{"id":1451,"date":"2023-01-13T12:48:33","date_gmt":"2023-01-13T17:48:33","guid":{"rendered":"https:\/\/mathvoices.ams.org\/mathmedia\/?p=1451"},"modified":"2023-01-13T14:03:17","modified_gmt":"2023-01-13T19:03:17","slug":"math-digests-december-2022","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-december-2022\/","title":{"rendered":"Math Digests December 2022"},"content":{"rendered":"<ul>\n<li><a href=\"#1\">A brief history of statistics in football: why actual goals remain king in predicting who will win<\/a><\/li>\n<li><a href=\"#2\">The Brain Uses Calculus to Control Fast Movements<\/a><\/li>\n<li><a href=\"#3\">Is It Actually Impossible to &#8220;Square the Circle?&#8221;<\/a><\/li>\n<li><a href=\"#4\">Mathematical Alarms Could Help Predict and Avoid Climate Tipping Points<\/a><\/li>\n<li><a href=\"#5\">How to win the gift-stealing game Bad Santa, according to a mathematician<\/a><\/li>\n<\/ul>\n<hr \/>\n<h3><a id=\"1\" href=\"https:\/\/theconversation.com\/a-brief-history-of-statistics-in-football-why-actual-goals-remain-king-in-predicting-who-will-win-194314\"><span style=\"font-weight: 400\">A brief history of statistics in football: why actual goals remain king in predicting who will win<\/span><\/a><\/h3>\n<p><i><span style=\"font-weight: 400\">The Conversation<\/span><\/i><span style=\"font-weight: 400\">, December 30, 2022.<\/span><\/p>\n<p><span style=\"font-weight: 400\">What kind of information is most helpful to predicting the outcome of a soccer game? There is lots of data you could try to incorporate in a prediction, but the best information comes from one of the humblest measurements: The number of goals historically scored by each team. Other kinds of data have been analyzed, like amount of time in possession of the ball, or number and quality of goal-scoring opportunities. But these don\u2019t add much to predictions about who will win, writes Laurence Shaw in <\/span><i><span style=\"font-weight: 400\">The Conversation<\/span><\/i><span style=\"font-weight: 400\">. That\u2019s because only scored goals have a clear and certain impact on the game\u2019s outcome. This failure of flashy statistics reminds us of what really matters for mathematical prediction. <\/span><span style=\"font-weight: 400\">\u201cA model that only uses goals to predict future games may seem remarkably simple, but its effectiveness lies in understanding what makes for good statistical analysis: high quality data, and lots of it,\u201d writes Shaw.<\/span><\/p>\n<p><b>Classroom Activities:<\/b> <i><span style=\"font-weight: 400\">statistics, modeling<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) Shaw writes: \u201cAs far back as 1968, <\/span><a href=\"https:\/\/www.jstor.org\/stable\/2343726?origin=JSTOR-pdf\"><span style=\"font-weight: 400\">a statistical study<\/span><\/a><span style=\"font-weight: 400\"> was unable to find any link between shots, possession or passing moves and the outcomes of football matches.\u201d Does this surprise you? Why or why not?<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Mid level) In chess, the Elo rating (named after <\/span><a href=\"https:\/\/en.wikipedia.org\/wiki\/Arpad_Elo\"><span style=\"font-weight: 400\">Arpad Elo<\/span><\/a><span style=\"font-weight: 400\">) can be used to predict who will win a game. The Elo rating is even simpler than the model discussed in the article: It only depends on whether you win, lose, or draw games. Check out <\/span><a href=\"https:\/\/www.318chess.com\/elo.html\"><span style=\"font-weight: 400\">this table<\/span><\/a><span style=\"font-weight: 400\"> showing how the difference between your rating and your opponent\u2019s rating corresponds to the chance that you\u2019ll win a game.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">If your rating is 1800, and your opponent\u2019s is 1600, what are the chances you\u2019ll win?<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">What if your rating is 1500 and your opponent\u2019s is 1600?<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><i><span style=\"font-weight: 400\">\u2014Leila Sloman<\/span><\/i><\/p>\n<hr \/>\n<h3><a id=\"2\" href=\"https:\/\/www.quantamagazine.org\/the-brain-uses-calculus-to-control-fast-movements-20221128\/\"><span style=\"font-weight: 400\">The Brain Uses Calculus to Control Fast Movements<\/span><\/a><\/h3>\n<p><i><span style=\"font-weight: 400\">Quanta Magazine<\/span><\/i><span style=\"font-weight: 400\">, November 28<\/span><\/p>\n<p><span style=\"font-weight: 400\">Much like a computer algorithm or instruction booklet, the fewer steps your brain takes to complete a task, the faster the results. And whether you&#8217;re a soccer player chasing a ball or a rat who just spotted a snack, you benefit from quick dialogue between your senses and muscles. In this article for <\/span><i><span style=\"font-weight: 400\">Quanta Magazine,<\/span><\/i><span style=\"font-weight: 400\"> Kevin Hartnett discusses a new study of how the brain calculates motor control. Just thinking \u201cstop\u201d or \u201cgo\u201d is not enough. \u201cIf you just take stop signals and feed them into [motor control region], the animal will stop, but the mathematics tell us that the stop won\u2019t be fast enough,\u201d a neuroscientist tells Hartnett. The real trick to quicker reflexes is about calculus.<\/span><\/p>\n<p><b>Classroom Activities: <\/b><i><span style=\"font-weight: 400\">calculus, rate of change<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) Introduce students to calculus with this video: <\/span><a href=\"https:\/\/www.youtube.com\/watch?v=4fqVT-DfpGg\"><span style=\"font-weight: 400\">How to use calculus in real life<\/span><\/a><span style=\"font-weight: 400\">.\u00a0<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Mid level) The article explains that movements depend on a \u201crate of change\u201d<\/span><span style=\"font-weight: 400\"> between opposing inhibitory and excitatory signals. Imagine a simplified scenario where there is only one type of signal. In each pair below, choose which scenario has the largest rate of change.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">A signal of 100 units followed 1 second later by a signal of \u201399 units <\/span><b>or <\/b><span style=\"font-weight: 400\">A signal of 100 units followed 1 second later by a signal of 10 units<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">A signal of 50 units followed 10 seconds later by a signal of 30 units <\/span><b>or <\/b><span style=\"font-weight: 400\">A signal of 50 units followed 1 second later by a signal of 30 units<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><i><span style=\"font-weight: 400\">\u2014Max Levy<\/span><\/i><\/p>\n<hr \/>\n<h3><a id=\"3\" href=\"https:\/\/www.discovermagazine.com\/the-sciences\/is-it-actually-impossible-to-square-the-circle\"><span style=\"font-weight: 400\">Is It Actually Impossible to &#8220;Square the Circle?&#8221;<\/span><\/a><\/h3>\n<p><i><span style=\"font-weight: 400\">Discover<\/span><\/i><span style=\"font-weight: 400\">, December 16, 2022<\/span><\/p>\n<p><span style=\"font-weight: 400\">A new paper has made progress on the \u201csquaring the circle\u201d problem, a question that dates back to the ancient Greeks. The modern form of the problem is to cut a circle into pieces and rearrange those pieces into a square, without any gaps or overlaps. (The ancient Greek version was to construct a square with the same area as a given circle using only a compass and straightedge. It is now known to be impossible.) This problem is difficult, but using advanced techniques, mathematicians have cut the circle into complex pieces that make it work. This year, Andr\u00e1s M\u00e1th\u00e9, Jonathan Noel, and Oleg Pikhurko found yet another way to square the circle, as Stephen Ornes writes for <\/span><i><span style=\"font-weight: 400\">Discover<\/span><\/i><span style=\"font-weight: 400\">. Still, there\u2019s no easy solution: they use a mind-boggling number of pieces \u2014 around <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=10%5E%7B200%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"10^{200}\" class=\"latex\" \/><\/span><span style=\"font-weight: 400\">\u00a0of them.<\/span><\/p>\n<p><b>Classroom Activities:<\/b> <i><span style=\"font-weight: 400\">algebra, geometry, irrational numbers, pi, golden ratio<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Algebra, Geometry) The ancient Greek problem of \u201csquaring the circle\u201d was impossible because <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cpi&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;pi\" class=\"latex\" \/> is a kind of number called a transcendental number, as mathematician James Grime explains in a video for Numberphile.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Watch the <\/span><a href=\"https:\/\/youtu.be\/CMP9a2J4Bqw\"><span style=\"font-weight: 400\">Numberphile video<\/span><\/a><span style=\"font-weight: 400\">. Notice that the video discusses three kinds of numbers: <\/span><a href=\"https:\/\/mathworld.wolfram.com\/ConstructibleNumber.html\"><span style=\"font-weight: 400\">constructible numbers<\/span><\/a><span style=\"font-weight: 400\">, <\/span><a href=\"https:\/\/mathworld.wolfram.com\/IrrationalNumber.html\"><span style=\"font-weight: 400\">irrational numbers<\/span><\/a><span style=\"font-weight: 400\">, and <\/span><a href=\"https:\/\/mathworld.wolfram.com\/TranscendentalNumber.html\"><span style=\"font-weight: 400\">transcendental numbers<\/span><\/a><span style=\"font-weight: 400\">.\u00a0<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">According to the video, are there any irrational numbers that are constructible? Are there any transcendental numbers that are constructible?<\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Algebra, Geometry) Read <\/span><a href=\"https:\/\/www.quantamagazine.org\/when-math-gets-impossibly-hard-20200914\/\"><span style=\"font-weight: 400\">this Quanta Magazine article<\/span><\/a><span style=\"font-weight: 400\"> by Dave Richeson about some other impossible problems in math.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Geometry) Show that if a cube has side length $1$, then a second cube with twice its volume has side length $\\sqrt[3]{2}$. Based on the article, why does this make \u201cdoubling the cube\u201d impossible?<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Algebra) Richeson writes about a problem from ancient Greece involving the golden ratio <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cphi+%3D+%281+%2B+%5Csqrt%7B5%7D%29%2F2&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;phi = (1 + &#92;sqrt{5})\/2\" class=\"latex\" \/>. It can only be solved if there is a number <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=L&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"L\" class=\"latex\" \/> for which <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=1%2FL&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"1\/L\" class=\"latex\" \/> and <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cphi%2FL&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;phi\/L\" class=\"latex\" \/> are both integers.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Prove that <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cphi&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;phi\" class=\"latex\" \/> is a root of the polynomial <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=x%5E2+-+x+-+1&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"x^2 - x - 1\" class=\"latex\" \/>.<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Advanced) Is <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cphi&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;phi\" class=\"latex\" \/> rational or irrational? What does this say about the ancient Greeks\u2019 problem?<\/span><\/li>\n<\/ul>\n<\/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 id=\"4\" href=\"https:\/\/insideclimatenews.org\/news\/27122022\/climate-tipping-points-2\/\"><span style=\"font-weight: 400\">Mathematical Alarms Could Help Predict and Avoid Climate Tipping Points<\/span><\/a><\/h3>\n<p><i><span style=\"font-weight: 400\">Inside Climate News<\/span><\/i><span style=\"font-weight: 400\">, December 27, 2022.<\/span><\/p>\n<p><span style=\"font-weight: 400\">With effects of climate change already apparent, scientists fear the catastrophic effects of \u201ctipping points\u201d \u2014 points at which environmental damage creates a self-perpetuating cycle. In a recent paper, scientists studied how to <\/span><span style=\"font-weight: 400\">predict when these tipping points will occur, reports Charlie Miller in this article. If this research can help stop humanity from crossing a tipping point, huge amounts of environmental destruction could be prevented. But as scientist Michael Oppenheimer told <\/span><i><span style=\"font-weight: 400\">Inside Climate News<\/span><\/i><span style=\"font-weight: 400\">, \u201c\u200b\u200bDon\u2019t expect clear answers anytime soon \u2026 The awesome complexity of the problem remains, and in fact we could already have passed a tipping point without knowing it.\u201d<\/span><\/p>\n<p><b>Classroom Activities:\u00a0<\/b><em>equilibria, climate modeling<\/em><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) Miller writes: \u201cThe study\u2019s authors use the analogy of a chair to illustrate tipping points and early warning signals. A chair can be tilted so it balances on two legs, and in this state could fall to either side. Balanced at this tipping point, it will react dramatically to the smallest push.\u201d Identify whether the following systems have tipping points, and if so, what they are:<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">A ball rolling around on a hilltop<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">A ball rolling around inside a bowl<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">A rocket launching into space<\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Mid Level) For more on math and climate prediction, read <\/span><a href=\"https:\/\/plus.maths.org\/content\/climate-modelling-made-easy\"><span style=\"font-weight: 400\">&#8220;Climate modelling made easy&#8221;<\/span><\/a><span style=\"font-weight: 400\">, by Chris Budd for <\/span><i><span style=\"font-weight: 400\">Plus Magazine<\/span><\/i><span style=\"font-weight: 400\">.<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Calculus) Miller mentions melting ice caps as a system subject to a tipping point. For more on this process, advanced students can read Marianne Freiberger\u2019s article <\/span><a href=\"https:\/\/plus.maths.org\/content\/maths-and-climate-change-melting-arctic\"><span style=\"font-weight: 400\">&#8220;Maths and climate change: the melting Arctic&#8221;<\/span><\/a><span style=\"font-weight: 400\">, also from <\/span><i><span style=\"font-weight: 400\">Plus Magazine.<\/span><\/i><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: right\"><i><span style=\"font-weight: 400\">\u2014Leila Sloman<\/span><\/i><\/p>\n<hr \/>\n<h3><a id=\"5\" href=\"https:\/\/theconversation.com\/how-to-win-the-gift-stealing-game-bad-santa-according-to-a-mathematician-196483\"><span style=\"font-weight: 400\">How to win the gift-stealing game Bad Santa, according to a mathematician<\/span><\/a><\/h3>\n<p><i><span style=\"font-weight: 400\">The Conversation<\/span><\/i><span style=\"font-weight: 400\">, December 15, 2022<\/span><\/p>\n<p><span style=\"font-weight: 400\">If you\u2019re a competitive person, you probably feel that the whole point of playing a game is to win \u2014 even at Christmas party games. In the game Bad Santa (also known as White Elephant or Yankee Swap), each person brings an anonymous gift, and gets a chance to open a gift from the pool of presents. The twist is that people can steal each other&#8217;s gifts. Some lucky players take advantage of this twist to snag their favorite item in the bunch, but others may end up with their least favorite. \u201cIt\u2019s a good alternative to buying a gift for everyone, and a great way to ruin friendships,\u201d writes Joel Gilmore, a mathematician from Griffith University who wrote about the game for <\/span><i><span style=\"font-weight: 400\">The Conversation.<\/span><\/i><span style=\"font-weight: 400\"> If you want to win the best presents next year, it helps to understand favorable strategies. In this article, Gilmore describes running computer simulations of the game to find the most fair rules and the most successful strategies.<\/span><\/p>\n<p><b>Classroom Activities: <\/b><i><span style=\"font-weight: 400\">simulations, optimization<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(All levels) Play a simplified game of Bad Santa similar to Gilmore\u2019s model. Form groups of 10 students. For the gift pool, take 10 cards from a regular card deck. Higher numbered cards represent better gifts. Choose a set of rules from the article and play the game. Students should feel free to use strategies discussed in the article as well.\u00a0<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Mid level) Discuss the results of the Bad Santa card game.\u00a0<\/span>\n<ul>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Who feels their strategy worked, and why?\u00a0<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Whose strategy did <\/span><i><span style=\"font-weight: 400\">not <\/span><\/i><span style=\"font-weight: 400\">work, and why?\u00a0<\/span><\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">Who felt that they had no control over their win or loss and why?<\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400\"><span style=\"font-weight: 400\">(Mid level) Change the rules and repeat. If the results are different, explain why you think the rules helped or hurt.<\/span><\/li>\n<\/ul>\n<p style=\"text-align: right\"><i><span style=\"font-weight: 400\">\u2014Max Levy<\/span><\/i><\/p>\n<hr \/>\n<h3>Some more of this month&#8217;s headlines<\/h3>\n<ul>\n<li><a href=\"https:\/\/theconversation.com\/ada-lovelaces-skills-with-language-music-and-needlepoint-contributed-to-her-pioneering-work-in-computing-193930\">Ada Lovelace\u2019s skills with language, music and needlepoint contributed to her pioneering work in computing<\/a><br \/>\n<em>The Conversation,<\/em> December 8, 2022<\/li>\n<li><a href=\"https:\/\/www.scientificamerican.com\/article\/6-marvelous-math-stories-from-2022\/\">6 Marvelous Math Stories from 2022<\/a><br \/>\n<em>Scientific American,\u00a0<\/em>December 12, 2022<\/li>\n<li><a href=\"https:\/\/www.nytimes.com\/2022\/12\/05\/technology\/chatgpt-ai-twitter.html\">The Brilliance and Weirdness of ChatGPT<\/a><br \/>\n<em>The New York Times<\/em>, December 5, 2022<\/li>\n<li><a href=\"https:\/\/www.grunge.com\/1126681\/who-was-george-dantzig-uc-berkeleys-real-life-will-hunting\/\">Who was George Dantzig, UC Berkeley&#8217;s Real Life Will Hunting?<\/a><br \/>\n<em>The Grunge<\/em>, December 10, 2022<\/li>\n<li><a href=\"https:\/\/english.elpais.com\/science-tech\/2022-12-10\/gh-hardy-remembering-britains-eccentric-and-brilliant-mathematician.html\">G.H. Hardy: Remembering Britain\u2019s eccentric and brilliant mathematician<\/a><br \/>\n<em>El Pa\u00eds,\u00a0<\/em>December 10, 2022<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>A brief history of statistics in football: why actual goals remain king in predicting who will win The Brain Uses Calculus to Control Fast Movements Is It Actually Impossible to &#8220;Square the Circle?&#8221; Mathematical Alarms Could Help Predict and Avoid Climate Tipping Points How to win the gift-stealing game Bad<span class=\"more-link\"><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/math-digests-december-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_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":[6,176,175,35,37,109,53,114,65,177,84],"class_list":["entry","author-leilasloman","post-1451","post","type-post","status-publish","format-standard","category-math-in-the-media-digests","tag-algebra","tag-climate-modeling","tag-equilibria","tag-geometry","tag-golden-ratio","tag-irrational-numbers","tag-mathematical-modeling","tag-optimization","tag-pi","tag-simulations","tag-statistics"],"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\/1451","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=1451"}],"version-history":[{"count":10,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/1451\/revisions"}],"predecessor-version":[{"id":1465,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/1451\/revisions\/1465"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/media?parent=1451"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/categories?post=1451"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/tags?post=1451"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}