This month’s topics:
An ancient arithmetic error
Kristina Killgrove’s “Babylonian tablet preserves student’s 4,000-year-old geometry mistake” was posted on LiveScience on December 2. Killgrove writes about a small clay tablet preserved in the Ashmolean Museum, Oxford. The tablet was excavated in Kish (modern-day Iraq) in 1931 and is dated to the Old Babylonian period, between 1900 BCE and 1600 BCE. As Killgrove tells us, it is “one of two dozen examples of ancient Babylonian mathematics homework” found during that dig.
Deciphering the inscriptions on this tablet can teach us about the Old Babylonian numbering system; specifically, the student’s error points to one of that system’s main shortcomings.
The mathematically inclined inhabitants of Kish in those days used a base-60 place-value system (works like our base-10, except that the “digits” run from 1 to 59). The digits themselves were written in cuneiform, using an essentially base-10 notation, as follows. In this student’s rendering, a vertical cone-shaped mark represented 1 and a diagonal mark represented 10; so at the left side of the tablet we read 1 52 30. This corresponds to $1\times 60^2 + 52\times 60 + 30$, or 6,750 in our notation, and gives the length of the base of the triangle. Similarly, above the triangle the student has written 3 45, i.e. $3\times 60 + 45$ or 225 as a decimal; traditionally, this number would represent the height of the triangle. The number inside the triangle should correspond to its area, which is half the base times the height, or $6750\times 225/2=759,375$ in decimal, which amounts to $3\times 216,000 + 30\times 3,600 + 56\times 60 + 15$; so in base 60, the area is 3 30 56 15. But inside the triangle the student has written the second “digit” as 8. The next one is mostly obscured and the last one is readable as ending in 5. This is obviously not correct (and the smudge may have been the teacher’s version of a red X). What could have gone wrong in this straightforward calculation?
Eleanor Robson, a British expert in Old Babylonian mathematics, published this tablet in the journal SCIAMVS in 2004; there, she reverse-engineered the student’s answer to find the source of the error. Her guess for the culprit: the Old Babylonians’ lack of a symbol for zero. When zero occurred in a multi-place number, Babylonians simply left that space blank. In Robson’s analysis, the student would have broken up the calculation by writing 1 52 30 as 1 00 00 + 50 00 + 2 00 + 30, and taken the sum of the corresponding products by 3 45. This gives$$1~ 52~ 30 \times 3~ 45 = 3~ 45~ 00~ 00 + 3~ 7~ 30~00 + 7~ 30~00 + 1~ 52~ 30.$$But to add the numbers correctly the same powers of 60 must line up, so somewhere the student should have written the following: (I’m using ${\underline ~}$ for a blank space.)
$\begin{array}{ccccccccccr}
1 & {\underline ~} & {\underline ~} & \times & 3 & 45& =& 3 & 45 & {\underline~} & {\underline ~}\\
& 50 & {\underline ~}& \times & 3 & 45& = &3 & 7 & 30 & {\underline ~}\\
& 2 & {\underline ~}& \times & 3 & 45& =& & 7 & 30 & {\underline ~}\\
& & 30 & \times & 3 & 45 & = & & 1 & 52 & 30\\ \hline
1 & 52 & 30 & \times & 3 & 45& = &7& 1 & 52 & 30\end{array}$
which divided by 2 gives the correct answer $3~30~56~15$.
In Robson’s reconstruction, the student miscopied the first product as $~3 ~{\underline ~}~ 45~{\underline ~}~$, and obtained
$\begin{array}{cccc}
3 & {\underline ~} & 45 & {\underline~} \\
3 & 7 & 30 & {\underline ~} \\
& 7 & 30 & {\underline ~} \\
& 1 & 52 & 30 \\ \hline
6 & 17 & 37 & 30\end{array}$
which divided by 2 gives $3~~8~48~45$, an answer compatible with what can be read on the tablet.
This is clearly a very easy mistake to make: If the space between two consecutive “digits” is slightly too large, it can be read as a zero. One reason why the Babylonians were able to work with this potentially dangerous notation for so many centuries (up to around 300 BC) may be the large size of their base. Whereas on average one out of 10 of the digits in one of our numbers will be a zero, the chance is only one out of 60 if you are writing in base-60.
Knot theory for Cat’s Cradle games
An article published on December 4 in the Journal of the Royal Society Interface uses notation and concepts from knot theory to classify string figures and to study their geographic distribution. String figures are the product of a game involving a long loop of string woven by the ten fingers on a pair of human hands. They are “among the most common forms of play and entertainment across cultural traditions worldwide,” according to the authors. “Cat’s Cradle” is one variant, practiced in English-speaking countries. One of the most widespread string figures is called “Jacob’s Ladder.”
String figures’ intricacy and specificity make them, like language, a cultural marker transmitted from one generation to the next. They have long drawn the attention of anthropologists—the article includes a description by Alfred Russel Wallace of how, to pass the time on a rainy day around 1860 in the Malay Archipelago, he showed the “cat’s cradle” to some young Dyak tribesmen. (“Greatly to my surprise, they knew all about it, and more than I did.”) But to systematically compare examples from distant cultures, scientists needed a way to reliably encode a string figure. Until now, there was no such code.
This new article presents a solution: the “Gauss code,” which can be calculated from a two-dimensional drawing of the figure. The Gauss code is used in the mathematical study of knots, but it can be applied equally well to study complicated unknotted loops such as string diagrams; its first appearance is in the Nachlass (unpublished papers) of Carl Friedrich Gauss (1777-1855), dating to around 1840.
A Gauss code for a planar projection of a loop, knotted or unknotted, is recorded as follows. Choose a starting point on the loop, and a direction of travel along the loop. Trace the loop in that direction from your starting point, and number each crossing as it first occurs. When you are done, you are ready to write down the Gauss code. To do that, make one more complete pass. This will involve encountering each crossing twice: one time you will be crossing under, and the other time you will be crossing over. As you move along the loop, record the numbers as you encounter them, but with a minus sign if you are going under at that crossing. You will end up with a string of $2n$ integers for an $n$-crossing figure, with each integer in the string occurring once with, and once without, a minus sign. This is a Gauss code. Notice that the code you have recorded depends on the starting point and direction that you chose. For a figure with $n$ crossings, there can be up to $2n$ distinct codes—2 possible directions multiplied by $n$ choices of where to start.
The authors of the Interface paper examine 826 string figures from 92 different cultural groups, and compare them using the Gauss code. But the fact that one string figure can have multiple Gauss codes presented a problem. If two figures were the same but assigned different codes, their similarity would be overlooked. To obviate this problem, the authors assign to an $n$-crossing string figure the set of all $2n$ possible codes. This integer string of $2n$ codes is what they call the figure’s “Gauss code sequence.”
With an unambiguous numerical token representing each string figure, the authors could mechanically scan the entire corpus looking for complete or partial matches. They found 83 classes of recurring designs; 380 of the 826 figures belonged to one of these classes, and 7 of the classes contained 10 or more individual figures. The authors remark: “The widely dispersed occurrence of some string figure designs reflects a deep ancestry not only of the practice in general but of particular designs.”
The authors focus in particular on the Jacob’s Ladder, which occurs in 26 different cultural groups, distributed over six continents. They make the following observations.
- The Jacob’s Ladder is quite complex, much more than the average string figure; its construction ends with “an unusual extension move … that novices often find unintuitive and difficult to execute.”
- Most Jacob’s Ladders are made with precisely the same sequence of steps. This is not the case with string figures in general. In particular, one of the first pairs of moves is executed right hand first in all the samples examined. Furthermore, in all but one sample, the move is made with the middle finger, whereas the index would work just as well.
- If the design had evolved independently in several locations, one would expect a variety of forms—”pseudo-Jacob’s Ladders” which look almost the same. The corpus contains a few pseudo-Jacob’s Ladders, but many fewer than could be expected from independent evolution.
They conclude that “the ubiquity, complexity, and congruence of Jacob’s ladder across cultures combined indicate a shared and likely very old ancestry” and suggest that it and other string figures “may be deeply woven into the fabric of human evolution.”
$\zeta(3)$ in Scientific American
Leonhard Euler (1707-1783) proved the amazing identity $$ 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6},$$ which Bernhard Riemann (1826-1866) generalized to the study of $$1 + \frac{1}{2^x} + \frac{1}{3^x} + \cdots$$ for an arbitrary real or complex number $x$. Riemann represented this sum by $\zeta(x)$, defining the “zeta function.” This function has come to occupy a special place at the heart of mathematics. Its complex roots are the subject of the Riemann hypothesis, possibly the most famous unsolved mathematical problem of our time. The numerical evidence overwhelmingly suggests the hypothesis is true, but no one has been able to prove it. A proof would have far-reaching consequences, especially in number theory, since the hypothesis is intimately related to the distribution of prime numbers.
In Riemann’s notation, Euler’s identity was $\zeta(2)=\frac{\pi^2}{6}$. Euler also gave similar but more complicated formulas for $\zeta(4)$, $\zeta(6)$, $\zeta(8)$, and so on—the values of $\zeta(x)$ for even integers $x$. These values were always of the form (rational number) $\times$ (power of $\pi$). Since $\pi$ is known to be transcendental, these are all irrational numbers.
This left questions about the value of $\zeta(x)$ when $x$ is odd. It had been known since Oresme (c. 1320-1382) that the sum $1 + \frac{1}{2} + \frac{1}{3} + \cdots$, i.e. $\zeta(1)$, does not have a finite limit, but what about $\zeta(3)$? The series is convergent; otherwise, nothing else was known until 1978, when Roger Apéry (1916-1994) announced his proof that $\zeta(3)$ is in fact irrational. An article by Manon Bischoff in Spektrum der Wissenschaften, translated and posted by Scientific American on December 19, 2024, recounts the meeting at which Apéry shared his achievement. Its subtitle calls that occasion “one of the most bizarre events in the history of mathematics.”
Bishoff describes Apéry as “a French mathematician who was relatively unknown and in his 60s at the time.” Indeed, Apéry, after a very brilliant beginning (a graduate of the École Normale Supérieure, he had shared first place at the national Agrégation examination in mathematics), was then aged 61 and working at a provincial university on Italian-style algebraic geometry problems, a rather unfashionable field at the time. (See MacTutor for a complete biography.) His presentation did not help his case. In Bischoff’s account, one of the first items was the “previously unknown” series representation $$\zeta(3) = \frac{5}{2}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^3 {2n \choose n}}.$$ When Apéry was questioned about it, he reportedly claimed that such formulas “grow in my garden.”
The lecture was attended by the Dutch-Australian number theorist Alfred van der Poorten (1942-2010), who wrote a detailed “informal report” for the Mathematical Intelligencer: “Though there had been earlier reports of his claiming a proof, scepticism was general. The lecture tended to strengthen this view to rank disbelief.” But van der Poorten goes on to say that Henri Cohen (now emeritus at Bordeaux), also in the audience, had gotten the point and later that evening convinced him that “Professor Apéry had indeed found a quite miraculous and magnificent demonstration of the irrationality of $\zeta(3)$.” (He also tells us that Apéry’s improbable series expression was actually “quite well known” to the experts and had appeared in print.)
Along with a sketch of the history of the zeta function, Bischoff mentions its very intriguing apparition in quantum physics, where it turns out that our constant plays a special role. More details are at the end of a 2019 Universe report by Walter Dittrich (Tübingen), who writes that the number $\zeta(3)$ is “of immense value for people doing calculations in QED [Quantum Electrodynamics], e.g. the second- and third-order (in $\alpha$) radiative corrections of the electron (and muon) anomalous magnetic moment, which is one of the best-measured and -calculated numbers in all of physics.”
—Tony Phillips, Stony Brook University