{"id":2494,"date":"2024-07-26T08:00:12","date_gmt":"2024-07-26T12:00:12","guid":{"rendered":"https:\/\/mathvoices.ams.org\/mathmedia\/?p=2494"},"modified":"2024-07-25T17:40:32","modified_gmt":"2024-07-25T21:40:32","slug":"math-digests-june-2024","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-june-2024\/","title":{"rendered":"Math Digests June 2024"},"content":{"rendered":"<ul>\n<li><a href=\"#1\">The mathematical structure of the Supreme Court in <i>Politico<\/i><\/a><\/li>\n<li><a href=\"#2\">How to calculate the time to &#8220;catch &#8217;em all&#8221; in Numberphile<\/a><\/li>\n<li><a href=\"#3\">Cosmological predictions in <i>Astronomy<\/i><\/a><\/li>\n<li><a href=\"#4\">Population prediction in Japan in <i>AP<\/i><\/a><\/li>\n<li><a href=\"#5\">Kelvin&#8217;s space-filling octahedron in <i>STV News<\/i><\/a><\/li>\n<\/ul>\n<hr \/>\n<h3><a id=\"1\" href=\"https:\/\/www.politico.com\/news\/magazine\/2024\/06\/02\/supreme-court-justice-math-00152188\">Opinion | Using Math to Analyze the Supreme Court Reveals an Intriguing Pattern<\/a><\/h3>\n<p><em>Politico<\/em>, June 2, 2024.<\/p>\n<p><a href=\"https:\/\/nysba.org\/6-to-3-the-impact-of-the-supreme-courts-conservative-super-majority\/\">Much is made<\/a> of the conservative-liberal divide on the Supreme Court. But in this article, legal commentator Sarah Isgur and economist Dean Jens argue \u2014 using math! \u2014 that the conservative group (currently comprising six justices) should be viewed as two distinct groups. Isgur and Jens argue that the main difference between these groups is institutionalism and show some of the numbers that they used in their analysis.<\/p>\n<p><strong>Classroom Activities: <\/strong><em>linear algebra, statistics<\/em><\/p>\n<ul>\n<li>(Mid level) Read the article, then navigate to the graphic titled \u201cThree Groupings on the Supreme Court Show Through in the 2023 Term.\u201d\n<ul>\n<li>Describe, in your own words, the meaning of the number in top left corner of the graphic.<\/li>\n<li>Examine the row representing Justice Sotomayor. Calculate the mean and variance of the data in this row. Calculate the mean and variance of the data in Sotomayor\u2019s row <em>within<\/em> each of the clusters proposed by Isgur and Jens (this will give a different mean and variance for each of the three clusters.)<\/li>\n<li>Randomly select two other justices besides Sotomayor and repeat the exercise on the data from their rows. Do your results support the clustering proposed by Isgur and Jens? Why or why not? As a class, vote on whether you agree with the 3-3-3 clustering.<\/li>\n<\/ul>\n<\/li>\n<li>(Mid level, Linear algebra) Isgur and Jens used a technique called <em>principal component analysis<\/em>. Principal component analysis involves finding eigenvalues and eigenvectors of a matrix of data.\n<ul>\n<li>Consider a hypothetical court that has three justices. Let $\\mathbf{j}_i$ be the vector of how Justice $i$ voted on five separate cases, with each vote being either \u201cyes\u201d or \u201cno.\u201d Suppose<br \/>\n$$\\mathbf{j}_1 = \\begin{bmatrix} \\text{yes} \\\\ \\text{yes} \\\\ \\text{no}\u00a0 \\\\ \\text{no}\u00a0 \\\\ \\text{yes}\u00a0 \\end{bmatrix},\\; \\mathbf{j}_2 = \\begin{bmatrix} \\text{yes} \\\\ \\text{no}\u00a0 \\\\ \\text{no}\u00a0 \\\\ \\text{no}\u00a0 \\\\ \\text{yes}\u00a0 \\end{bmatrix},\\; \\mathbf{j}_3 = \\begin{bmatrix} \\text{no} \\\\ \\text{no}\u00a0 \\\\ \\text{yes}\u00a0 \\\\ \\text{yes}\u00a0 \\\\ \\text{no}\u00a0 \\end{bmatrix} $$<br \/>\nFind the matrix whose $(i,j)$-th entry is the percentage of the time that Justice $i$ and Justice $j$ agree with one another. (The analog to the plot shown in the article.) Find the eigenvalues and eigenvectors of $M$.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Leila Sloman<\/em><em>\u00a0<\/em><\/p>\n<hr \/>\n<h3><a id=\"2\" href=\"https:\/\/www.youtube.com\/watch?v=K79aOe-F0Mk\">Pok\u00e9mon and Geometric Distributions<\/a><\/h3>\n<p><em>Numberphile,<\/em> June 13, 2024.<\/p>\n<p>Suppose that if you roll the number $N$ on a six-sided die, you win $N$ dollars. How much money would you expect to earn in a single roll? This scenario is an example of \u201cexpected value\u201d in statistics. In this <em>Numberphile <\/em>video, we learn that expected value equals the sum of each outcome multiplied by its probability of occurring. This statistical method is useful in games such as Pok\u00e9mon, where we can calculate the expected number of encounters it takes to \u201ccatch \u2018em all\u201d using geometric series.<\/p>\n<p><strong>Classroom Activities:<\/strong> <em>expected value, geometric distributions<\/em><\/p>\n<ul>\n<li>(All levels) Based on the rules outlined above, would you expect to win or lose money with the dice-rolling game if playing the game cost $30 for 9 rolls?\n<ul>\n<li>How much money would you win or lose? Show your work.<\/li>\n<\/ul>\n<\/li>\n<li>(Mid level) Watch the video. Explain in your own words:\n<ul>\n<li>Why does the mathematician in the video use an \u201cinfinite sum\u201d when calculating expected value?<\/li>\n<li>What is a geometric distribution?<\/li>\n<\/ul>\n<\/li>\n<li>(Mid level) How many encounters would it take to catch 5 different Pok\u00e9mon? Assume that each Pok\u00e9mon has equal likelihood of appearing. Show each step of your work.<\/li>\n<li>(High level) Write out each step of the generalized example from the video of expected encounters for any number n of Pok\u00e9mon. Explain why the mathematician compares the harmonic sequence to $1\/x$.\n<ul>\n<li>Why does integrating $1\/x$ give a useful approximation for expected value?<\/li>\n<li>How many encounters do you expect it would take to catch 500 Pok\u00e9mon?<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Max Levy<\/em><\/p>\n<hr \/>\n<h3><a id=\"3\" href=\"https:\/\/www.astronomy.com\/science\/how-we-can-understand-our-universe-through-math\/\">How we can understand our universe through math<\/a><\/h3>\n<p><em>Astronomy<\/em>, June 1, 2024.<\/p>\n<p>In the early 20th century, Albert Einstein presented a new theory of gravity that applied not just to our planet or our solar system, but to the universe. Einstein\u2019s theory included mathematical terms for the familiar attractive forces of gravity, as well as for repulsive forces that prevented his mathematical universe from imploding due to gravity. This repulsion was represented by a cosmological constant, lambda. Scientists didn\u2019t know what lambda represented in the physical world until many years later. \u201cFor physicists and mathematicians who work with these equations today, lambda represents dark energy,\u201d wrote Steve Nadis and Shing-Tung Yau for <em>Astronomy.<\/em> In this article, Nadis and Yau describe how Einstein\u2019s theory was connected to dark energy, as well as how mathematics predicted other important physics discoveries like black holes.<\/p>\n<p><strong>Classroom Activities:<\/strong> <em>mathematical physics<\/em><\/p>\n<ul>\n<li>(Mid level) Read the overview of <a href=\"https:\/\/ny.pbslearningmedia.org\/resource\/buac20-912-sci-ess-wwt-galaxies-lp\/galaxies-and-the-history-of-the-universe-lesson-plan\/\">this \u201cGalaxies and the History of the Universe\u201d lesson<\/a> from PBS.\n<ul>\n<li>Complete the problems 1-6 in the <a href=\"https:\/\/static.pbslearningmedia.org\/media\/media_files\/ecd0b29c-b0a7-4b59-adb3-ea5e3f7e5598\/2b539b74-a418-44e0-98bf-b2f24bc6daff.pdf\">worksheet<\/a> on your own.<\/li>\n<li>For problem 7, discuss each response as a class.<\/li>\n<\/ul>\n<\/li>\n<li>(All levels) Watch this video <a href=\"https:\/\/www.youtube.com\/watch?v=xodtfM1r9FA&amp;ab_channel=ScienceClicEnglish\">introducing the math of general relativity<\/a>.\n<ul>\n<li>Define in your own words: spacetime; worldlines; coordinate system<\/li>\n<li>Propose three different 2D coordinate systems to track the progress of an airplane arriving in New York from Los Angeles. Explain the differences of each system.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Max Levy<\/em><\/p>\n<hr \/>\n<h3><a id=\"4\" href=\"https:\/\/apnews.com\/article\/japan-birth-rate-declining-population-82662ea061286cb907fd7d071e5b0b9b\">Japan&#8217;s birth rate falls to a record low as the number of marriages also drops<\/a><\/h3>\n<p><em>AP News,<\/em> June 5, 2024.<\/p>\n<p>Japan\u2019s population is shrinking. As the population of Japan ages, young people are starting fewer and smaller families. Women in Japan average about 1.2 babies each in their lifetime, based on recent data \u2014 about 25% lower than the U.S. birth rate and about 45% below the global average. The new data makes 2023 the eighth consecutive year that Japan\u2019s birth rate reached a new low. In this article for <em>AP News<\/em>, Mari Yamaguchi writes about the implications of low birth-rates on future economic predictions.<\/p>\n<p><strong>Classroom Activities:<\/strong> <em>data analysis, compounding<\/em><\/p>\n<ul>\n<li>(All levels) Based only on the numbers above, calculate the U.S. and global birth rates.<\/li>\n<li>(Mid level) Read <a href=\"https:\/\/mathworld.wolfram.com\/CompoundInterest.html\">this MathWorld resource<\/a> to learn more about compounding. Answer the following questions:\n<ul>\n<li>If Japan\u2019s birth rate continues to fall 5.6% annually, when will the birth rate fall below 1?<\/li>\n<li>By what percent would Japan\u2019s birth rate need to increase annually to catch up to the global average within:\n<ul>\n<li>10 years?<\/li>\n<li>5 years?<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Max Levy<\/em><\/p>\n<hr \/>\n<h3><a id=\"5\" href=\"https:\/\/news.stv.tv\/west-central\/university-of-glasgow-celebrates-lord-kelvin-bicentenary-with-series-of-events-including-art-exhibition\">University of Glasgow to celebrate 200th anniversary of Lord Kelvin&#8217;s birth<\/a><\/h3>\n<p><em>STV News<\/em>, June 24, 2024.<\/p>\n<p>June 26 was Lord Kelvin\u2019s 200<sup>th<\/sup> birthday, and in celebration, the university where he spent most of his career launched a two-week exhibit titled \u201cLord Kelvin: Beyond Absolute Zero.\u201d Two paintings made as part of the bicentennial highlight a project of Kelvin&#8217;s that was decidedly mathematical. <a href=\"https:\/\/gregorharvie.com\/Kelvin_installation.html\">In 1887<\/a>, Kelvin proposed a shape which tiles all of three-dimensional space while keeping its surface area as small as possible. His project was misguided \u2014 he <a href=\"https:\/\/gregorharvie.com\/Kelvin_installation.html\">wanted to model<\/a> an \u201cether\u201d through which light could travel, when in fact no such ether exists \u2014 but he made long-lasting progress on the underlying math problem. It took 106 years to <a href=\"https:\/\/mathworld.wolfram.com\/KelvinsConjecture.html\">find a shape<\/a> that both fills 3D space and has a smaller surface area than Kelvin\u2019s.<\/p>\n<p><strong>Classroom Activities: <\/strong><em>tessellation, geometry<\/em><\/p>\n<ul>\n<li>(All levels) A shape is called <em>space-filling <\/em>if copies of it can be packed together with no gaps or spaces in between the copies. For example, cubes are space-filling: You can stack them on top of and next to one another with no gaps. Are the following shapes space-filling?\n<ul>\n<li>Rectangular prisms\n<p><figure id=\"attachment_2499\" aria-describedby=\"caption-attachment-2499\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-2499 size-medium\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2024\/07\/Screen-Shot-2024-07-25-at-4.05.54-PM.png?resize=300%2C147&#038;ssl=1\" alt=\"\" width=\"300\" height=\"147\" srcset=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2024\/07\/Screen-Shot-2024-07-25-at-4.05.54-PM.png?resize=300%2C147&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2024\/07\/Screen-Shot-2024-07-25-at-4.05.54-PM.png?resize=768%2C376&amp;ssl=1 768w, https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2024\/07\/Screen-Shot-2024-07-25-at-4.05.54-PM.png?resize=465%2C228&amp;ssl=1 465w, https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2024\/07\/Screen-Shot-2024-07-25-at-4.05.54-PM.png?resize=695%2C340&amp;ssl=1 695w, https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2024\/07\/Screen-Shot-2024-07-25-at-4.05.54-PM.png?w=870&amp;ssl=1 870w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-2499\" class=\"wp-caption-text\">Rectangular prism, drawn by Leila Sloman in TikZ.<\/figcaption><\/figure><\/li>\n<li>Spheres<\/li>\n<li>Equilateral tetrahedrons (shapes where each of 4 sides is an equilateral triangle).\n<figure id=\"attachment_2507\" aria-describedby=\"caption-attachment-2507\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2507\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2024\/07\/Screen-Shot-2024-07-25-at-5.20.18-PM.png?resize=300%2C260&#038;ssl=1\" alt=\"\" width=\"300\" height=\"260\" srcset=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2024\/07\/Screen-Shot-2024-07-25-at-5.20.18-PM.png?resize=300%2C260&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2024\/07\/Screen-Shot-2024-07-25-at-5.20.18-PM.png?resize=465%2C403&amp;ssl=1 465w, https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2024\/07\/Screen-Shot-2024-07-25-at-5.20.18-PM.png?w=496&amp;ssl=1 496w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-2507\" class=\"wp-caption-text\">Equilateral tetrahedron, drawn by Leila Sloman in TikZ.<\/figcaption><\/figure>\n<p>(Note for teachers: This example may be hard to visualize without physical tetrahedrons for students to play with.)<\/li>\n<\/ul>\n<\/li>\n<li>(All levels) Kelvin thought his shape had the smallest surface area possible for a space-filling shape. <a href=\"https:\/\/mathworld.wolfram.com\/Space-FillingPolyhedron.html\">Here is a list<\/a> of space-filling polyhedra. Calculate the surface area of:\n<ul>\n<li>A regular hexagonal prism (all edge lengths are the same) of volume 1.<\/li>\n<li>A cube of volume 1.<\/li>\n<li>Using formulas (9) and (10) <a href=\"https:\/\/mathworld.wolfram.com\/TruncatedOctahedron.html\">here<\/a>, calculate the surface area of a truncated octahedron whose volume is 1. Kelvin\u2019s shape was a curved version of this shape.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><em>\u2014Leila Sloman<\/em><em>\u00a0<\/em><\/p>\n<hr \/>\n<h3><strong>Some more of this month\u2019s math headlines<\/strong><\/h3>\n<ul>\n<li><a href=\"https:\/\/www.theguardian.com\/science\/article\/2024\/jun\/24\/can-you-solve-it-try-this-triple-tetris-teaser\">Can you solve it? Try this triple Tetris teaser<\/a><br \/>\n<em>The Guardian<\/em>, June 24, 2024.<\/li>\n<li><a href=\"https:\/\/www.theengineer.co.uk\/content\/news\/texas-team-uses-ai-to-prevent-power-outages\">Texas team uses AI to prevent power outages<\/a><br \/>\n<em>The Engineer<\/em>, June 24, 2024.<\/li>\n<li><a href=\"https:\/\/www.newscientist.com\/article\/2432195-mathematicians-discover-impossible-problem-in-super-mario-games\/\">Mathematicians discover impossible problem in Super Mario games<\/a><br \/>\n<em>New Scientist<\/em>, June 13, 2024.<\/li>\n<li><a href=\"https:\/\/www.popularmechanics.com\/science\/a61042424\/mathematicians-rethinking-equal-sign\/\">Mathematicians Are Suddenly Rethinking the Equal Sign<\/a><br \/>\n<em>Popular Mechanics<\/em>, June 12, 2024.<\/li>\n<li><a href=\"https:\/\/www.scientificamerican.com\/article\/ai-will-become-mathematicians-co-pilot\/\">AI Will Become Mathematicians\u2019 \u2018Co-Pilot\u2019<\/a><br \/>\n<em>Scientific American<\/em>, June 8, 2024.<\/li>\n<li><a href=\"https:\/\/www.scientificamerican.com\/article\/prime-number-puzzle-has-stumped-mathematicians-for-more-than-a-century\/\">Prime Number Puzzle Has Stumped Mathematicians for More Than a Century<\/a><br \/>\n<em>Scientific American<\/em>, June 7, 2024.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The mathematical structure of the Supreme Court in Politico How to calculate the time to &#8220;catch &#8217;em all&#8221; in Numberphile Cosmological predictions in Astronomy Population prediction in Japan in AP Kelvin&#8217;s space-filling octahedron in STV News Opinion | Using Math to Analyze the Supreme Court Reveals an Intriguing Pattern Politico,<span class=\"more-link\"><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-june-2024\/\">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_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":"","jetpack_post_was_ever_published":false},"categories":[2],"tags":[346,19,168,344,35,237,345,84,347],"class_list":["entry","author-leilasloman","post-2494","post","type-post","status-publish","format-standard","category-math-in-the-media-digests","tag-compounding","tag-data-analysis","tag-expected-value","tag-geometric-distributions","tag-geometry","tag-linear-algebra","tag-mathematical-physics","tag-statistics","tag-tessellation"],"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\/2494","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=2494"}],"version-history":[{"count":11,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/2494\/revisions"}],"predecessor-version":[{"id":2508,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/2494\/revisions\/2508"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/media?parent=2494"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/categories?post=2494"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/tags?post=2494"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}