{"id":1148,"date":"2022-09-06T17:52:27","date_gmt":"2022-09-06T21:52:27","guid":{"rendered":"https:\/\/mathvoices.ams.org\/mathmedia\/?p=1148"},"modified":"2022-09-12T11:00:34","modified_gmt":"2022-09-12T15:00:34","slug":"math-digests-august-2022","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-august-2022\/","title":{"rendered":"Math Digests August 2022"},"content":{"rendered":"<h3><a href=\"https:\/\/www.nytimes.com\/2022\/07\/05\/science\/maryna-viazovska-math.html\">Maryna Viazovska: Second to none in any dimension.<\/a><\/h3>\n<p><i><span style=\"font-weight: 400\">New York Times<\/span><\/i><span style=\"font-weight: 400\">, July 5, 2022<\/span><\/p>\n<p><span style=\"font-weight: 400\">Math helps us make the most out of space. Sometimes that involves familiar dimensions: Geometric rules can dictate how to stack oranges in a three-dimensional box or arrange tiles on a two-dimensional plane so that as much space as possible is used up. Other times, mathematicians answer these questions in higher dimensions. This summer, mathematician Maryna Viazovska of the Swiss Federal Institute in Zurich was awarded the Fields Medal\u2014considered by many to be math\u2019s highest honor\u2014for proofs of sphere packing in space of dimensions 8 and 24. The work represents an extreme twist on a 400-year-old conjecture that stacking bowling balls in a pyramid fills nearly 75% of the available space. Viazovska is only the second woman to win a Fields Medal. \u201cI feel sad that I\u2019m only the second woman,\u201d Viazovska told the Times. \u201cI hope it will change in the future.\u201d In this article for the <\/span><i><span style=\"font-weight: 400\">New York Times<\/span><\/i><span style=\"font-weight: 400\">, Kenneth Chang describes Viazovska\u2019s work and gives more context on her momentous award.<\/span><\/p>\n<p><b>Classroom activities:<\/b> <i><span style=\"font-weight: 400\">stacking, geometry, higher dimensions<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) In high-dimensional spaces, like those that Viazovska worked in, it\u2019s tough to visualize geometric objects. For more information about math and geometry in higher dimensions, watch <\/span><a href=\"https:\/\/www.youtube.com\/watch?v=4TI1onWI_IM&amp;ab_channel=TheLazyEngineer\"><span style=\"font-weight: 400\">\u201cA Journey into the 4th Dimension\u201d<\/span><\/a><span style=\"font-weight: 400\">.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">In the video we learn that we can imagine higher dimensions by picturing their projection onto a lower dimension. Can you draw a 2D projection of the following 3D shapes? 1) Sphere, 2) rectangular prism, 3) triangular prism from the top, 4) triangular prism from the side.<\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Mid level) Demonstrate a type of sphere packing with a small box (such as a shoebox) and ping pong balls. (The box should have straight sides and the balls should all have the same diameter.) Cover the bottom layer of the box with as many ping pong balls as possible.\u00a0<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">How many can you fit in that one layer? What is the area of the floor of the box? Looking from the top down, what fraction of the area is empty space? (Hint: calculate the total area covered by ping pong balls.)<\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(High level) Now add a second layer of ping pong balls on top of the first. Place them such that you are maximizing the number of ping pong balls that will fit in two layers.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">What is the total volume of ping pong balls in that space? What total volume spans the bottom of the first layer to the top of the second? (Measure the height of the second layer from the bottom of the box.) What fraction of the volume is empty space?<\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Look at Figure 1 of <\/span><a href=\"https:\/\/www.ams.org\/journals\/notices\/201702\/rnoti-p102.pdf\"><span style=\"font-weight: 400\">this article<\/span><\/a><span style=\"font-weight: 400\"> about sphere packing. Do your arrangements from the last exercise resemble the photos? Do the packing densities (the fraction of volume taken up by ping pong balls) match the numbers in Table 1 for dimensions <\/span><i><span style=\"font-weight: 400\">n <\/span><\/i><span style=\"font-weight: 400\">= 1 and 2?<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Figure 2 shows the densities of the best packings mathematicians have found in each dimension. What do you notice about the chart? Are you surprised by its appearance?<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Max Levy<\/em><\/p>\n<hr \/>\n<h3><a href=\"https:\/\/www.irishexaminer.com\/opinion\/columnists\/arid-40946021.html\">Centenary of Irish scholar&#8217;s death in Alps sadly overlooked<\/a><\/h3>\n<p><i><span style=\"font-weight: 400\">Irish Examiner, <\/span><\/i><span style=\"font-weight: 400\">August 24, 2022.<\/span><\/p>\n<p><span style=\"font-weight: 400\">As the buzz surrounding Maryna Viazovska\u2019s Fields Medal win demonstrated this summer, the granting of one of math\u2019s most prestigious prizes to a woman remains, unfortunately, big news. But over a century ago, women like Sophie Bryant were already breaking barriers in mathematics: When Bryant earned her Ph.D. in 1884, no other women in the United Kingdom or Ireland had done so. In this article, Clodagh Finn sketches the accomplishments of Bryant, who tragically died in a hiking accident 100 years ago this August.<\/span><\/p>\n<p><b>Classroom Activities:<\/b><i><span style=\"font-weight: 400\"> geometry, trigonometry, calculus<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) For a more in-depth look at Sophie Bryant and her mathematical work, read <\/span><a href=\"https:\/\/chalkdustmagazine.com\/biographies\/biography-sophie-bryant\/\"><span style=\"font-weight: 400\">this 2017 biography<\/span><\/a><span style=\"font-weight: 400\"> by Patricia Rothman for <\/span><i><span style=\"font-weight: 400\">Chalkdust Magazine<\/span><\/i><span style=\"font-weight: 400\">.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Now, choose another mathematician who broke gender barriers to research and write a 500-word report about. You may find <\/span><a href=\"https:\/\/mathwomen.agnesscott.org\/women\/chronol.htm\"><span style=\"font-weight: 400\">this list<\/span><\/a><span style=\"font-weight: 400\"> of women mathematicians from Agnes Scott College helpful for getting started.<\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Bryant\u2019s 1884 paper, the first by a woman for the London Mathematical Society, studied the shapes of cells within honeycombs. Read <\/span><a href=\"https:\/\/nautil.us\/why-nature-prefers-hexagons-4458\/\"><span style=\"font-weight: 400\">this article by Philip Ball for <\/span><i><span style=\"font-weight: 400\">Nautilus<\/span><\/i><\/a> <span style=\"font-weight: 400\">on that same topic.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Trigonometry) Ball writes: \u201c<\/span><span style=\"font-weight: 400\">Hexagonal cells require the least total length of wall, compared with triangles or squares of the same area.\u201d Suppose you have a regular hexagon, an equilateral triangle, and a square, each with total area 1. Calculate their perimeters. Hint: A hexagon is made up of 6 equilateral triangles. Why?<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Trigonometry) Now suppose you have a regular polygon with <\/span><i><span style=\"font-weight: 400\">n <\/span><\/i><span style=\"font-weight: 400\">sides and total area 1. What\u2019s the perimeter?<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Calculate the perimeter of the area 1 10-gon, 100-gon, and 1000-gon. What shape of area 1 do you think has the smallest perimeter? (Calculus) Prove it.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\">\u2014<em>Leila Sloman<\/em><\/p>\n<hr \/>\n<h3><a href=\"https:\/\/www.sciencenews.org\/article\/sea-urchin-skeletons-patterns-structure\">Sea urchin skeletons\u2019 splendid patterns may strengthen their structure<\/a><\/h3>\n<p><i><span style=\"font-weight: 400\">Science News, <\/span><\/i><span style=\"font-weight: 400\">August 22, 2022.<\/span><\/p>\n<p><span style=\"font-weight: 400\">Packing problems seem to be a theme in math news this month. A new paper in <\/span><i><span style=\"font-weight: 400\">Journal of the Royal Society Interface<\/span><\/i><span style=\"font-weight: 400\"> shows that sea urchin skeletons resemble a mathematical packing pattern called a Voronoi pattern. These patterns look like a web or mesh whose holes, or \u201ccells\u201d, appear somewhat irregular. But the cell shapes actually obey strict rules. Within each cell is a \u201cseed\u201d that governs the cell\u2019s shape, writes Rachel Crowell for <\/span><i><span style=\"font-weight: 400\">Science News<\/span><\/i><span style=\"font-weight: 400\">. Cells must hug their seeds tightly\u2014if you sit down anywhere on the Voronoi pattern, the closest seed should be in the same cell as you. In this article, Crowell explains the rules of the Voronoi pattern, and some of the engineering benefits this structure may offer both the sea urchin and, potentially, human technology.<\/span><\/p>\n<p><b>Classroom Activities: <\/b><em>geometry, <\/em><i><span style=\"font-weight: 400\">Voronoi patterns<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) Try out computer scientist Alex Beutel\u2019s <\/span><a href=\"http:\/\/alexbeutel.com\/webgl\/voronoi.html\"><span style=\"font-weight: 400\">interactive Voronoi diagram generator<\/span><\/a><span style=\"font-weight: 400\">.<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) For more on Voronoi patterns, read <\/span><a href=\"https:\/\/blogs.scientificamerican.com\/observations\/voronoi-tessellations-and-scutoids-are-everywhere\/\"><span style=\"font-weight: 400\">this <\/span><i><span style=\"font-weight: 400\">Scientific American <\/span><\/i><span style=\"font-weight: 400\">article by Susan D&#8217;Agostino<\/span><\/a><span style=\"font-weight: 400\">.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Now, make your own Voronoi pattern: Draw three dots anywhere you like on a piece of paper, and treat these as the \u201cseeds\u201d for the cells. Try to figure out where the boundaries of the cells should be! Come up with a guess and write a brief justification.<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Once you have your guess, read <\/span><a href=\"https:\/\/plus.maths.org\/content\/maths-minute-voronoi-diagrams\"><span style=\"font-weight: 400\">this tutorial<\/span><\/a><span style=\"font-weight: 400\"> on making a Voronoi diagram from <\/span><i><span style=\"font-weight: 400\">Plus Magazine<\/span><\/i><span style=\"font-weight: 400\">. What did you learn from their solution? Does it match your guess?<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\">\u2014<em>Leila Sloman<\/em><\/p>\n<hr \/>\n<h3><a href=\"https:\/\/www.independent.co.uk\/voices\/how-many-holes-straw-debate-twitter-b2145201.html\"><span style=\"font-weight: 400\">How many holes are there in a straw? The answer may surprise you<\/span><\/a><\/h3>\n<p><i><span style=\"font-weight: 400\">The Independent<\/span><\/i><span style=\"font-weight: 400\">, August 15, 2022<\/span><\/p>\n<p><span style=\"font-weight: 400\">Here\u2019s a question you might not know is contentious: how many holes are there in an ordinary drinking straw? You could say that a straw has two holes\u2014one on each end\u2014but you could also say that the empty space inside the straw is just one long hole. In an article for <\/span><i><span style=\"font-weight: 400\">The Independent<\/span><\/i><span style=\"font-weight: 400\">, mathematician Kit Yates examines the straw question through the lens of the mathematical field of topology. Yates compares topological shapes to objects made of dough, which can be stretched, pulled, or squished without being fundamentally changed. To topologists, a drinking glass is the same as a plate, a <\/span><a href=\"https:\/\/en.wikipedia.org\/wiki\/Topology#\/media\/File:Mug_and_Torus_morph.gif\"><span style=\"font-weight: 400\">mug is the same as a donut<\/span><\/a><span style=\"font-weight: 400\">, and a pair of binoculars is the same as a pair of glasses. And a straw can be compressed down into a ring\u2014which has just one hole.<\/span><\/p>\n<p><b>Classroom activities:<\/b> <i><span style=\"font-weight: 400\">topology, geometry<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) Look at some objects in the room and try to determine how many holes they have, topologically. Try, for example, to determine the number of holes in a book, a rubber band, a water bottle, and a pair of pants.\u00a0<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Pair up with another student and discuss your answers. If you disagree about the number of holes in a particular object, work together to figure out the right number.<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Scavenger hunt: Find an object that has one hole, an object that has two holes, and an object that has three or more holes.<\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) Read the section on cylinders in this article about <\/span><a href=\"https:\/\/chalkdustmagazine.com\/features\/topological-tic-tac-toe\/\"><span style=\"font-weight: 400\">topological tic-tac-toe<\/span><\/a><span style=\"font-weight: 400\">, then play a few games of cylindrical tic-tac-toe with another student.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Advanced) Try some of the tic-tac-toe puzzles in the article. Note: the puzzles get progressively harder!<\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Geometry, Precalculus) Watch this <\/span><a href=\"https:\/\/youtu.be\/AmgkSdhK4K8\"><span style=\"font-weight: 400\">video about using topology to solve problems<\/span><\/a><span style=\"font-weight: 400\"> by 3Blue1Brown.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Draw an ellipse on a piece of paper and find an inscribed rectangle inside the ellipse.<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Advanced) Draw a few different closed loops and try to find an inscribed rectangle in each one. Discuss: Why is this hard to do, even after watching the video? Notice that the video proves that there is always a solution to the inscribed rectangle problem, but doesn\u2019t show how to construct a solution for any particular loop. (A proof that demonstrates that a solution exists without giving a method for finding it is sometimes called an \u201cexistence proof.\u201d)<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><i><span style=\"font-weight: 400\">\u2014Tamar Lichter Blanks<\/span><\/i><\/p>\n<hr \/>\n<h3><a href=\"https:\/\/www.theguardian.com\/science\/2022\/jul\/25\/can-you-solve-it-blockbusters\"><span style=\"font-weight: 400\">Can you solve it? Blockbusters!<\/span><\/a><\/h3>\n<p><i><span style=\"font-weight: 400\">The Guardian<\/span><\/i><span style=\"font-weight: 400\">, July 25, 2022<\/span><\/p>\n<p><span style=\"font-weight: 400\">In a game show from the 1980s called <\/span><i><span style=\"font-weight: 400\">Blockbusters<\/span><\/i><span style=\"font-weight: 400\">, participants try to connect two sides of a map of tiles. Each tile on the map is an identical hexagon. Tiles on two opposing edges are blue (one team\u2019s color), and the other opposing edges are white (for another team). When participants answer questions correctly, a tile between the edges becomes white or blue\u2014a point for their team. The goal: connect your team\u2019s two edges with a path of hexagons of the same color. In this article, puzzle expert Alex Bellos describes the Blockbusters game and imagines a diamond shaped map (shown below). Bellos poses a mathematical riddle. How many configurations of the map will contain a path connecting the two blue edges? The solution requires a little math, but it mostly uses a clever trick of logic.<\/span><\/p>\n<p><b>Classroom activities: <\/b><i><span style=\"font-weight: 400\">probability, logic puzzles, percolation<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) Assuming there are 100 non-edge tiles, solve the riddle. Compare your answer to the <\/span><a href=\"https:\/\/www.theguardian.com\/science\/2022\/jul\/25\/did-you-solve-it-blockbusters\"><span style=\"font-weight: 400\">solution published here<\/span><\/a><span style=\"font-weight: 400\">.<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) Play Blockbusters in class using <\/span><a href=\"https:\/\/lingolex.com\/games\/hexagons\/\"><span style=\"font-weight: 400\">this online version<\/span><\/a><span style=\"font-weight: 400\">. (Find some <\/span><a href=\"https:\/\/lingolex.com\/games\/hexagons\/questions.php\"><span style=\"font-weight: 400\">default questions<\/span><\/a><span style=\"font-weight: 400\"> here.)<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) Demonstrate percolation as introduced in the article. What you\u2019ll need: 1 strainer or sieve; 1 bowl or cup with an opening large enough to hold up the strainer; 3 materials of different size (coffee grounds, sand, small rocks, little rocks, marbles, or something similar). Fill the strainer with one material. Place on top of the cup\/bowl. Pour a cup of water over the material in the strainer. Measure how long it takes for the entire fluid to percolate. Repeat for each material. What do you notice about the relationship between fluid flow and material size?<\/span><\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Max Levy<\/em><\/p>\n<hr \/>\n<h3>Some more of this month&#8217;s math headlines:<\/h3>\n<ul>\n<li><a href=\"https:\/\/www.newscientist.com\/article\/0-octonions-the-strange-maths-that-could-unite-the-laws-of-nature\/\">Octonions: The strange maths that could unite the laws of nature<\/a><br \/>\n<em>New Scientist, <\/em>August 16, 2022<\/li>\n<li><a href=\"https:\/\/www.sciencenews.org\/article\/grill-burger-math-time-calculations\">Here\u2019s the quickest way to grill burgers, according to math<\/a><br \/>\n<em>Science News, <\/em>July 26, 2022<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Maryna Viazovska: Second to none in any dimension. New York Times, July 5, 2022 Math helps us make the most out of space. Sometimes that involves familiar dimensions: Geometric rules can dictate how to stack oranges in a three-dimensional box or arrange tiles on a two-dimensional plane so that as<span class=\"more-link\"><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-august-2022\/\">Read More &rarr;<\/a><\/span><\/p>\n","protected":false},"author":13,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[2],"tags":[143,35,141,146,147,73,140,145,142,144],"class_list":["entry","author-leilasloman","post-1148","post","type-post","status-publish","format-standard","category-math-in-the-media-digests","tag-calculus","tag-geometry","tag-higher-dimensions","tag-logic-puzzles","tag-percolation","tag-probability","tag-stacking","tag-topology","tag-trigonometry","tag-voronoi-patterns"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/1148","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/users\/13"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/comments?post=1148"}],"version-history":[{"count":6,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/1148\/revisions"}],"predecessor-version":[{"id":1160,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/1148\/revisions\/1160"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/media?parent=1148"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/categories?post=1148"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/tags?post=1148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}