{3}, 2),$ the circles $C_5( -\frac{7}{8}, 2),$ $C_5( \frac{2}{3}, 2),$ $C_5( 2, 2),$ and $C_5( \frac{9}{2}, 2),$ are all equal to our original circle $C_5( \frac{7}{3}, 2)$.
As a result of the above theorem, we have another surprising fact about 5-adic circles:
Theorem.
Any two 5-adic circles are disjoint or concentric.
If the intersection of $C_5( \frac{a}{b}, r)$ and $C_5( \frac{c}{d}, t)$ is nonempty, then the circle with the smaller radius (without loss of generality, $t$) is contained in the circle with the larger radius: $ C_5( \frac{c}{d}, t) \subseteq C_5( \frac{a}{b}, r).$
This is very different from Archimedean circles. For example, $C_{\mathrm{Arch} }( \frac{1}{2}, 1.3)$ and $C_\mathrm{Arch}( \frac{8}{3}, 1.5)$ are two non-concentric, non-disjoint Archimedean circles:
This never occurs under the 5-adic distance!
The surprising geometric properties of the 5-adic distance that we have already noticed have far-reaching effects. The notion of Archimedean distance is the starting point for the study of real manifolds: the real numbers $\mathbb{R}$ can be constructed as the completion of the rational numbers under the Archimedean distance. Then, real manifolds are defined to be topological spaces that look like $\mathbb{R}$ (or $\mathbb{R}^2$, or $\mathbb{R}^3$, ...) around every point, once one zooms in far enough. Below are illustrations of some 2-dimensional real manifolds: a Klein bottle and a torus.
What happens when we try to repeat this construction with the 5-adic distance instead of the Archimedean distance? That is, how can we create 5-adic manifolds? First, one can replicate the process of constructing the real numbers from the rational numbers, but with the 5-adic distance in place of the Archimedean distance. This produces the 5-adic numbers, $\mathbb{Q}_5$, and if 5 is replaced with another prime $p$ this construction will yield the $p$-adic numbers, $\mathbb{Q}_p$.
One possible construction of $p$-adic manifolds (the theory of $\mathbb{Q}_p$-analytic spaces, as described by Jean-Pierre Serre in Lie groups and Lie algebras) is very analogous to the construction of real manifolds. While the natural approach in some circumstances, $\mathbb{Q}_p$-analytic spaces leave also some things to be desired. Without getting into the technicalities, these inconveniences can be summed up perfectly in the words of John Tate, who referred to $\mathbb{Q}_p$-analytic spaces as "wobbly" analytic spaces!
This wobbliness fundamentally originates from our third observation about the $p$-adic distance: any two $p$-adic circles are disjoint or concentric. Because of this, there are many ways to write one open circle as a union of others (called an "open cover"). To fix the wobbliness of $\mathbb{Q}_p$-analytic spaces, there have been two main approaches. The first, pioneered by Tate in the 1960s (and then developed by many others) used ideas of Alexander Grothendieck to formally disallow some open covers. These $p$-adic manifolds are called rigid analytic spaces (in contrast with the original wobbly ones).
A second approach was developed in the 1990s by Roland Huber, who removed the wobbliness of $\mathbb{Q}_p$-analytic spaces essentially by adding in some "missing points." These $p$-adic manifolds are called adic spaces. Perfectoid spaces, powerful tools in number theory and algebraic geometry recently developed by Peter Scholze, are a special kind of adic space. In this sense, our geometric observations about triangles and circles under the 5-adic distance are the root of cutting-edge mathematical developments today.
Just because the 5-adic distance enjoys many different properties from the Archimedean distance, it doesn't mean that this idea of distance is either new or unusual.
The Archimedean distance is undoubtably older: it is named for the "Archimedean property" of this distance function, and Archimedes himself originally named this property after Eudoxus of Cnidus (who was born around 400 BC). However, the $5$-adic distance is also very old. The general concept of a $p$-adic distance (defined analogously to the 5-adic distance, for any prime $p$) was first formally studied by Kurt Hensel in his 1897 paper Über eine neue Begründung der Theorie der algebraischen Zahlen. Related ideas can be traced back at least a little further to the work of Ernst Kummer.
Additionally, by Ostrowski's Theorem, the only distance functions on the rational numbers are the Archimedean distance $d_\mathrm{Arch}$ and the $p$-adic distances $d_p$. Since there are infinitely many $p$-adic distance functions and only one Archimedean one, perhaps the Archimedean distance should be considered as the "unusual" one!
Joe Malkevitch
York College (CUNY)
Andrew Wiles came to the attention of the public and the mathematics community when he solved a problem that had eluded solution for hundreds of years, Fermat’s Last Theorem. He showed that there were no positive integer solutions to the equation
$$x^n + y^n = z^n$$
for $n$ a positive integer that is 3 or greater. This is a sharp contrast with what happens when $n =2$, where the corresponding equation has infinitely many solutions, $x=3$, $y=4$ and $z=5$ being a well known solution. While it is relatively easy to state the question, the proof Wiles used to resolve it is not easy to understand. The complex mathematics that Wiles needed (with assistance from his onetime PhD student Richard Taylor to complete the proof) did provide the answer to this long standing unsolved problem.
Maryna Viazovska (from Ukraine, 2022) came to attention of the mathematics community, and much less to that of the general public when she became the second woman to win the Fields Medal, one of the highest awards that a mathematician can win. She did very innovative work that gives insight into dense packing of identical spheres in higher dimensional spaces, a problem whose nature is still not fully understood or explored in arbitrary dimension.
Here I will pay tribute to someone who contributed to mathematics in a less meteoric way, Donald Warren Crowe (1927-2022). However, I hope you will find this window into mathematics a compelling one. Below are a sample of images of Don as he and his beard grayed:
Sample of photographs of Donald Crowe. Courtesy of the family of Donald Crowe and the Mathematics Department University of Wisconsin-Madison.
Modern mathematics extends greatly beyond the common wisdom that it is the subject concerned with numbers (arithmetic and algebra) and shape (geometry and topology). Another way to capture its domain is that it is the science of studying patterns. Relatively few mathematicians know the name Donald Warren Crowe (1927-2022), but he deserves to be better known for having helped put together a community of people interested in patterns, symmetry, art, and the presence of symmetry in the mathematics and artwork of cultures all over the globe and throughout the history of humankind.
Donald Crowe’s most cited work on MathSciNet (an abstracting source for research in mathematics run by the American Mathematical Society) is the book Symmetries of Culture, which he jointly developed with the anthropologist Dorothy Washburn.
The book was originally published in 1988 by Washington University Press but a version was relatively recently republished in 2020 by Dover Press. The book is notable in being both a fascinating blend of coffee-table book, with striking images of examples of patterns and designs from the artistic works of indigenous peoples, especially those who live in the southwestern part of the United States, and a tutorial about classifying symmetric patterns on fabrics or pottery.
The genesis of this book takes us along the path of Don Crowe’s life story, which I will now sketch briefly.
Don was born in Lincoln, Nebraska in 1927, his father having been an academic but not in mathematics. Don attended the University of Nebraska and the University of Minnesota, studying mathematics, physics and philosophy. He returned to Lincoln to get a Masters Degree. He eventually made his way to the University of Michigan, where his interest in geometry led him to do some work with the graph theorist (dots and lines geometric objects) Frank Harary (1921-2005). When the eminent geometer Harold Scott MacDonald Coxeter (1907-2003) came to the University of Michigan from the University of Toronto, Don found Coxeter’s mathematical interests appealing. So he followed Donald (as Coxeter was known to his friends) back to Toronto and eventually completed his doctoral dissertation for Michigan under Coxeter’s direction.
His thesis generalized ideas about polygons using the quaternion number system, where the multiplication operation does not obey the commutative law that $ab = ba$. It is noteworthy that during a period when few mathematicians called themselves geometers (in contrast to analysts, algebraists, topologists) Coxeter breathed new life into geometry, partly with his book Introduction to Geometry (1961, 2nd ed. 1968). Coxeter was the subject of the book King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry (2006) by Siobhan Roberts, who drew for her biography on thoughts about Coxeter and his work from John Horton Conway (1937-2020). Conway himself made important contributions to revitalizing geometry in the 20th century. Ironically, Coxeter will probably be most remembered not for his contributions to geometry but to algebra. Thus, many who know his name know it because of the notion of a Coxeter Group, which bears his name. In addition to Crowe, Coxeter had 16 other doctoral students. He had relatively few doctoral “grandchildren” because most of his doctoral students, while creative contributors to mathematics, found work at liberal arts colleges rather than schools granting doctorate degrees. Eventually Don became a professor of mathematics at the University of Wisconsin in Madison and taught geometry courses, though he also taught abstract algebra and group theory courses related to geometry.
In 1969, Don, together with his two University of Wisconsin colleagues Anatol Beck and Michael Bleicher, published a book for a general education course in mathematics called Excursions into Mathematics. One chapter that Don contributed to this book dealt with the remarkable formula, seemingly not discovered by any ancient civilization, due to Leonard Euler. This formula, often called Euler’s Polyhedral Formula, states that for a convex bounded 3-dimensional polyhedron
$$V\text{(number of vertices)} + F \text{(number of faces)} – E \text{(edges)} =2.$$
This formula also has a version for connected graphs drawn in the plane. While Excursions into Mathematics is not directly related to symmetry patterns, clearly Don had been thinking about the relationship between symmetry, geometry transformations, patterns and “regular” polyhedra. In 1973, Don had assisted Claudia Zaslavsky, a precollege mathematics teacher and mother of the distinguished mathematician Thomas Zaslavsky, by writing a chapter for her important book, Africa Counts. This chapter dealt with patterns in African art.
Don loved to travel and after completing his doctorate in 1959 he traveled to Nigeria to have a visiting professorship at University College in Ibadan, Nigeria. It was his experience in teaching in Nigeria that helped further his interests in art, mathematics and symmetry. While in Africa he participated in archeological digs in Ghana. He returned from Africa with samples of African fabrics that he collected while he was there. His great intellectual curiosity built on what he learned while in Nigeria. These experiences involving the decorative arts in the form of fabrics and raffia clothes would later cause him to become involved in writing scholarly articles in mathematics journals about, and promoting the use of, symmetry and pattern at the intersection of artworks and mathematics. For many people, understanding the geometry of patterns is an appealing alternative entry point to mathematics, rather than learning about how to carry out arithmetic and algebraic computations. The photos below show some samples of art that Crowe photographed or collected on trips that he took inside or outside the United States.
Many friezes, some on a fabric and some on pottery. Photos courtesy of the Crowe family.
These examples of symmetric artistic creations include subpieces that can be interpreted as a design in a band, strip, or frieze, in contrast to a designs that fill the whole plane, which are often said to be wallpaper patterns. (Of course, real world artists and artisans cannot fill an infinite strip, quadrant of a plane or the whole plane but often the substance of their work suggests that they are showing a finite portion of an idealized infinite canvas.) The cases of an infinite strip or the whole plane are of particular interest because, as seen above, they approximate symmetric patterns seen in textiles, on pottery or on the frieze of a building.
One of the most remarkable theorems of elementary geometry, though it is rarely mentioned in the mathematics education experiences Americans have in schools, is that if one has a pattern on a band, frieze, or strip (disregarding color if as often happens more than one color appears), regardless of the artistic content of what one sees, there are exactly 7 different patterns that are possible. That is, if one has a floral design, collection of letters, or animal design on a strip (frieze) it can be assigned a unique class to which it belongs. One way to construct frieze patterns and to help understand geometric transformations, symmetry, and shape is to use simple geometric figures and/or representations of letters of the alphabet to generate patterns. In the figure below, we have a part of a frieze of squares and another using the letter H (capital H rather than a lowercase h), but rendered in the spirit of graph theory using dots and edges.
Part of a frieze of squares and another using the letter H, but rendered in the spirit of graph theory using dots and edges (line segments).
In the first row we have a “run” along a strip of an implied infinite collection of squares, while in the second row we have an infinite collection of stylized representations of the letter H. Often the repeated shape, in the case that the frieze does not consist of one connected subject, is called a motif. While these two frieze patterns have different artistic content, they both have the same symmetry pattern. Each of these rows is a frieze which has as symmetries translations, vertical mirrors, horizontal mirrors, rotations of the whole frieze by 180 degrees, and glide reflections (a translation followed by a horizontal reflection).
T: translation
R: rotation by 180 degrees (or half-turn)
H: horizontal reﬂection
V: vertical reﬂection
G: glide reﬂection
Many systems of names have emerged for designating the 7 types of frieze patterns, but one can use the symbols above to note the presence of particular types of geometric symmetry. Again, it can be shown that in some cases the presence of certain symmetries must imply the presence of others.
Using the everyday artistic content of lower or upper case letters of the alphabet, one sees below representatives of the 7 types of frieze patterns that are possible.
….. L L L L ……translation
…..D D D D … translation, horizontal mirror
….V V V V ……translation, vertical mirror
……H H H H ……translation, vertical mirror, horizontal mirror
…..N N N N …. translation, half-turn (180 degree rotation)
….p b p b p b …translation, glide reflection
…..b p q d b p q d…… translation, glide reflection, vertical mirror
To help you check if you understand the system being used in classifying friezes, here is another collection of 7 patterns which includes each of the 7 patterns. Can you see which type each pattern below belongs to guided by the examples above?
…… S S S S S S S….
……d q d q d q d q……
…… M M M M M M ….
…….q d b p q d b p …
…….b b b b b b …….
…….C C C C C ….
………X X X X X…..
One approach to symmetry, which Donald Crowe emphasized for the benefit of those whose goal was to put a name on the type of pattern they were looking at, was to use a series of questions. In this way, after a small set of questions one could say that the pattern under consideration was pinned down. Unfortunately, as sometimes happens in scholarly areas that overlap in their interests, the names that crystallographers, anthropologists, and mathematicians use are not always the same. While one can convert between the different systems, there are pros and cons to the different naming systems. Suffice it to say that one does not need these naming systems to admire the beauty of real world examples illustrating symmetric patterns. Here are two examples of patterns teachers can use to entice students to see the nifty connections between patterns, art and mathematics.
Two patterns, one from an Egyptian tomb and the other Indian continent metal work. Images Courtesy of Wikipedia.
Relatively recently, Branko Grünbaum (1929-2018) and Geoffrey Shephard (1927-2016) showed that the seven types of frieze patterns can be refined to 15 types of frieze patterns in the case where the motif of the pattern was discrete (rather than one connected design that could be broken up into a “fundamental region” that was repeated by translation). For two examples of connected frieze patterns, consult the figure below.
Crowe also attempted to popularize and make geometers aware of frieze patterns generated from arrays of numbers—a somewhat different riff on the notation of a frieze pattern due to the innovative work of John Conway and Coxeter. It turns out there is a surprising connection between this kind of frieze pattern and subdividing a convex polygon into triangles in different ways.
One reason Donald Crowe achieved fame was that he was one of the 17 doctoral students of the remarkable mathematician Harold Scott MacDonald Coxeter. During a period where geometry had been eclipsed by other branches of mathematics, Coxeter was praised for work and writing that suggested that geometry was not dead and that there were many important geometric issues still worthy of attention by researchers, teachers and students. Often important mathematical ideas and notions are born in a geometric setting but reach their full maturity when the geometric problems can be treated using algebraic techniques. Thus, it is not an accident that Euclid (synthetic geometry) came before Descartes (analytical geometry).
The term ethnomathematics describes the branch of mathematics that deals with mathematical ideas inspired by, or born of, the mathematics and art that has been developed in many cultures over long periods of time. Examples of Don’s contributions to this part of mathematics are papers listed in the references about Bakuba art and the fact that his interests in this area inspired other people to teach and write about these topics. For example, Darrah Chavey was one of Don’s doctoral students, and Chavey collaborated with Philip Straffin at Beloit College in Wisconsin on research in ethnomathematics and teaching courses related to this topic.
As noted at the start, Don’s most cited publication is joint with the anthropologist Dorothy Washburn. Washburn, who obtained her doctorate at Harvard in anthropology, developed an approach to frieze patterns and wallpaper patterns that was independent of the work on these matters in the mathematical community. Her work illustrated the idea that what seems like theoretical mathematical ideas can with ingenuity be applied in novel settings. Washburn invented a method of using patterns as a way to understand cultural diffusion. Thus, in two neighboring villages, say in West Africa, one might see different patterns used in the textiles of these villages during a certain time period. However, at a later time period one might see patterns from the first village appear in the work of the artisans of the other village. Careful analysis of patterns might help provide insight into trade patterns that would allow one to try to infer information about how different tribal groups interacted over time.
Don’s collaborations with Dorothy Washburn and Claudia Zaslavsky were typical of his attempts to further the cause of mathematics and its applications as a scholar, teacher and educator. His legacy will live on!
Donald Warren Crowe was a mensch. He attempted to make the world a better place and help others be the best people they could be. It was my great good fortune to have had him as a teacher, mentor (he was my doctoral thesis advisor (1968)), and friend. I miss him every day.
Amit, M., & Abu Qouder, F. (2015). Bedouin Ethnomathematics: How integrating cultural elements into mathematics classrooms impacts motivation, self-esteem, and achievement. Proceedings of the PME, 39, 24- 31.
Ascher, Marcia, Ethnomathematics, New York: Chapman & Hall/CRC, 1998.
Broline, D. and D. W. Crowe and I. M. Isaacs, The geometry of frieze patterns, Geometriae Dedicata, 3 (1974) 171–176.
Campbell, Paul J., and Darrah P. Chavey. Tchuka ruma solitaire. UMAP Journal 16, no. 4 (1995) 343-365.
Cromwell, P., Polyhedra, Cambridge U. Press, London, 1997.
Crowe, D., The geometry of African art, I. Bakuba art, Journal of Geometry 1 (1971) 169-182.
Crowe, D., The geometry of African art, II. A catalog of Benin patterns, Historia Mathematica 2 (1975) 253-271.
Crowe, D., The geometry of African art, III: The smoking pipes of Begho, in The Geometric Vein, (Coxeter Festschrift), C. Davis et al., (eds.), Springer-Verlag, New York, 1981.
Crowe, D., Tongan symmetries, in Science of Pacific Island Peoples, Part IV, Education, Language, Patterns and Policy, J. Morrison, P. Garaghty, and L. Crowl, (eds.), Suva: Institute of Pacific Studies, 1994.
Crowe, D. and D. Nagy, Cakaudrove-style masi kesa of Fiji, Ars Textrina 18 (1992) 119-155.
Crowe, D. and R. Torrence, Admiralty Islands spear decorations: A minicatalog of pmm patterns, Symmetry: Culture and Science 4 (1993) 385-396.
Crowe, D. and D. Washburn, Groups and geometry in the ceramic art of San Ildefonso, Algebras, Groups and Geometries 3 (1985) 263-277.
Gerdes, Paulus (1991a), Ethnogeometrie, Franzbecker Verlag, Hildesheim.
Gerdes, Paulus (1991b), Sobre o Despertar do Pensamento Geométrico, Universidade Federal de Paraná, Curitiba.
Gerdes, Paulus (1999), Geometry from Africa: Mathematical and Educational Explorations, The Mathematical Association of America, Washington.
Gerdes, Paulus (2000), Le cercle et le carré: Créativité géométrique, artistique, et symbolique de vannières et vanniers d’Afrique, d’Amérique, d’Asie et d’Océanie, L’Harmattan, Paris.
Gerdes, Paulus (2002), The Awakening of Geometrical Thought in Early Culture, MEP Press, Minneapolis MN.
Grünbaum, B.. and Z. Grünbaum, G. Shephard, Symmetry in Moorish and Other Ornaments, Comp. and Math. with Appl., 12 (1986) 641-653.
Grünbaum, B. and G. Shephard, Tilings and Patterns, Freeman. New York, 1987.
Hargettai, I., (ed.), Symmetry1: Unifying Human Understanding, Pergamon, Oxford, 1986.
Hargettai, I., (ed.), Symmetry2, Pergamon, Oxford, 1989.
Hargittai, I., (ed.), Fivefold Symmetry, World Scientific, Singapore, 1992.
Jablan, S., Mirror generated curves, Symmetry: Culture and Science, 6 (1995) 275-278.
Kappraff, J., Connections, The Geometric Bridge between Art and Science, McGraw Hill, 1990.
Schattschneider, D. The plane symmetry groups. Their recognition and notation, Amer. Math. Monthly 85 (1978) 439-450.
Schattschneider, D., Tiling the plane with congruent pentagons, Mathematics Magazine 51 (1978) 29-44.
Schattschneider, D., Will it tile? Try the Conway criterion!, Mathematics Magazine 53 (1980) 224-233.
Schattschneider, Doris. “In Black and white: how to create perfectly colored symmetric patterns.” Computers and Mathematics with Applications, 12B no. 3/4 (1986): 673 – 695. Also in Symmetry: Unifying Human Understanding, edited by István Hargittai, pp. 673 – 695. New York: Pergamon, 1986.
Schattschneider, D., The Polya-Escher connection, Mathematics Magazine 60 (1987) 293-298.
Schattschneider, D., Visions of Symmetry, W. H. Freeman, New York, 1990.
Schattschneider, D., Escher: A mathematician in spite of himself, in The Lighter Side of Mathematics, R. Guy, and R. Woodrow, (eds.), Mathematical Association of America, 1994, p. 91-100. (Reprinted from Structural Topology 15 (1988) 9-22).
Schattschneider, D. and W. Walker, M.C. Escher Kaleidocycles, Pomegranate Artbooks, Rohnert Park, 1987.
Schreiber, P., Art and Architecture, in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, I. Grattan-Guiness, (ed.), Routledge, London, 1994, p. 1593-1611.
Senechal, M. and G. Fleck, (eds.), Patterns of Symmetry, U. Massachusetts Press, Amherst, 1974.
Senechal, M., Point groups and color symmetry, Z. Kristall., 142 (1975) 1-23.
Senechal, M., Color groups, Disc. Appl. Math., 1 (1979) 51-73.
Stevens, P., Handbook of Regular Patterns, MIT Press, Cambridge, 1981.
Stewart, I. and M. Golubitsky, Fearful Symmetry – Is God a Geometer?, Blackwell, Oxford, 1992.
Torday, E. and T.A. Joyce, Notes Ethnographiques sur les peuples communement appeles Bakuba, ainsi que sur les peuplades apparentées. – Les Bushongo. Annales, Serie 3, Ethnographie, Anthropologie. Documents Ethnographiques concernant les populations du Congo Belge, Tome II, Fascicule I, Musee Royal du Congo Belge, Bruxelles, 1910.
Washburn, Dorothy K., Symmetry Analysis of Yurok, Karok, and Hupa Indian Basket Designs. Empirical Studies of the Arts 4:19-45, 1986.
Washburn, Dorothy K., Style, Classiﬁcation and Ethnicity: Design Categories on Bakuba Rafﬁa Cloth. Transactions of the American Philosophical Society, Vol. 80, Pt. 3.
Washburn, Dorothy K., Perceptual Anthropology: the Cultural Salience of Symmetry. American Anthropologist 101(3):547-562, 1999.
Washburn, Dorothy K., and Crowe, Donald W. Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. Seattle: University of Washington Press, 1988.
Washburn, Dorothy K. and Donald W. Crowe, Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. University of Washington Press, Seattle, 1988.
Washburn, Dorothy K., and Crowe, Donald W., eds. Symmetry Comes of Age: the Role of Pattern in Culture. University of Washington Press, Seattle, 2004.
Zaslavsky, C., Africa Counts: Number and Pattern in African Culture, Lawrence Hill Books, Brooklyn, 1973.
]]>Courtney Gibbons
Hamilton College
It was a dark and stormy night… Okay, it was probably more like 3:30 in the afternoon on a crisp fall day back when I was teaching Calc 1 for the first time as “Professor Gibbons,” and I was looking through my colleagues’ past syllabi to see what problems they liked to assign. One of the problems sent me off on a tangent (pun intended!) because it evocatively named the rational function $y = \frac{1}{1+x^2}$ “The Witch of Agnesi” and, in this problem and others in subsequent chapters, proceeded to use differential calculus to tease out its secrets.
In fact, the Witch is one of a family of plane curves, $x^2 y + 4c^2y – 8c^3 = 0$, parametrized by $c$ (just take $c = \frac{1}{2}$).
I don’t want to spoil anyone’s calculus homework, but Evelyn Lamb has written a nice blog about the Witch over at Roots of Unity. (Okay, one spoiler: the Italian for “curve” is “versiera” while “witch” is “avversiera” – so the name of the curve is a mistranslation at best, or one of those groan-inducing mathematician puns at worst.)
My encounter with the Witch left me thinking about the many ways we discuss polynomials (and rational functions) with our college students, and I wanted to share some of those perspectives in this column.
A math student’s first encounter with polynomials often comes hot on the heels of the definition of a function. In this context, we meet our friends the single-variable polynomials $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ and spend a lot of time with the functions that plot lines ($y = mx + b$) and parabolas ($y = ax^2 + bx + c$). Implicit in this introduction is that the coefficients belong to the real numbers.
There are lots of ways to write down a line, and thinking back to high school algebra, you might remember that you and your math class best buddy might look at the line plotted below and come up with different point-slope equations.
You write down $y – 5 = \frac{1}{2}(x – 6)$ while your pal writes down $y – 3 = \frac{1}{2}(x-2)$. These should be the same because we say two functions are equal if they make the exact same input-output assignments (and in the case of functions we can plot in the $xy$-plane, this means they sketch out the same graph). Now, if you and your high school math pal wanted to check that your lines are the same in a different way, you’d each wrangle your line into slope-intercept form and make sure they match (in this case, you’d both end up with the single-variable polynomial expression $\frac{1}{2}x + 2$).
Okay, but. That last part? That relies on the definition of polynomial equality (two polynomials are equal if they are equal coefficient by coefficient) matching the definition of function equality, and that doesn’t always work!
Consider the field $F$ with two elements, $0$ and $1$, with addition and multiplication defined modulo $2$. That is, $0 + 0 = 0 = 1+1$ and $0+1 = 1 = 1+0$, while $0\cdot 1 = 1\cdot 0 = 0\cdot 0 = 0$ and $1\cdot 1 = 1$. We’re going to take our polynomial coefficients from this field.
As functions, $f(x) = x^2 + 1$ and $g(x) = x+1$ take elements of $F$ and assign them to elements of $F$ the same way. Indeed, $f(0) = g(0) = 1$ and $f(1) = g(1) = 0$. So, as functions, $f(x)$ and $g(x)$ are equal! But as polynomials, they are not, because when we do a coefficient comparison, we see that $f$ has a nonzero coefficient for $x^2$ while $g$ does not. (In this weird little example, we actually have that $f = g^2$.)
It’s a fair question to ask why we would bother with a special definition of polynomial equality if it’s going to pull sneaky tricks like this. Polynomials, it turns out, are useful for more than just input-output assignments!
I will admit to taking a certain joy in teaching partial fraction decomposition in calculus. There are pedagogical arguments in favor and against including it in the syllabus, but my enthusiasm is epicurean: I like that students are seeing an example of a basis for a pretty weird vector space before they have taken linear algebra!
If your integrand contains the rational function $y = \frac{p(x)}{(x+2)^2(x^2 +1)}$, you may remember that you can decompose it by first performing polynomial long division to write $\frac{p(x)}{(x+2)^2(x^2+1)} = q(x) + \frac{r(x)}{(x+2)^2(x^2+1)}$ and then break the “reduced” rational function down into a sum of the form $$\frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{Cx}{x^2+1} + \frac{D}{x^2+1}.$$ This technique works because $\frac{1}{x+2}$, $\frac{1}{(x+2)^2}$, $\frac{x}{x^2+1}$, and $\frac{1}{x^2+1}$ form a basis for the vector space of rational functions with denominator $(x+2)^2(x^2 +1)$. One technique for finding the constants $A$, $B$, $C$, and $D$ is to use that rational functions are, well, functions! If two expressions really are the same functions, they’ll have the same input-output assignments. So, trying some handy values for $x$ (who doesn’t love $x = 0$?) gives us leverage to solve for the missing constants.
More generally, many abstract mathematical objects have features that you might want to collect (or count), and you might want to stash your collection (or count) somewhere. For example, the characteristic polynomial $p(x) = |A – Ix|$ of a matrix $A$ stashes the eigenvalues of $A$ in its linear factors! The entries in a Young tableau are stashed as the coefficients of a Schur polynomial (read all about it)! It’s a bonus when these polynomials turn out to be invariants, and even better when you can specialize to all sorts of other polynomial invariants, as you can with the Tutte polynomial of a graph (or link, or matroid, or…)!
In marshalling my resources for this blog post, I spent a little time reviewing the AMS Notices “What is…?” collection. There are some recent entries that piqued my interest and might pique yours, too: a column about multiple orthogonal polynomials and another about Sobolev orthogonal polynomials.
And, finally, Florian Cajori writes this entry in his A History of Mathematics (available for free through Project Gutenberg):
Maria Gaetana Agnesi (1718–1799) of Milan, distinguished as a linguist, mathematician, and philosopher, filled the mathematical chair at the University of Bologna during her father’s sickness. In 1748 she published her Instituzioni Analitiche, which was translated into English in 1801. The “witch of Agnesi” or “versiera” is a plane curve containing a straightline, $x = 0$, and a cubic, $(\frac{y}{c})^2 + 1 = \frac{c}{x}$.
It’s a terse entry for a rather remarkable person. Indeed, by 7 she had mastered Greek, Hebrew, and Latin (having already mastered French by 5); at 9, she defended higher education for women in her father’s salon. After her father died, she left her mathematical and scientific endeavors to care for dying women. Scientific American has a nice biography column for those who want to learn more!
Allechar Serrano López
Harvard University
The Jordan curve theorem is a result in topology that states that every Jordan curve (a plane simple closed curve) divides the plane into an “inside” region enclosed by the curve and an “outside” region. We can think of a plane simple closed curve as a closed loop that does not intersect with itself. The theorem feels true: it intuitively makes sense and we do not have to spend several minutes trying to convince ourselves it’s true. We have seen this theorem in action in our lives… this is why fences work, right?
In order to state the theorem in a formal manner, we need a definition:
A Jordan curve $C$ is a simple closed curve in $\mathbb{R}^2$. We can construct such a curve as the image of a continuous map $\phi: [0,1] \rightarrow \mathbb{R}^2$ such that:
Here, condition (1) makes sure that we have a loop, and condition (2) ensures that our loop does not have any self-intersection points.
Then we can state the theorem as follows:
Let $C$ be a Jordan curve in the plane $\mathbb{R}^2$. Then its complement, $\mathbb{R}^2\setminus C$, consists of two connected components. One of these components is bounded (the interior) and the other one is unbounded (the exterior), and the curve $C$ is the boundary of each component.
So, the Jordan curve separates $\mathbb{R}^2$ into two pieces: an inside region (which has a finite area) and an outside region.
If we have a point on the plane and draw a closed curve, we can try to figure out if the curve encloses the point (and how many times it goes around it) or if it doesn’t. First, we start by giving the curve an orientation; mathematicians chose a long time ago that counterclockwise is the way to go, so the winding number is positive if the curve encloses the point counterclockwise. The winding number of a closed curve around a given point is an integer that counts the number of times that the curve goes around the point. If the curve does not encircle the point, then the winding number at that point is 0. The winding number of a Jordan curve around a point in its interior is 1 (or -1 if we travel in the other direction!)
While the definition of winding numbers seems straightforward, they come up in some of the advanced undergraduate- and graduate-level courses like differential geometry and complex analysis. In complex analysis, they came to haunt me disguised as line integrals. In fact, Stokes’ theorem and the residue theorem are related to the Jordan curve theorem.
I was first introduced to the Jordan curve theorem in my algebraic topology class. Like most definitions and theorems, it was introduced in clinical detail: as part of a long list of results with no mention of a context or why would anyone care about it. However, mathematics presents itself in different ways to different groups of people, so I would like to discuss the Jordan curve theorem as part of an intrinsic human activity: storytelling.
The Chokwe people live in Southwestern Africa and they are known for their art, which they also employ in their storytelling. They have a tradition of drawing figures in the sand, these are known as lusona (plural: sona), to illustrate their stories. The sona illustrate fables, games, riddles, proverbs, and stories. Each lusona starts with a series of evenly-spaced dots, in a rectangular array, and the drawing consists of lines weaving in and out around the dots.
The storyteller draws and narrates simultaneously while keeping the audience engaged. The sona and the stories accompanying them played an important role in the passing down of knowledge and traditions from one generation to the next, but many of them were lost due to colonization and slavery. What we know about the sona today comes from documentation kept by missionaries.
The mukanda is a rite of passage for boys into adulthood and it begins when the chief of a Chokwe village and his counselors decide that there is a sufficiently large group of children to carry out the rite. The mukanda is a camp enclosed by a fence with huts for the boys; the length of the stay at the camp varies from one year to three years. In the mukanda, they learn rituals, stories, and how to make masks, and they can return home after the prescribed education is complete. Kalelwa, a spirit who is incarnated by a mask of the same name, is who gives the signal for the coming and going from the mukanda, and mothers are not allowed to see their sons while they are going through the rite of passage.
There are several sona referring to the mukanda. A lusona which is a continuous closed curve with no self-intersections includes a story where the line of dots are the children involved in the rite of passage, the two higher dots are the guardians of the camp, and the lower dots represent people who are not involved in the ceremony. The children and the guardians are inside the camp and so they are in the bounded connected component while people not participating in the ceremony are in the unbounded connected component.
The mukanda exemplifies a topological concern of the Chokwe: distinguish between the inside of the mukanda (where the children are) and the outside world, that is, the need to determine two regions with a common boundary (which is exactly the issue that the Jordan curve theorem addresses!).
Bill Casselman
University of British Columbia
Commercial transactions on the internet are invariably passed through a process that hides them from unauthorized parties, using RSA public key encryption (named after the authors of the method) to hide details best kept hidden from prying spies.
I’ll remind you how this works, in very basic terms. Two people want to communicate with each other. The messages they exchange pass over a public network, so that they must assume that anybody at all can read them. They therefore want to express these messages—to encrypt them—in such a way that they are nonsense, for all practical purposes, to those not in their confidence.
The basic idea of RSA, and perhaps all public key encryption schemes, is that each person—say A— willing to receive messages chooses a set of private keys and computes from these a set of public keys, which are then made available at large. The point is that computing the public key from the private one is straightforward, whereas going from public to private is very, very difficult. Anyone—say B—who wishes to send A a message uses the public key to encode a message, before he sends it. Once it is in this enciphered form it can—one hopes!—be reconstituted only by knowing A’s private keys.
In this, and perhaps in all public key encryption systems, messages are made into a sequence of integers, and the enciphered message is then computed also as a sequence of integers. The difficulty of reading these secret message depends strongly on the difficulty of carrying out certain mathematical tasks.
In the RSA scheme, one of the two components in the public key data a person publishes is the product of two large prime numbers, and reading a message sent using this key requires knowing these factors. As long as such factoring is impractical, only person A can do this. This factoring has been so far a very difficult computational problem, but as Tony Phillips explained in two earlier columns, it seems very possible that at some point in the not-so-distant future quantum computers will be able to apply impressively efficient parallel computational strategies to make it easy. (If it doesn’t become possible, it won’t be because a lot of clever people aren’t trying. Quantum computing will have benefits beyond the ability to read other people’s mail.)
In order to deal with this apparently looming threat, the NIST (National Institute of Standards and Technology) has optimistically engaged in what it calls a “Post-Quantum Cryptography (PQC) standardization process”, with the aim of finding difficult mathematical problems even quantum computers cannot handle. Preliminary results were announced in the summer of 2022. Mathematically, the most intriguing of the new proposals use lattices for message encryption. Lattices comprise a topic of great theoretical interest since the late 18th century. That’s what this column is about.
Suppose $u$ and $v$ to be two vectors in the plane. The lattice they generate is the set of all vectors of the form $au +bv$ with integers $a$, $b$. These make up a discrete set of points, evenly spaced.
The choice of vectors $u$, $v$ partition the whole plane into copies of the parallelogram they span:
But there are many other pairs of vectors that generate the same lattice. In this figure, the pair $U = 4u + v$, $V = 5u+v$:
Geometrically, a generating pair has the property that the parallelogram they span doesn’t contain any lattice points inside it.
There is a simple way to generate such pairs. Let $T$ be a double array of integers
$$ T = \left[ \matrix { a & b \cr c & d \cr } \right ] \hbox{ for example } \left[ \matrix { 4 & 1 \cr 5 & 1 \cr } \right ] \, . $$
This is a matrix of size $2 \times 2$. Impose the condition that its determinant $ad-bc$ be $\pm 1$ (as it is here). Then $au+bv$, $cu+dv$ will generate the same lattice, and all generating pairs are found in this way.
The point of requiring determinant $1$ is that the relationship is reversible. From
$$ U = au + bv, \qquad V = cu + dv $$
we can solve to get
$$ u = dU—bV, \qquad v = -cU + aV \, , $$
as you can check by substitution.
These things are best dealt with as matrix equations:
$$ \left[ \matrix { U \cr V \cr } \right ] = \left[ \matrix { a & b \cr c & d \cr } \right ]\left[ \matrix { u \cr v \cr } \right ] , \qquad \left[ \matrix { u \cr v \cr } \right ] = \left[ \matrix { a & b \cr c & d \cr } \right ]^{-1}\left[ \matrix { U \cr V \cr } \right ] = \left[ \matrix { \phantom{-}d & -b \cr -c & \phantom{-}a \cr } \right ]\left[ \matrix { U \cr V \cr } \right ] \, , $$ and in the example $$ \left[ \matrix { u \cr v \cr } \right ] = \left[ \matrix { \phantom{-}1 & -1 \cr -5 & \phantom{-}4 \cr } \right ]\left[ \matrix { U \cr V \cr } \right ] , \quad u = U—V, \; v = V—5U \, , $$
so that $u$ and $v$ are in the lattice generated by $U$, $V$.
Every vector $w$ in the plane is a linear combination $au +bv$ of $u$ and $v$, with $a$, $b$ arbitrary real numbers. If
$$ \eqalign{ w &= \left[ \matrix { x & y \cr } \right] \cr u &= \left[ \matrix { u_{x} & u_{y} \cr } \right] \cr v &= \left[ \matrix { v_{x} & v_{y} \cr } \right] \, . \cr } $$
then we write
$$ \eqalign { w &= au + bv \cr x &= a u_{x} + b v_{x} \cr y &= a u_{y} + b v_{y} \cr } $$
which means that in order to find what $a$ and $b$ are we have to solve two equations in two unknowns. The efficient way to do this to write it as a matrix equation
$$ \left[ \matrix { x & y \cr } \right] = \left[ \matrix { a & b \cr } \right] \left[ \matrix { u_{x} & u_{y} \cr v_{x} & v_{y} } \right] $$
which gives us
$$ \left[ \matrix { a & b \cr } \right] = \left[ \matrix { x & y \cr } \right] \left[ \matrix { u_{x} & u_{y} \cr v_{x} & v_{y} } \right]^{-1}. $$
If $a$ and $b$ are integers, then $w$ will be in the lattice generated by $u$ and $v$, but otherwise not.
Many of the new encryption methods rely on a problem that is easy to state, but not quite so easy to solve: Given a lattice in the plane and a point $P$ in the plane, what is the point in the lattice closest to $P$?
The difficulty of answering this question depends strongly on how the lattice is specified. The best situation is that in which it is given by two generators that are nearly orthogonal. The parallelogram they span can be used to partition the plane, and every point in the plane will belong to a copy of this basic one.
For example, suppose that the lattice is generated by $u = \left[\matrix { 5 & 1 \cr } \right]$, $v = \left[\matrix {-3 & 7 \cr } \right] $. Let $P = \left[\matrix { 9 & 10 \cr } \right]$. The formula from the previous section tells us that $P = a u + b v$ with
$$ \left[\matrix { a & b \cr } \right] = \left[\matrix { 9 & 10 \cr } \right]\left[\matrix {\phantom{-}5 & 1 \cr -3 & 7 \cr } \right]^{-1} \sim \left[\matrix { 2.24 & 1.11 \cr } \right] \, . $$
The important fact now is that the nearest point in the lattice will be one of the four nearest vertices of that copy. Finding which one is quite quick. In the example just above, the nearest lattice point is $2u + v = \left[\matrix { 7 & 9 \cr } \right]$.
In the figure on the left (called a Voronoi diagram), each point in the lattice is surrounded by the region of points closest to it. On the right, this is overlaid by the span of good generators. You can verify by looking that what I write is true.
Thus, if the lattice is specified in terms of nearly orthogonal generators, we are in good shape. (In fact, this could be taken as the definition of ‘nearly orthogonal’.) But suppose we are given a pair of generators that are nowhere orthogonal. We can still find which parallelogram we are in, and we can find the nearest vertex of that parallelogram easily, but now that vertex may be far from the nearest lattice point.
To summarize: we can answer the question about the nearest lattice point, if we have in hand a good pair of generators of the lattice, but if not we can expect trouble. We shall see in a moment what this has to do with cryptography.
There are several different ways to send encrypted messages using lattices. The simplest is known as GGH, after the authors of the paper that originated it. It has turned out that it is not secure enough to use in its basic form, but the mathematics it incorporates is also used in other, better schemes, so it is worth looking at.
All schemes for public key messaging depend on some way to transform messages into arrays of integers. In my examples, I’ll use the Roman alphabet and the standard ASCII code (acronym of American Standard Code for Information Interchange). This assigns to each letter and each punctuation symbol a number in the open range $[0, 256 = 2^{8})$.
With this encoding, the message “Hello!” becomes the sequence 72 101 108 108 111 33 . But I’ll make this in turn into a sequence of larger numbers by assembling them into groups of four, and then making each group a number in the range $[0, 4294967296=2^{32})$:
$$ \eqalign { 1,819,043,144 &= 72 + 101\cdot 2^8 + 108\cdot 2^{16} + 108 \cdot 2^{24} \cr 8,559 &= 111 + 33 \cdot 2^8 \, . \cr } $$
For the moment, I’ll look only at messages of $8$ characters, so this gives us an array $[m, n]$ of two large integers, here $\left[ \matrix{1819043144 & 8559} \right]$.
Of course the array $[m, n]$ can be turned back into the original message very easily. For example, if you divide $1,819,043,144$ by $256$ you get a remainder of $72$, which you convert to “H”. Continue:
$$ \eqalign { 1819043144 &= 72 + 256\cdot 7105637 \cr 7105637 &= 101 + 256 \cdot 27756 \cr 27756 &= 101 + 256\cdot 108 \cr 108 &= 108\, . \cr } $$
We want to transform this un-secret message into one that is a bit more secret. There are three steps to making this possible: (1) The person—say A—who is to receive messages makes up and posts some public data needed by people who want to write to her. (2) A person—say B—who writes must use these data to make up a publicly unreadable message. (3) Person A applies her private key to read the message.
Some preliminary work has to be done. A person—say A—who wants to receive messages first chooses a pair of vectors in the plane with integral coordinates. These shouldn’t be too small, and they should be nearly orthogonal. For example, say $ u = \left[\matrix { 5 & 1 \cr } \right]$, $v = \left[\matrix {-3 & 7 \cr } \right] $, and in a picture:
They generate a lattice, made up of all integral linear combinations $a u + bv $ with $a$ and $b$ integers:
As we have seen, the vectors $u$ and $v$ are a good set of generators of the lattice. But, also as we have seen, there are lots of other pairs of vectors that generate it. For maximum security, she should choose two that are not at all orthogonal:
This pair make up A’s public key, which fits into a matrix:
$$ W = \left[ \matrix {22 & 12 \cr 17 & 11 \cr } \right ] $$
When person B wants to send a message to A, he first encodes it as an array $m$ of two integers in the range $[0, 2^{32})$. He then looks up A’s key $W$, which is a $2 \times 2$ matrix. He also chooses a small random vector $r$, computes
$$ M = m \cdot W + r $$
and sends it to A.
Why do we expect that no unauthorized person will be able to reverse this computation? The vector $m \cdot W$ is in A’s lattice. Since $r$ is small, it is the vector in the lattice that is closest to $M$. But since we don’t know a good nearly orthogonal generating pair for the lattice, computing the closest point, as we have seen, is not quite trivial.
But $A$ does have in hand a good generating pair of vectors, and she can therefore compute $m\cdot W$, then $m$, quite easily.
Here is an outline of how things go, with messages of any length.
Let’s take up again the earlier example, in which A’s secret key was the matrix
$$ T = \left[ \matrix { \phantom{-}5 & 1 \cr -3 & 7 } \right ] \, . $$
She chooses to multiply it by
$$ U = \left[ \matrix { 5 & 1 \cr 4 & 1 } \right ] $$
to get her public key
$$ W = UT = \left[\matrix { 22 & 12 \cr 17 & 11 \cr } \right] \,. $$
Suppose B wants to tell her “Hello!”. As we have seen, applying ASCII code produces as raw text the integer array $m = \left[ \matrix{1819043144 & 8559} \right]$. He multiplies it on the right by $W$, and adds a small vector $r$:
$$ M = m \cdot W + \left[ \matrix{2 & -1} \right] = \left[\matrix { 9095190045 & 1819103056 \cr } \right] \, . $$
Admittedly, this $r$ is a bit small. I should say, the role of $r$ is crucial, since without it this whole business just amounts to a really complicated substitution cipher of the kind read by Sherlock Holmes.
When $A$ gets this, she calculates $$ M \cdot T^{-1} = \left [ \matrix { 1819043144.29 & 8558.82 } \right ] $$ which she converts (correctly) to $\left [ \matrix { 1819043144 & 8559 } \right ]$ and translates to “Hello!”.
We have seen that reading messages in this business amounts to finding closest lattice points, and that this in turn depends on finding good generators of a lattice. It turns out that in low dimensions there are extremely efficient ways to do this. It is exactly this problem that arose in the classification of integral quadratic forms, a branch of number theory. Finding closest vectors for arrays of length $2$ is particularly easy, due to a well known ‘reduction algorithm’ due to the eighteenth century French mathematician Lagrange. It was later made well known by work of Karl Friedrich Gauss, to whom it is sometimes attributed.
A similar algorithm for arrays of length $3$ was found by the nineteenth century German mathematician Gotthold Eisenstein (which seems to have been incorporated in the computer algebra system SageMath). There are very general algorithms known for all dimensions, but they run more and more slowly as the length of arrays increases, and are not really practical.
Sadly, even for long messages the GGH scheme was shown by Phong Nguyen not to be very secure. But schemes that do depend on the difficulty of finding good generating sets of lattice vectors in high dimensions still look quite plausible.
Are quantum computers really in sight?
Chapter 7 is about lattices.
Lattices in higher dimensions are an active topic of research.