CINXE.COM

<!DOCTYPE html> <html lang="en-US"> <head> <meta charset="UTF-8" /> <title>Welcome to the Fold – Feature Column</title> <meta name='robots' content='max-image-preview:large' /> <style>img:is([sizes="auto" i], [sizes^="auto," i]) { contain-intrinsic-size: 3000px 1500px }</style> <script type='text/x-mathjax-config'> MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], ['\$','\$']], processEscapes: true } }); </script>  <meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="generator" content="Hoot Business 2.11.1" /> <link rel='dns-prefetch' href='//cdn.jsdelivr.net' /> <link rel='dns-prefetch' href='//stats.wp.com' /> <link rel='dns-prefetch' href='//fonts.googleapis.com' /> <link rel='dns-prefetch' href='//v0.wordpress.com' /> <link rel='dns-prefetch' href='//widgets.wp.com' /> <link rel='dns-prefetch' href='//s0.wp.com' /> <link rel='dns-prefetch' href='//0.gravatar.com' /> <link rel='dns-prefetch' href='//1.gravatar.com' /> <link rel='dns-prefetch' href='//2.gravatar.com' /> <link rel='preconnect' href='//i0.wp.com' /> <link rel='preconnect' href='//c0.wp.com' /> <link rel="alternate" type="application/rss+xml" title="Feature Column » Feed" href="https://mathvoices.ams.org/featurecolumn/feed/" /> <link rel="alternate" type="application/rss+xml" title="Feature Column » Comments Feed" href="https://mathvoices.ams.org/featurecolumn/comments/feed/" /> <link rel="alternate" type="application/rss+xml" title="Feature Column » Welcome to the Fold Comments Feed" href="https://mathvoices.ams.org/featurecolumn/2024/08/01/welcome-to-the-fold/feed/" /> <link rel="preload" href="https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/library/fonticons/webfonts/fa-solid-900.woff2" as="font" crossorigin="anonymous"> <link rel="preload" href="https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/library/fonticons/webfonts/fa-regular-400.woff2" as="font" crossorigin="anonymous"> <link rel="preload" href="https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/library/fonticons/webfonts/fa-brands-400.woff2" as="font" crossorigin="anonymous"> <script> window._wpemojiSettings = {"baseUrl":"https:\/\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/","ext":".png","svgUrl":"https:\/\/s.w.org\/images\/core\/emoji\/15.0.3\/svg\/","svgExt":".svg","source":{"concatemoji":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-includes\/js\/wp-emoji-release.min.js?ver=6.7.2"}}; /*! This file is auto-generated */ !function(i,n){var o,s,e;function c(e){try{var t={supportTests:e,timestamp:(new Date).valueOf()};sessionStorage.setItem(o,JSON.stringify(t))}catch(e){}}function p(e,t,n){e.clearRect(0,0,e.canvas.width,e.canvas.height),e.fillText(t,0,0);var t=new Uint32Array(e.getImageData(0,0,e.canvas.width,e.canvas.height).data),r=(e.clearRect(0,0,e.canvas.width,e.canvas.height),e.fillText(n,0,0),new Uint32Array(e.getImageData(0,0,e.canvas.width,e.canvas.height).data));return t.every(function(e,t){return e===r[t]})}function u(e,t,n){switch(t){case"flag":return n(e,"\ud83c\udff3\ufe0f\u200d\u26a7\ufe0f","\ud83c\udff3\ufe0f\u200b\u26a7\ufe0f")?!1:!n(e,"\ud83c\uddfa\ud83c\uddf3","\ud83c\uddfa\u200b\ud83c\uddf3")&&!n(e,"\ud83c\udff4\udb40\udc67\udb40\udc62\udb40\udc65\udb40\udc6e\udb40\udc67\udb40\udc7f","\ud83c\udff4\u200b\udb40\udc67\u200b\udb40\udc62\u200b\udb40\udc65\u200b\udb40\udc6e\u200b\udb40\udc67\u200b\udb40\udc7f");case"emoji":return!n(e,"\ud83d\udc26\u200d\u2b1b","\ud83d\udc26\u200b\u2b1b")}return!1}function f(e,t,n){var r="undefined"!=typeof WorkerGlobalScope&&self instanceof WorkerGlobalScope?new OffscreenCanvas(300,150):i.createElement("canvas"),a=r.getContext("2d",{willReadFrequently:!0}),o=(a.textBaseline="top",a.font="600 32px Arial",{});return e.forEach(function(e){o[e]=t(a,e,n)}),o}function t(e){var t=i.createElement("script");t.src=e,t.defer=!0,i.head.appendChild(t)}"undefined"!=typeof Promise&&(o="wpEmojiSettingsSupports",s=["flag","emoji"],n.supports={everything:!0,everythingExceptFlag:!0},e=new Promise(function(e){i.addEventListener("DOMContentLoaded",e,{once:!0})}),new Promise(function(t){var n=function(){try{var e=JSON.parse(sessionStorage.getItem(o));if("object"==typeof e&&"number"==typeof e.timestamp&&(new Date).valueOf()<e.timestamp+604800&&"object"==typeof e.supportTests)return e.supportTests}catch(e){}return null}();if(!n){if("undefined"!=typeof Worker&&"undefined"!=typeof OffscreenCanvas&&"undefined"!=typeof URL&&URL.createObjectURL&&"undefined"!=typeof Blob)try{var e="postMessage("+f.toString()+"("+[JSON.stringify(s),u.toString(),p.toString()].join(",")+"));",r=new Blob([e],{type:"text/javascript"}),a=new Worker(URL.createObjectURL(r),{name:"wpTestEmojiSupports"});return void(a.onmessage=function(e){c(n=e.data),a.terminate(),t(n)})}catch(e){}c(n=f(s,u,p))}t(n)}).then(function(e){for(var t in e)n.supports[t]=e[t],n.supports.everything=n.supports.everything&&n.supports[t],"flag"!==t&&(n.supports.everythingExceptFlag=n.supports.everythingExceptFlag&&n.supports[t]);n.supports.everythingExceptFlag=n.supports.everythingExceptFlag&&!n.supports.flag,n.DOMReady=!1,n.readyCallback=function(){n.DOMReady=!0}}).then(function(){return e}).then(function(){var e;n.supports.everything||(n.readyCallback(),(e=n.source||{}).concatemoji?t(e.concatemoji):e.wpemoji&&e.twemoji&&(t(e.twemoji),t(e.wpemoji)))}))}((window,document),window._wpemojiSettings); </script> <style id='wp-emoji-styles-inline-css'> img.wp-smiley, img.emoji { display: inline !important; border: none !important; box-shadow: none !important; height: 1em !important; width: 1em !important; margin: 0 0.07em !important; vertical-align: -0.1em !important; background: none !important; padding: 0 !important; } </style> <link rel='stylesheet' id='wp-block-library-css' href='https://c0.wp.com/c/6.7.2/wp-includes/css/dist/block-library/style.min.css' media='all' /> <style id='wp-block-library-theme-inline-css'> .wp-block-audio :where(figcaption){color:#555;font-size:13px;text-align:center}.is-dark-theme .wp-block-audio :where(figcaption){color:#ffffffa6}.wp-block-audio{margin:0 0 1em}.wp-block-code{border:1px solid #ccc;border-radius:4px;font-family:Menlo,Consolas,monaco,monospace;padding:.8em 1em}.wp-block-embed :where(figcaption){color:#555;font-size:13px;text-align:center}.is-dark-theme .wp-block-embed :where(figcaption){color:#ffffffa6}.wp-block-embed{margin:0 0 1em}.blocks-gallery-caption{color:#555;font-size:13px;text-align:center}.is-dark-theme .blocks-gallery-caption{color:#ffffffa6}:root :where(.wp-block-image figcaption){color:#555;font-size:13px;text-align:center}.is-dark-theme :root :where(.wp-block-image figcaption){color:#ffffffa6}.wp-block-image{margin:0 0 1em}.wp-block-pullquote{border-bottom:4px solid;border-top:4px solid;color:currentColor;margin-bottom:1.75em}.wp-block-pullquote cite,.wp-block-pullquote footer,.wp-block-pullquote__citation{color:currentColor;font-size:.8125em;font-style:normal;text-transform:uppercase}.wp-block-quote{border-left:.25em solid;margin:0 0 1.75em;padding-left:1em}.wp-block-quote cite,.wp-block-quote footer{color:currentColor;font-size:.8125em;font-style:normal;position:relative}.wp-block-quote:where(.has-text-align-right){border-left:none;border-right:.25em solid;padding-left:0;padding-right:1em}.wp-block-quote:where(.has-text-align-center){border:none;padding-left:0}.wp-block-quote.is-large,.wp-block-quote.is-style-large,.wp-block-quote:where(.is-style-plain){border:none}.wp-block-search .wp-block-search__label{font-weight:700}.wp-block-search__button{border:1px solid #ccc;padding:.375em .625em}:where(.wp-block-group.has-background){padding:1.25em 2.375em}.wp-block-separator.has-css-opacity{opacity:.4}.wp-block-separator{border:none;border-bottom:2px solid;margin-left:auto;margin-right:auto}.wp-block-separator.has-alpha-channel-opacity{opacity:1}.wp-block-separator:not(.is-style-wide):not(.is-style-dots){width:100px}.wp-block-separator.has-background:not(.is-style-dots){border-bottom:none;height:1px}.wp-block-separator.has-background:not(.is-style-wide):not(.is-style-dots){height:2px}.wp-block-table{margin:0 0 1em}.wp-block-table td,.wp-block-table th{word-break:normal}.wp-block-table :where(figcaption){color:#555;font-size:13px;text-align:center}.is-dark-theme .wp-block-table :where(figcaption){color:#ffffffa6}.wp-block-video :where(figcaption){color:#555;font-size:13px;text-align:center}.is-dark-theme .wp-block-video :where(figcaption){color:#ffffffa6}.wp-block-video{margin:0 0 1em}:root :where(.wp-block-template-part.has-background){margin-bottom:0;margin-top:0;padding:1.25em 2.375em} </style> <link rel='stylesheet' id='mediaelement-css' href='https://c0.wp.com/c/6.7.2/wp-includes/js/mediaelement/mediaelementplayer-legacy.min.css' media='all' /> <link rel='stylesheet' id='wp-mediaelement-css' href='https://c0.wp.com/c/6.7.2/wp-includes/js/mediaelement/wp-mediaelement.min.css' media='all' /> <style id='jetpack-sharing-buttons-style-inline-css'> .jetpack-sharing-buttons__services-list{display:flex;flex-direction:row;flex-wrap:wrap;gap:0;list-style-type:none;margin:5px;padding:0}.jetpack-sharing-buttons__services-list.has-small-icon-size{font-size:12px}.jetpack-sharing-buttons__services-list.has-normal-icon-size{font-size:16px}.jetpack-sharing-buttons__services-list.has-large-icon-size{font-size:24px}.jetpack-sharing-buttons__services-list.has-huge-icon-size{font-size:36px}@media print{.jetpack-sharing-buttons__services-list{display:none!important}}.editor-styles-wrapper .wp-block-jetpack-sharing-buttons{gap:0;padding-inline-start:0}ul.jetpack-sharing-buttons__services-list.has-background{padding:1.25em 2.375em} </style> <style id='classic-theme-styles-inline-css'> /*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} </style> <style id='global-styles-inline-css'> :root{--wp--preset--aspect-ratio--square: 1;--wp--preset--aspect-ratio--4-3: 4/3;--wp--preset--aspect-ratio--3-4: 3/4;--wp--preset--aspect-ratio--3-2: 3/2;--wp--preset--aspect-ratio--2-3: 2/3;--wp--preset--aspect-ratio--16-9: 16/9;--wp--preset--aspect-ratio--9-16: 9/16;--wp--preset--color--black: #000000;--wp--preset--color--cyan-bluish-gray: #abb8c3;--wp--preset--color--white: #ffffff;--wp--preset--color--pale-pink: #f78da7;--wp--preset--color--vivid-red: #cf2e2e;--wp--preset--color--luminous-vivid-orange: #ff6900;--wp--preset--color--luminous-vivid-amber: #fcb900;--wp--preset--color--light-green-cyan: #7bdcb5;--wp--preset--color--vivid-green-cyan: #00d084;--wp--preset--color--pale-cyan-blue: #8ed1fc;--wp--preset--color--vivid-cyan-blue: #0693e3;--wp--preset--color--vivid-purple: #9b51e0;--wp--preset--color--accent: #316dcd;--wp--preset--color--accent-font: #ffffff;--wp--preset--gradient--vivid-cyan-blue-to-vivid-purple: linear-gradient(135deg,rgba(6,147,227,1) 0%,rgb(155,81,224) 100%);--wp--preset--gradient--light-green-cyan-to-vivid-green-cyan: linear-gradient(135deg,rgb(122,220,180) 0%,rgb(0,208,130) 100%);--wp--preset--gradient--luminous-vivid-amber-to-luminous-vivid-orange: linear-gradient(135deg,rgba(252,185,0,1) 0%,rgba(255,105,0,1) 100%);--wp--preset--gradient--luminous-vivid-orange-to-vivid-red: linear-gradient(135deg,rgba(255,105,0,1) 0%,rgb(207,46,46) 100%);--wp--preset--gradient--very-light-gray-to-cyan-bluish-gray: linear-gradient(135deg,rgb(238,238,238) 0%,rgb(169,184,195) 100%);--wp--preset--gradient--cool-to-warm-spectrum: linear-gradient(135deg,rgb(74,234,220) 0%,rgb(151,120,209) 20%,rgb(207,42,186) 40%,rgb(238,44,130) 60%,rgb(251,105,98) 80%,rgb(254,248,76) 100%);--wp--preset--gradient--blush-light-purple: linear-gradient(135deg,rgb(255,206,236) 0%,rgb(152,150,240) 100%);--wp--preset--gradient--blush-bordeaux: linear-gradient(135deg,rgb(254,205,165) 0%,rgb(254,45,45) 50%,rgb(107,0,62) 100%);--wp--preset--gradient--luminous-dusk: linear-gradient(135deg,rgb(255,203,112) 0%,rgb(199,81,192) 50%,rgb(65,88,208) 100%);--wp--preset--gradient--pale-ocean: linear-gradient(135deg,rgb(255,245,203) 0%,rgb(182,227,212) 50%,rgb(51,167,181) 100%);--wp--preset--gradient--electric-grass: linear-gradient(135deg,rgb(202,248,128) 0%,rgb(113,206,126) 100%);--wp--preset--gradient--midnight: linear-gradient(135deg,rgb(2,3,129) 0%,rgb(40,116,252) 100%);--wp--preset--font-size--small: 13px;--wp--preset--font-size--medium: 20px;--wp--preset--font-size--large: 36px;--wp--preset--font-size--x-large: 42px;--wp--preset--spacing--20: 0.44rem;--wp--preset--spacing--30: 0.67rem;--wp--preset--spacing--40: 1rem;--wp--preset--spacing--50: 1.5rem;--wp--preset--spacing--60: 2.25rem;--wp--preset--spacing--70: 3.38rem;--wp--preset--spacing--80: 5.06rem;--wp--preset--shadow--natural: 6px 6px 9px rgba(0, 0, 0, 0.2);--wp--preset--shadow--deep: 12px 12px 50px rgba(0, 0, 0, 0.4);--wp--preset--shadow--sharp: 6px 6px 0px rgba(0, 0, 0, 0.2);--wp--preset--shadow--outlined: 6px 6px 0px -3px rgba(255, 255, 255, 1), 6px 6px rgba(0, 0, 0, 1);--wp--preset--shadow--crisp: 6px 6px 0px rgba(0, 0, 0, 1);}:where(.is-layout-flex){gap: 0.5em;}:where(.is-layout-grid){gap: 0.5em;}body .is-layout-flex{display: flex;}.is-layout-flex{flex-wrap: wrap;align-items: center;}.is-layout-flex > :is(*, div){margin: 0;}body .is-layout-grid{display: grid;}.is-layout-grid > :is(*, div){margin: 0;}:where(.wp-block-columns.is-layout-flex){gap: 2em;}:where(.wp-block-columns.is-layout-grid){gap: 2em;}:where(.wp-block-post-template.is-layout-flex){gap: 1.25em;}:where(.wp-block-post-template.is-layout-grid){gap: 1.25em;}.has-black-color{color: var(--wp--preset--color--black) !important;}.has-cyan-bluish-gray-color{color: var(--wp--preset--color--cyan-bluish-gray) !important;}.has-white-color{color: var(--wp--preset--color--white) !important;}.has-pale-pink-color{color: var(--wp--preset--color--pale-pink) !important;}.has-vivid-red-color{color: var(--wp--preset--color--vivid-red) !important;}.has-luminous-vivid-orange-color{color: var(--wp--preset--color--luminous-vivid-orange) !important;}.has-luminous-vivid-amber-color{color: var(--wp--preset--color--luminous-vivid-amber) !important;}.has-light-green-cyan-color{color: var(--wp--preset--color--light-green-cyan) !important;}.has-vivid-green-cyan-color{color: var(--wp--preset--color--vivid-green-cyan) !important;}.has-pale-cyan-blue-color{color: var(--wp--preset--color--pale-cyan-blue) !important;}.has-vivid-cyan-blue-color{color: var(--wp--preset--color--vivid-cyan-blue) !important;}.has-vivid-purple-color{color: var(--wp--preset--color--vivid-purple) !important;}.has-black-background-color{background-color: var(--wp--preset--color--black) !important;}.has-cyan-bluish-gray-background-color{background-color: var(--wp--preset--color--cyan-bluish-gray) !important;}.has-white-background-color{background-color: var(--wp--preset--color--white) !important;}.has-pale-pink-background-color{background-color: var(--wp--preset--color--pale-pink) !important;}.has-vivid-red-background-color{background-color: var(--wp--preset--color--vivid-red) !important;}.has-luminous-vivid-orange-background-color{background-color: var(--wp--preset--color--luminous-vivid-orange) !important;}.has-luminous-vivid-amber-background-color{background-color: var(--wp--preset--color--luminous-vivid-amber) !important;}.has-light-green-cyan-background-color{background-color: var(--wp--preset--color--light-green-cyan) !important;}.has-vivid-green-cyan-background-color{background-color: var(--wp--preset--color--vivid-green-cyan) !important;}.has-pale-cyan-blue-background-color{background-color: var(--wp--preset--color--pale-cyan-blue) !important;}.has-vivid-cyan-blue-background-color{background-color: var(--wp--preset--color--vivid-cyan-blue) !important;}.has-vivid-purple-background-color{background-color: var(--wp--preset--color--vivid-purple) !important;}.has-black-border-color{border-color: var(--wp--preset--color--black) !important;}.has-cyan-bluish-gray-border-color{border-color: var(--wp--preset--color--cyan-bluish-gray) !important;}.has-white-border-color{border-color: var(--wp--preset--color--white) !important;}.has-pale-pink-border-color{border-color: var(--wp--preset--color--pale-pink) !important;}.has-vivid-red-border-color{border-color: var(--wp--preset--color--vivid-red) !important;}.has-luminous-vivid-orange-border-color{border-color: var(--wp--preset--color--luminous-vivid-orange) !important;}.has-luminous-vivid-amber-border-color{border-color: var(--wp--preset--color--luminous-vivid-amber) !important;}.has-light-green-cyan-border-color{border-color: var(--wp--preset--color--light-green-cyan) !important;}.has-vivid-green-cyan-border-color{border-color: var(--wp--preset--color--vivid-green-cyan) !important;}.has-pale-cyan-blue-border-color{border-color: var(--wp--preset--color--pale-cyan-blue) !important;}.has-vivid-cyan-blue-border-color{border-color: var(--wp--preset--color--vivid-cyan-blue) !important;}.has-vivid-purple-border-color{border-color: var(--wp--preset--color--vivid-purple) !important;}.has-vivid-cyan-blue-to-vivid-purple-gradient-background{background: var(--wp--preset--gradient--vivid-cyan-blue-to-vivid-purple) !important;}.has-light-green-cyan-to-vivid-green-cyan-gradient-background{background: var(--wp--preset--gradient--light-green-cyan-to-vivid-green-cyan) !important;}.has-luminous-vivid-amber-to-luminous-vivid-orange-gradient-background{background: var(--wp--preset--gradient--luminous-vivid-amber-to-luminous-vivid-orange) !important;}.has-luminous-vivid-orange-to-vivid-red-gradient-background{background: var(--wp--preset--gradient--luminous-vivid-orange-to-vivid-red) !important;}.has-very-light-gray-to-cyan-bluish-gray-gradient-background{background: var(--wp--preset--gradient--very-light-gray-to-cyan-bluish-gray) !important;}.has-cool-to-warm-spectrum-gradient-background{background: var(--wp--preset--gradient--cool-to-warm-spectrum) !important;}.has-blush-light-purple-gradient-background{background: var(--wp--preset--gradient--blush-light-purple) !important;}.has-blush-bordeaux-gradient-background{background: var(--wp--preset--gradient--blush-bordeaux) !important;}.has-luminous-dusk-gradient-background{background: var(--wp--preset--gradient--luminous-dusk) !important;}.has-pale-ocean-gradient-background{background: var(--wp--preset--gradient--pale-ocean) !important;}.has-electric-grass-gradient-background{background: var(--wp--preset--gradient--electric-grass) !important;}.has-midnight-gradient-background{background: var(--wp--preset--gradient--midnight) !important;}.has-small-font-size{font-size: var(--wp--preset--font-size--small) !important;}.has-medium-font-size{font-size: var(--wp--preset--font-size--medium) !important;}.has-large-font-size{font-size: var(--wp--preset--font-size--large) !important;}.has-x-large-font-size{font-size: var(--wp--preset--font-size--x-large) !important;} :where(.wp-block-post-template.is-layout-flex){gap: 1.25em;}:where(.wp-block-post-template.is-layout-grid){gap: 1.25em;} :where(.wp-block-columns.is-layout-flex){gap: 2em;}:where(.wp-block-columns.is-layout-grid){gap: 2em;} :root :where(.wp-block-pullquote){font-size: 1.5em;line-height: 1.6;} </style> <link rel='stylesheet' id='events-manager-css' href='https://mathvoices.ams.org/featurecolumn/wp-content/plugins/events-manager/includes/css/events-manager.min.css?ver=6.3' media='all' /> <link rel='stylesheet' id='stcr-font-awesome-css' href='https://mathvoices.ams.org/featurecolumn/wp-content/plugins/subscribe-to-comments-reloaded/includes/css/font-awesome.min.css?ver=6.7.2' media='all' /> <link rel='stylesheet' id='stcr-style-css' href='https://mathvoices.ams.org/featurecolumn/wp-content/plugins/subscribe-to-comments-reloaded/includes/css/stcr-style.css?ver=6.7.2' media='all' /> <link rel='stylesheet' id='jetpack_likes-css' href='https://c0.wp.com/p/jetpack/14.3/modules/likes/style.css' media='all' /> <link rel='stylesheet' id='hootbiz-googlefont-css' href='https://fonts.googleapis.com/css2?family=Fira%20Sans:ital,wght@0,300;0,400;0,500;0,600;0,700;0,800;1,400;1,700&family=Oswald:ital,wght@0,400&display=swap' media='all' /> <link rel='stylesheet' id='font-awesome-css' href='https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/library/fonticons/font-awesome.css?ver=5.15.4' media='all' /> <link rel='stylesheet' id='jetpack-subscriptions-css' href='https://c0.wp.com/p/jetpack/14.3/modules/subscriptions/subscriptions.css' media='all' /> <link rel='stylesheet' id='hoot-style-css' href='https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/style.css?ver=2.11.1' media='all' /> <link rel='stylesheet' id='hoot-wpblocks-css' href='https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/include/blocks/wpblocks.css?ver=2.11.1' media='all' /> <style id='hoot-wpblocks-inline-css'> .hgrid { max-width: 1080px; } a { color: #316dcd; } a:hover { color: #25529a; } .accent-typo { background: #316dcd; color: #ffffff; } .invert-typo { color: #ffffff; } .enforce-typo { background: #ffffff; } body.wordpress input[type="submit"], body.wordpress #submit, body.wordpress .button { background: #316dcd; color: #ffffff; } body.wordpress input[type="submit"]:hover, body.wordpress #submit:hover, body.wordpress .button:hover, body.wordpress input[type="submit"]:focus, body.wordpress #submit:focus, body.wordpress .button:focus { background: #25529a; color: #ffffff; } h1, h2, h3, h4, h5, h6, .title, .titlefont { font-family: "Fira Sans", sans-serif; text-transform: none; } #main.main,#header-supplementary,.below-header { background: #ffffff; } #topbar { background: #316dcd; color: #ffffff; } #topbar.js-search .searchform.expand .searchtext { background: #316dcd; } #site-logo.logo-border { border-color: #316dcd; } .header-aside-search.js-search .searchform i.fa-search { color: #316dcd; } #site-title { font-family: "Oswald", sans-serif; text-transform: uppercase; } .site-logo-with-icon #site-title i { font-size: 50px; } .site-logo-mixed-image img { max-width: 200px; } .site-title-line em { color: #316dcd; } .site-title-line mark { background: #316dcd; color: #ffffff; } .site-title-heading-font { font-family: "Fira Sans", sans-serif; } .menu-items ul { background: #ffffff; } .menu-items > li.current-menu-item:not(.nohighlight), .menu-items > li.current-menu-ancestor, .menu-items > li:hover,.menu-items ul li.current-menu-item:not(.nohighlight), .menu-items ul li.current-menu-ancestor, .menu-items ul li:hover { background: #ffffff; } .menu-items > li.current-menu-item:not(.nohighlight) > a, .menu-items > li.current-menu-ancestor > a, .menu-items > li:hover > a,.menu-items ul li.current-menu-item:not(.nohighlight) > a, .menu-items ul li.current-menu-ancestor > a, .menu-items ul li:hover > a { color: #316dcd; } .more-link { color: #316dcd; } .more-link a { background: #ffffff; color: #316dcd; } .more-link a:hover { background: #ffffff; color: #25529a; } .main-content-grid,.widget,.frontpage-area { margin-top: 45px; } .widget,.frontpage-area { margin-bottom: 45px; } .frontpage-area.module-bg-highlight, .frontpage-area.module-bg-color, .frontpage-area.module-bg-image { padding: 45px 0; } .footer .widget { margin: 30px 0; } .js-search .searchform.expand .searchtext { background: #ffffff; } #infinite-handle span,.lrm-form a.button, .lrm-form button, .lrm-form button[type=submit], .lrm-form #buddypress input[type=submit], .lrm-form input[type=submit],.widget_newsletterwidget input.tnp-submit[type=submit], .widget_newsletterwidgetminimal input.tnp-submit[type=submit],.wc-block-components-button, .woocommerce #respond input#submit.alt, .woocommerce a.button.alt, .woocommerce button.button.alt, .woocommerce input.button.alt,.widget_breadcrumb_navxt .breadcrumbs > .hoot-bcn-pretext { background: #316dcd; color: #ffffff; } .woocommerce nav.woocommerce-pagination ul li a:focus, .woocommerce nav.woocommerce-pagination ul li a:hover { color: #25529a; } .woocommerce div.product .woocommerce-tabs ul.tabs li:hover,.woocommerce div.product .woocommerce-tabs ul.tabs li.active { background: #316dcd; } .woocommerce div.product .woocommerce-tabs ul.tabs li:hover a, .woocommerce div.product .woocommerce-tabs ul.tabs li:hover a:hover,.woocommerce div.product .woocommerce-tabs ul.tabs li.active a { color: #ffffff; } .wc-block-components-button:hover, .woocommerce #respond input#submit.alt:hover, .woocommerce a.button.alt:hover, .woocommerce button.button.alt:hover, .woocommerce input.button.alt:hover { background: #25529a; color: #ffffff; } .widget_newsletterwidget input.tnp-submit[type=submit]:hover, .widget_newsletterwidgetminimal input.tnp-submit[type=submit]:hover { background: #25529a; color: #ffffff; } .widget_breadcrumb_navxt .breadcrumbs > .hoot-bcn-pretext:after { border-left-color: #316dcd; } :root .has-accent-color,.is-style-outline>.wp-block-button__link:not(.has-text-color), .wp-block-button__link.is-style-outline:not(.has-text-color) { color: #316dcd; } :root .has-accent-background-color,.wp-block-button__link,.wp-block-button__link:hover,.wp-block-search__button,.wp-block-search__button:hover, .wp-block-file__button,.wp-block-file__button:hover { background: #316dcd; } :root .has-accent-font-color,.wp-block-button__link,.wp-block-button__link:hover,.wp-block-search__button,.wp-block-search__button:hover, .wp-block-file__button,.wp-block-file__button:hover { color: #ffffff; } :root .has-accent-font-background-color { background: #ffffff; } @media only screen and (max-width: 969px){ .mobilemenu-fixed .menu-toggle, .mobilemenu-fixed .menu-items { background: #ffffff; } .sidebar { margin-top: 45px; } .frontpage-widgetarea > div.hgrid > [class*="hgrid-span-"] { margin-bottom: 45px; } } </style> <link rel='stylesheet' id='sharedaddy-css' href='https://c0.wp.com/p/jetpack/14.3/modules/sharedaddy/sharing.css' media='all' /> <link rel='stylesheet' id='social-logos-css' href='https://c0.wp.com/p/jetpack/14.3/_inc/social-logos/social-logos.min.css' media='all' /> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/jquery.min.js" id="jquery-core-js"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/jquery-migrate.min.js" id="jquery-migrate-js"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/ui/core.min.js" id="jquery-ui-core-js"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/ui/mouse.min.js" id="jquery-ui-mouse-js"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/ui/sortable.min.js" id="jquery-ui-sortable-js"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/ui/datepicker.min.js" id="jquery-ui-datepicker-js"></script> <script id="jquery-ui-datepicker-js-after"> jQuery(function(jQuery){jQuery.datepicker.setDefaults({"closeText":"Close","currentText":"Today","monthNames":["January","February","March","April","May","June","July","August","September","October","November","December"],"monthNamesShort":["Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec"],"nextText":"Next","prevText":"Previous","dayNames":["Sunday","Monday","Tuesday","Wednesday","Thursday","Friday","Saturday"],"dayNamesShort":["Sun","Mon","Tue","Wed","Thu","Fri","Sat"],"dayNamesMin":["S","M","T","W","T","F","S"],"dateFormat":"MM d, yy","firstDay":1,"isRTL":false});}); </script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/ui/resizable.min.js" id="jquery-ui-resizable-js"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/ui/draggable.min.js" id="jquery-ui-draggable-js"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/ui/controlgroup.min.js" id="jquery-ui-controlgroup-js"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/ui/checkboxradio.min.js" id="jquery-ui-checkboxradio-js"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/ui/button.min.js" id="jquery-ui-button-js"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/jquery/ui/dialog.min.js" id="jquery-ui-dialog-js"></script> <script id="events-manager-js-extra"> var EM = {"ajaxurl":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-admin\/admin-ajax.php","locationajaxurl":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-admin\/admin-ajax.php?action=locations_search","firstDay":"1","locale":"en","dateFormat":"yy-mm-dd","ui_css":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-content\/plugins\/events-manager\/includes\/css\/jquery-ui\/build.min.css","show24hours":"","is_ssl":"1","autocomplete_limit":"10","calendar":{"breakpoints":{"small":560,"medium":908,"large":false}},"datepicker":{"format":"Y-m-d"},"search":{"breakpoints":{"small":650,"medium":850,"full":false}},"bookingInProgress":"Please wait while the booking is being submitted.","tickets_save":"Save Ticket","bookingajaxurl":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-admin\/admin-ajax.php","bookings_export_save":"Export Bookings","bookings_settings_save":"Save Settings","booking_delete":"Are you sure you want to delete?","booking_offset":"30","bb_full":"Sold Out","bb_book":"Book Now","bb_booking":"Booking...","bb_booked":"Booking Submitted","bb_error":"Booking Error. Try again?","bb_cancel":"Cancel","bb_canceling":"Canceling...","bb_cancelled":"Cancelled","bb_cancel_error":"Cancellation Error. Try again?","txt_search":"Search","txt_searching":"Searching...","txt_loading":"Loading..."}; </script> <script src="https://mathvoices.ams.org/featurecolumn/wp-content/plugins/events-manager/includes/js/events-manager.js?ver=6.3" id="events-manager-js"></script> <link rel="https://api.w.org/" href="https://mathvoices.ams.org/featurecolumn/wp-json/" /><link rel="alternate" title="JSON" type="application/json" href="https://mathvoices.ams.org/featurecolumn/wp-json/wp/v2/posts/2044" /><link rel="EditURI" type="application/rsd+xml" title="RSD" href="https://mathvoices.ams.org/featurecolumn/xmlrpc.php?rsd" /> <meta name="generator" content="WordPress 6.7.2" /> <link rel="canonical" href="https://mathvoices.ams.org/featurecolumn/2024/08/01/welcome-to-the-fold/" /> <link rel='shortlink' href='https://mathvoices.ams.org/featurecolumn/?p=2044' /> <link rel="alternate" title="oEmbed (JSON)" type="application/json+oembed" href="https://mathvoices.ams.org/featurecolumn/wp-json/oembed/1.0/embed?url=https%3A%2F%2Fmathvoices.ams.org%2Ffeaturecolumn%2F2024%2F08%2F01%2Fwelcome-to-the-fold%2F" /> <link rel="alternate" title="oEmbed (XML)" type="text/xml+oembed" href="https://mathvoices.ams.org/featurecolumn/wp-json/oembed/1.0/embed?url=https%3A%2F%2Fmathvoices.ams.org%2Ffeaturecolumn%2F2024%2F08%2F01%2Fwelcome-to-the-fold%2F&format=xml" /> <style>img#wpstats{display:none}</style> <meta name="description" content="A natural question in the context of origami mathematics is: What if we make the paper infinitely large? Welcome to the Fold Sara Chari Saint Mary's College of Maryland Adriana Salerno Bates College and the National Science Foundation Origami—from the Japanese words for “fold” (oru) and “paper” (kami)—is the art of folding paper to make…" />  <meta property="og:type" content="article" /> <meta property="og:title" content="Welcome to the Fold" /> <meta property="og:url" content="https://mathvoices.ams.org/featurecolumn/2024/08/01/welcome-to-the-fold/" /> <meta property="og:description" content="A natural question in the context of origami mathematics is: What if we make the paper infinitely large? Welcome to the Fold Sara Chari Saint Mary’s College of Maryland Adriana Salerno Bates …" /> <meta property="article:published_time" content="2024-08-01T04:01:27+00:00" /> <meta property="article:modified_time" content="2024-07-31T14:52:33+00:00" /> <meta property="og:site_name" content="Feature Column" /> <meta property="og:image" content="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/07/cropped-FC1380x500x2.png?fit=1200%2C250&ssl=1" /> <meta property="og:image:width" content="1200" /> <meta property="og:image:height" content="250" /> <meta property="og:image:alt" content="Feature Column" /> <meta property="og:locale" content="en_US" /> <meta name="twitter:site" content="@amermathsoc" /> <meta name="twitter:text:title" content="Welcome to the Fold" /> <meta name="twitter:image" content="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/07/cropped-FC1380x500x2.png?fit=1200%2C250&ssl=1&w=640" /> <meta name="twitter:image:alt" content="Feature Column" /> <meta name="twitter:card" content="summary_large_image" />  <link rel="icon" href="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2022/02/cropped-ams2.png?fit=32%2C32&ssl=1" sizes="32x32" /> <link rel="icon" href="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2022/02/cropped-ams2.png?fit=192%2C192&ssl=1" sizes="192x192" /> <link rel="apple-touch-icon" href="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2022/02/cropped-ams2.png?fit=180%2C180&ssl=1" /> <meta name="msapplication-TileImage" content="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2022/02/cropped-ams2.png?fit=270%2C270&ssl=1" /> <style id="wp-custom-css"> body { font-family: Helvetica, sans-serif; color: #25282B; background-color: white; line-height: 1.75em; } summary { cursor: pointer; } .list-inline { padding-left:0; margin-left:-5px; list-style:none } .headline.img-headline { display: none; } /*.entry-title {display:none;} .entry-byline{display:none;}*/ .list-inline>li{ display:inline-block; padding-right:5px; padding-left:5px} </style> </head> <body class="post-template-default single single-post postid-2044 single-format-standard hoot-business wordpress ltr en en-us parent-theme multisite blog-2 logged-out custom-background custom-header singular singular-post singular-post-2044" dir="ltr" itemscope="itemscope" itemtype="https://schema.org/Blog"> <a href="#main" class="screen-reader-text">Skip to content</a> <div id="page-wrapper" class=" hgrid site-boxed page-wrapper sitewrap-wide-right sidebarsN sidebars1 hoot-cf7-style hoot-mapp-style hoot-jetpack-style hoot-sticky-sidebar"> <header id="header" class="site-header header-layout-primary-menu header-layout-secondary-none tablemenu" role="banner" itemscope="itemscope" itemtype="https://schema.org/WPHeader"> <div id="header-primary" class=" header-part header-primary header-primary-menu"> <div class="hgrid"> <div class="table hgrid-span-12"> <div id="branding" class="site-branding branding table-cell-mid"> <div id="site-logo" class="site-logo-text"> <div id="site-logo-text" class="site-logo-text site-logo-text-small"><div id="site-title" class="site-title" itemprop="headline"><a href="https://mathvoices.ams.org/featurecolumn" rel="home" itemprop="url"><span class="blogname">Feature Column</span></a></div></div> </div> </div> <div id="header-aside" class=" header-aside table-cell-mid header-aside-menu header-aside-menu-fixed"> <div class="screen-reader-text">Primary Navigation Menu</div> <nav id="menu-primary" class="menu nav-menu menu-primary mobilemenu-fixed mobilesubmenu-click" role="navigation" itemscope="itemscope" itemtype="https://schema.org/SiteNavigationElement"> <a class="menu-toggle" href="#"><span class="menu-toggle-text">Menu</span><i class="fas fa-bars"></i></a> <ul id="menu-primary-items" class="menu-items sf-menu menu"><li id="menu-item-703" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-home menu-item-703"><a href="http://mathvoices.ams.org/featurecolumn"><span class="menu-title">Home</span></a></li> <li id="menu-item-704" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-home menu-item-704"><a href="https://mathvoices.ams.org/featurecolumn/"><span class="menu-title">About</span></a></li> <li id="menu-item-705" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-705"><a href="https://mathvoices.ams.org/featurecolumn/archive/"><span class="menu-title">Feature Column Archive</span></a></li> </ul> </nav> </div> </div> </div> </div> </header> <div id="main" class=" main"> <div class="entry-featured-img-headerwrap loop-meta-withbg loop-meta-staticbg-nocrop"><img class="entry-headerimg" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/07/cropped-FC1380x500x2.png?fit=1380%2C288&ssl=1"></div> <div class="hgrid main-content-grid"> <main id="content" class="content hgrid-span-8 has-sidebar layout-wide-right " role="main"> <div id="content-wrap" class=" content-wrap theiaStickySidebar"> <div id="loop-meta" class=" loop-meta-wrap pageheader-bg-default loop-meta-withtext"> <div class="hgrid"> <div class=" loop-meta hgrid-span-12" itemscope="itemscope" itemtype="https://schema.org/WebPageElement"> <div class="entry-header"> <h1 class=" loop-title entry-title" itemprop="headline">Welcome to the Fold</h1> <div class=" loop-description" itemprop="text"><div class="entry-byline"> <div class="entry-byline-block entry-byline-cats"> <span class="entry-byline-label">In:</span> <a href="https://mathvoices.ams.org/featurecolumn/category/2024/" rel="category tag">2024</a>, <a href="https://mathvoices.ams.org/featurecolumn/category/columnists/adriana-salerno/" rel="category tag">Adriana Salerno</a>, <a href="https://mathvoices.ams.org/featurecolumn/category/algebra-and-number-theory/" rel="category tag">Algebra and Number Theory</a>, <a href="https://mathvoices.ams.org/featurecolumn/category/math-and-arts-humanities/" rel="category tag">Math and Arts/Humanities</a>, <a href="https://mathvoices.ams.org/featurecolumn/category/columnists/sara-chari/" rel="category tag">Sara Chari</a> </div> <div class="entry-byline-block entry-byline-tags"> <span class="entry-byline-label">Tagged:</span> <a href="https://mathvoices.ams.org/featurecolumn/tag/lattices-groups/" rel="tag">lattices (groups)</a>, <a href="https://mathvoices.ams.org/featurecolumn/tag/origami/" rel="tag">origami</a>, <a href="https://mathvoices.ams.org/featurecolumn/tag/rings/" rel="tag">rings</a> </div></div></div> </div> </div> </div> </div> <article id="post-2044" class="entry author-uwhitcher post-2044 post type-post status-publish format-standard has-post-thumbnail category-147 category-adriana-salerno category-algebra-and-number-theory category-math-and-arts-humanities category-sara-chari tag-lattices-groups tag-origami tag-rings" itemscope="itemscope" itemtype="https://schema.org/BlogPosting" itemprop="blogPost"> <div class="entry-content" itemprop="articleBody"> <div class="entry-the-content"> <p><span id="pullQuote"><em>A natural question in the context of origami mathematics is: What if we make the paper infinitely large?</em></span></p> <h1 class="headlineText">Welcome to the Fold</h1> <p><b>Sara Chari<br /> Saint Mary's College of Maryland</b></p> <p><b>Adriana Salerno<br /> Bates College and the National Science Foundation</b></p> <p>Origami—from the Japanese words for “fold” (oru) and “paper” (kami)—is the art of folding paper to make certain shapes, with the simple constraints that one is not allowed to cut or glue any part of the paper. Advances in origami design have yielded some amazing <a href="https://erikdemaine.org/curved/MoMath/">artistic</a>, and even <a href="https://www.youtube.com/watch?v=Ly3hMBD4h5E">scientific </a>results! </p> <p>If you’ve ever tried your hand at origami, you will know that once you make a particular shape, you can unfold the paper to see all the creases (lines on the paper, which may or may not intersect). Crease patterns also tell you how you should fold paper to obtain certain shapes. Here are some images (made by a former Bates College student, Wuyue Zhou), of a paper crane with a few initial steps. </p> <div align="center"><img data-recalc-dims="1" fetchpriority="high" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/cranes.jpg?resize=600%2C448&ssl=1" alt="Four stages of folding an origami crane: blank paper, two diagonal folds, three folds, and the finished crane." width="600" height="448" class="aligncenter size-full wp-image-2065" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/cranes.jpg?w=600&ssl=1 600w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/cranes.jpg?resize=300%2C224&ssl=1 300w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/cranes.jpg?resize=465%2C347&ssl=1 465w" sizes="(max-width: 600px) 100vw, 600px" /><br /> <em>Wuyue Zhou</em></div> <p>And just for fun, here is an origami seashell folded by <a href="https://www.rose-hulman.edu/academics/faculty/riehl-manda-riehl.html">Manda Riehl</a> and its corresponding crease pattern:</p> <div align="center"> <img data-recalc-dims="1" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-riehl-square.jpg?resize=294%2C300&ssl=1" alt="Origami folded into a seashell with a square spiral point." width="294" height="300" class="aligncenter size-medium wp-image-2075" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-riehl-square.jpg?resize=294%2C300&ssl=1 294w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-riehl-square.jpg?resize=465%2C474&ssl=1 465w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-riehl-square.jpg?resize=491%2C500&ssl=1 491w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-riehl-square.jpg?w=687&ssl=1 687w" sizes="(max-width: 294px) 100vw, 294px" /></p> <p><img data-recalc-dims="1" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-crease-riehl-square.jpg?resize=300%2C300&ssl=1" alt="Unfolded paper shows the crease pattern of the origami seashell." width="300" height="300" class="aligncenter size-medium wp-image-2074" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-crease-riehl-square.jpg?resize=300%2C300&ssl=1 300w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-crease-riehl-square.jpg?resize=150%2C150&ssl=1 150w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-crease-riehl-square.jpg?resize=768%2C768&ssl=1 768w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-crease-riehl-square.jpg?resize=465%2C465&ssl=1 465w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-crease-riehl-square.jpg?resize=500%2C500&ssl=1 500w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/seashell-crease-riehl-square.jpg?w=928&ssl=1 928w" sizes="(max-width: 300px) 100vw, 300px" /><br /> <em>Images and folds by Manda Riehl</em> </div> <p><a href="https://erikdemaine.org">Erik Demaine</a> is one of the leading experts in so called “mathematical” origami, which he describes on his website <a href="https://erikdemaine.org/folding/">Folding and Unfolding</a> as coming in two flavors: </p> <blockquote><p>“Origami mathematics</i> is a recent branch of mathematics (whose major study started circa 1980) that studies the properties of origami, such as what patterns you might get when you unfold a flat origami. <i>Computational origami</i> is an even more recent branch of computer science that explores algorithms for creating origami or solving paper-folding problems. This field began in North America with Robert Lang's work on TreeMaker (circa 1993).</p></blockquote> <p>A natural question in the context of origami mathematics is: What if we make the paper infinitely large? This leads to a variety of designs left by the folds and, more importantly, the points at which the folds intersect.</p> <p>Joe Buhler, Steve Butler, Warwick De Launay, and Ron Graham, inspired by a question of Demaine’s, first came up with the idea of an “origami construction” to describe these ideas more formally, in their 2012 paper <a href="https://mathweb.ucsd.edu/~ronspubs/12_04_origami.pdf">Origami Rings</a>. In this column, we will describe the main ideas behind these constructions, and some of the natural questions that arise. </p> <h1>Origami Constructions </h1> <p>Just like with traditional origami, we set up a few constraints for our mathematical origami: we are allowed two points on the plane to start, typically 0 and 1. These are called "seed points" and they will “grow” outward into our origami construction. We are also allowed a finite set of angles or directions at which we are allowed to "fold" the plane (read: draw a line). Each pair of folds intersects in a single point unless the corresponding lines are parallel. When a new point is created this way, it gets added to our collection of points, along with the seed points. </p> <p>So, if you have the line with angle $\alpha$ going through the point $p$, and the line with angle $\beta$ going through the point $q$, their intersection point (below) gets added to our set of points. </p> <div align="center"> <img data-recalc-dims="1" loading="lazy" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/intersection-1.png?resize=200%2C240&ssl=1" alt="The intersection of lines p and q defined by angles alpha and beta" width="200" height="240" class="aligncenter size-full wp-image-2063" /></div> <p>Folds must be made only through two points that have already been added to the collection, so at first the options are limited. As the process continues, there are more and more points and therefore, more and more lines that may be drawn. Below is a sequence, starting with the seed points and the angles 0, $\pi/4$ and $\pi/2$. </p> <div align="center"> <img data-recalc-dims="1" loading="lazy" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen0.png?resize=429%2C402&ssl=1" alt="Seed points" width="429" height="402" class="aligncenter size-full wp-image-2052" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen0.png?w=429&ssl=1 429w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen0.png?resize=300%2C281&ssl=1 300w" sizes="auto, (max-width: 429px) 100vw, 429px" /></p> <p><img data-recalc-dims="1" loading="lazy" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen1.png?resize=420%2C420&ssl=1" alt="First generated points" width="420" height="420" class="aligncenter size-full wp-image-2053" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen1.png?w=420&ssl=1 420w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen1.png?resize=300%2C300&ssl=1 300w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen1.png?resize=150%2C150&ssl=1 150w" sizes="auto, (max-width: 420px) 100vw, 420px" /></p> <p><img data-recalc-dims="1" loading="lazy" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen2.png?resize=398%2C395&ssl=1" alt="More points" width="398" height="395" class="aligncenter size-full wp-image-2054" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen2.png?w=398&ssl=1 398w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen2.png?resize=300%2C298&ssl=1 300w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen2.png?resize=150%2C150&ssl=1 150w" sizes="auto, (max-width: 398px) 100vw, 398px" /></p> <p><img data-recalc-dims="1" loading="lazy" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen3.png?resize=430%2C406&ssl=1" alt="The origami points are beginning to suggest a square lattice near the middle" width="430" height="406" class="aligncenter size-full wp-image-2055" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen3.png?w=430&ssl=1 430w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen3.png?resize=300%2C283&ssl=1 300w" sizes="auto, (max-width: 430px) 100vw, 430px" /></p> <p><img data-recalc-dims="1" loading="lazy" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen4.png?resize=435%2C440&ssl=1" alt="With even more points, the suggestion of a square lattice is definite." width="435" height="440" class="aligncenter size-full wp-image-2056" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen4.png?w=435&ssl=1 435w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen4.png?resize=297%2C300&ssl=1 297w" sizes="auto, (max-width: 435px) 100vw, 435px" /> </div> <p>This process continues infinitely, so it may feel strange to think about an end result, but it is often possible to describe the final set of points, and sometimes they even form a nice pattern (more on that later). From here on out, we will refer to this set of points as the <em>origami set</em>. </p> <div align="center"> <img data-recalc-dims="1" loading="lazy" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen5.png?resize=412%2C414&ssl=1" alt="After many iterations of the origami process, we see part of a square lattice." width="412" height="414" class="aligncenter size-full wp-image-2057" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen5.png?w=412&ssl=1 412w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen5.png?resize=300%2C300&ssl=1 300w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/gen5.png?resize=150%2C150&ssl=1 150w" sizes="auto, (max-width: 412px) 100vw, 412px" /><br /> <em>After many iterations of the origami process begun above, a pattern is obvious.</em> </div> <p>Once you start playing with the origami point construction algorithm, you might ask yourself many questions. For example, when do we get a nice “visual” pattern (as in the example above)? Do we get other “nice” properties? Since the plane can also be thought of as the complex numbers, you could also ask: when is this subset also a sub<i>ring</i>? Given a subring, is there a corresponding origami construction? What happens if we try this in higher dimensions? </p> <h1>Lattices or not?</h1> <p>Here are some examples of the first few iterations of origami constructions using different numbers of angles (made by some summer research students of Adriana’s at Bates). What do you notice? </p> <div align="center"> <img data-recalc-dims="1" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/interesting.png?w=600&ssl=1" alt="Three seed points produce part of a lattice after several iterations, while four seed points yield chaotic-looking clouds." /> </div> <p>It seems like the only “nice” example (at least visually) is the first one. In fact, Buhler, Butler, de Launay and Graham already showed that when there are 4 or more allowed angles, the origami set will be dense in the complex plane, meaning that for any point in the whole plane you wish to pick, you can find a point in the origami set that is as close to your original point as you would like. Another way to view this is for any circle you draw in the complex plane, there will be infinitely many points from your origami set within the circle. That’s a lot of points to keep track of! (Notice the “zoomed in” picture above.) </p> <p>Our first, “nice” example is actually a lattice. This is a special kind of discrete infinite subset of $\mathbb{R}^2$—or, more generally, $\mathbb{R}^n$. (If you want to dive deeper into the special properties of lattices, there have been some great recent Feature Columns <a href="https://mathvoices.ams.org/featurecolumn/tag/lattices-groups/">on the subject</a>.)</p> <p>But do you always get a lattice when you use three angles? The answer is yes, as long as one of your angles is 0, and this was <a href="https://arxiv.org/abs/1502.07995">proved by Dmitri Nedrenco</a> in 2015. </p> <h1>Origami Rings</h1> <p>Fun fact: when you Google “origami ring” what you find are things like this, which are probably super fun to make (we haven’t tried) but not the kind of “ring” we are talking about. </p> <div align="center"> <img data-recalc-dims="1" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/rose-ring-sophy-flickr.jpg?w=600&ssl=1" alt="Two flower-shaped rings made from origami on a white person's hand" /><br /> <em>Rose rings designed by Hyo Ahn and folded by <a href="https://www.flickr.com/photos/oirgami/">Origami Sophy</a>.</em> (<a href="https://creativecommons.org/licenses/by-sa/2.0/">CC BY-SA 2.0</a>) </div> <p>Which origami constructions lead to subrings of the complex numbers? This was in fact the central question in the Buhler et al. paper, and the answer is pretty cool—if your angle set forms a subgroup of the unit circle, then your origami set is a ring. Algebraic structure yields algebraic structure!</p> <p>A question they posed in the same paper was essentially the inverse—which rings can be “origami constructed”? Juergen Kritschgau, in his 2015 senior Bates thesis, proved that it is possible to obtain the ring of integers of an imaginary quadratic field through an origami construction (this was <a href="https://math.colgate.edu/~integers/r34/r34.pdf">published in the INTEGERS journal in 2017</a>).</p> <p>Finally, <a href="https://arxiv.org/abs/1804.10449">Florian Moller definitively answered this question in 2018</a>: he gives a set of criteria that determines precisely when the construction will be a ring. </p> <p><strong>Note:</strong> an origami construction can be both a lattice and a ring (like the very first example we showed, which ends up being the Gaussian integers), but these two “nice” features are independent. </p> <h1>Higher Dimensions</h1> <p>Since we live in a three dimensional world, one may wish to do mathematical origami in higher dimensions. The rules are the same except that the allowed angles need not be contained in the complex plane. In the image shown, the standard basis elements are called 1, i, and j. To the reader who wishes to think about multiplication, these belong to the 4-dimensional Hamilton quaternion algebra. Sadly, all four dimensions are needed in order to have a closed multiplication structure.</p> <div align="center"> <img data-recalc-dims="1" loading="lazy" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/cubes.png?resize=300%2C174&ssl=1" alt="A cube with corners determined by sums and differences of 1, i, and j" width="300" height="174" class="aligncenter size-full wp-image-2068" /> </div> <p>One major difference in higher dimensions is that the intersection of two lines is not guaranteed anymore if the lines are skew. Luckily, this does not pose any problems; we simply do not record an intersection point if it does not exist. This does, however, allow for the inclusion of an arbitrary number of "irrelevant" allowed angles that either do not intersect anything at all besides the seed points and other points already in the set. There is no limit to the number of angles to obtain a lattice, but there is a minimal number of angles that guarantee a lattice: 2<i>n</i>+1, where <i>n</i> is the dimension of the space. If carefully chosen, just one additional angle can force a lattice to become dense which is quite remarkable!</p> <p>Even though there is not a nice multiplication structure in 3 dimensions, 3-dimensional lattices are of interest to chemists alike crystallography. In particular, <a href="https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Book%3A_Introduction_to_Inorganic_Chemistry_(Wikibook)/06%3A_Metals_and_Alloys-_Structure_Bonding_Electronic_and_Magnetic_Properties/6.03%3A_Bravais_Lattices">Bravais lattices</a> have been classified and one may wish to study which of them can be obtained via mathematical origami.</p> <p>And what about even higher dimensions? We thought about this question, together with then-Bates student Deveena Banerjee, and found many other cool results. For more, <a href="https://arxiv.org/abs/2110.08119">see our paper</a>! </p> <h1>What’s next? </h1> <p>There really are so many options, since these ideas are so new. What we’ve described here is also not the only way to “mathematize” origami! For example, if you extract certain geometric “axioms” from origami, you can <a href="https://divisbyzero.com/2012/06/01/angle-trisection-using-origami/">trisect an angle</a>! We have also not discussed the area of <a href="https://langorigami.com/article/computational-origami/">computational origami</a>, and all the other scientific applications of origami. </p> <p>Let us end with an invitation: what is your favorite origami design? What does its crease pattern look like?</p> <div align="center"> <img data-recalc-dims="1" loading="lazy" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-crease-square-jill-m-bean.jpeg?resize=300%2C300&ssl=1" alt="An origami crease pattern on green paper." width="300" height="300" class="aligncenter size-medium wp-image-2076" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-crease-square-jill-m-bean.jpeg?resize=300%2C300&ssl=1 300w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-crease-square-jill-m-bean.jpeg?resize=150%2C150&ssl=1 150w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-crease-square-jill-m-bean.jpeg?resize=768%2C768&ssl=1 768w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-crease-square-jill-m-bean.jpeg?resize=465%2C465&ssl=1 465w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-crease-square-jill-m-bean.jpeg?resize=500%2C500&ssl=1 500w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-crease-square-jill-m-bean.jpeg?w=855&ssl=1 855w" sizes="auto, (max-width: 300px) 100vw, 300px" /></p> <p><img data-recalc-dims="1" loading="lazy" decoding="async" src="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-square-jill-m-bean.jpeg?resize=300%2C300&ssl=1" alt="The same piece of origami paper folded into a box with a red interior and green exterior." width="300" height="300" class="aligncenter size-medium wp-image-2077" srcset="https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-square-jill-m-bean.jpeg?resize=300%2C300&ssl=1 300w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-square-jill-m-bean.jpeg?resize=150%2C150&ssl=1 150w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-square-jill-m-bean.jpeg?resize=768%2C768&ssl=1 768w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-square-jill-m-bean.jpeg?resize=465%2C465&ssl=1 465w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-square-jill-m-bean.jpeg?resize=500%2C500&ssl=1 500w, https://i0.wp.com/mathvoices.ams.org/featurecolumn/wp-content/uploads/sites/2/2024/08/box-square-jill-m-bean.jpeg?w=852&ssl=1 852w" sizes="auto, (max-width: 300px) 100vw, 300px" /><br /> <em>Image and folds by Jill M. Bean.</em> </div> <div class="sharedaddy sd-sharing-enabled"><div class="robots-nocontent sd-block sd-social sd-social-icon-text sd-sharing"><h3 class="sd-title">Share this:</h3><div class="sd-content"><ul><li class="share-facebook"><a rel="nofollow noopener noreferrer" data-shared="sharing-facebook-2044" class="share-facebook sd-button share-icon" href="https://mathvoices.ams.org/featurecolumn/2024/08/01/welcome-to-the-fold/?share=facebook" target="_blank" title="Click to share on Facebook" ><span>Facebook</span></a></li><li class="share-x"><a rel="nofollow noopener noreferrer" data-shared="sharing-x-2044" class="share-x sd-button share-icon" href="https://mathvoices.ams.org/featurecolumn/2024/08/01/welcome-to-the-fold/?share=x" target="_blank" title="Click to share on X" ><span>X</span></a></li><li class="share-end"></li></ul></div></div></div><div class='sharedaddy sd-block sd-like jetpack-likes-widget-wrapper jetpack-likes-widget-unloaded' id='like-post-wrapper-202620863-2044-67be94ec59e0a' data-src='https://widgets.wp.com/likes/?ver=14.3#blog_id=202620863&post_id=2044&origin=mathvoices.ams.org&obj_id=202620863-2044-67be94ec59e0a&n=1' data-name='like-post-frame-202620863-2044-67be94ec59e0a' data-title='Like or Reblog'><h3 class="sd-title">Like this:</h3><div class='likes-widget-placeholder post-likes-widget-placeholder' style='height: 55px;'><span class='button'><span>Like</span></span> <span class="loading">Loading...</span></div><span class='sd-text-color'></span><a class='sd-link-color'></a></div> </div> </div> <div class="screen-reader-text" itemprop="datePublished" itemtype="https://schema.org/Date">2024-08-01</div> </article> <div id="loop-nav-wrap" class="loop-nav"><div class="prev">Previous Post: <a href="https://mathvoices.ams.org/featurecolumn/2024/07/01/looking-ahead/" rel="prev">Looking ahead</a></div><div class="next">Next Post: <a href="https://mathvoices.ams.org/featurecolumn/2024/09/01/payoffs-and-strategies/" rel="next">Payoffs and strategies</a></div></div> <section id="comments-template"> <div id="respond" class="comment-respond"> <h3 id="reply-title" class="comment-reply-title">Leave a Reply <small><a rel="nofollow" id="cancel-comment-reply-link" href="/featurecolumn/2024/08/01/welcome-to-the-fold/#respond" style="display:none;">Cancel reply</a></small></h3><form action="https://mathvoices.ams.org/featurecolumn/wp-comments-post.php" method="post" id="commentform" class="comment-form" novalidate><p class="comment-notes"><span id="email-notes">Your email address will not be published.</span> <span class="required-field-message">Required fields are marked <span class="required">*</span></span></p><p class="comment-form-comment"><label for="comment">Comment <span class="required">*</span></label> <textarea id="comment" name="comment" cols="45" rows="8" maxlength="65525" required></textarea></p><p><b>HTML tags are not allowed.</b></p><p class="comment-form-author"><label for="author">Name <span class="required">*</span></label> <input x-autocompletetype="name-full" id="author" name="author" type="text" value="" size="30" maxlength="245" autocomplete="name" required /></p> <p class="comment-form-email"><label for="email">Email <span class="required">*</span></label> <input x-autocompletetype="email" id="email" name="email" type="email" value="" size="30" maxlength="100" aria-describedby="email-notes" autocomplete="email" required /></p> <p class="comment-form-cookies-consent"><input id="wp-comment-cookies-consent" name="wp-comment-cookies-consent" type="checkbox" value="yes" /> <label for="wp-comment-cookies-consent">Save my name, email, and website in this browser for the next time I comment.</label></p> <p>60,853 Spambots Blocked by <a href="https://www.toddlahman.com/shop/simple-comments/" title="Simple Comments" target="_blank">Simple Comments</a></p> <p class="comment-subscription-form"><input type="checkbox" name="subscribe_comments" id="subscribe_comments" value="subscribe" style="width: auto; -moz-appearance: checkbox; -webkit-appearance: checkbox;" /> <label class="subscribe-label" id="subscribe-label" for="subscribe_comments">Notify me of follow-up comments by email.</label></p><p class="comment-subscription-form"><input type="checkbox" name="subscribe_blog" id="subscribe_blog" value="subscribe" style="width: auto; -moz-appearance: checkbox; -webkit-appearance: checkbox;" /> <label class="subscribe-label" id="subscribe-blog-label" for="subscribe_blog">Notify me of new posts by email.</label></p><p class='comment-form-subscriptions'><label for='subscribe-reloaded'><input style='width:30px' type='checkbox' name='subscribe-reloaded' id='subscribe-reloaded' value='yes' /> Notify me of followup comments via e-mail. You can also <a href='https://mathvoices.ams.org/featurecolumn/comment-subscriptions/?srp=2044&srk=670fa4bfeb139d13245dcdc12ca15773&sra=s&srsrc=f'>subscribe</a> without commenting.</label></p><p class="form-submit"><input name="submit" type="submit" id="submit" class="submit" value="Post Comment" /> <input type='hidden' name='comment_post_ID' value='2044' id='comment_post_ID' /> <input type='hidden' name='comment_parent' id='comment_parent' value='0' /> </p> <p><h3><noscript>JavaScript must be turned on to leave a comment.</noscript></h3></p> <input type='hidden' name='sec_pwd_field' id='pwd_field_sec' value='' /> <input type='hidden' name='sec_comment_ip' id='comment_ip_sec' value='' /></form> </div> </section> </div> </main> <aside id="sidebar-primary" class="sidebar sidebar-primary hgrid-span-4 layout-wide-right " role="complementary" itemscope="itemscope" itemtype="https://schema.org/WPSideBar"> <div class=" sidebar-wrap theiaStickySidebar"> <section id="search-2" class="widget widget_search"><div class="searchbody"><form method="get" class="searchform" action="https://mathvoices.ams.org/featurecolumn/" ><label class="screen-reader-text">Search</label><i class="fas fa-search"></i><input type="text" class="searchtext" name="s" placeholder="Type Search Term …" value="" /><input type="submit" class="submit" name="submit" value="Search" /><span class="js-search-placeholder"></span></form></div></section><section id="blog_subscription-2" class="widget widget_blog_subscription jetpack_subscription_widget"><h3 class="widget-title"><span>Subscribe to Feature Column via Email</span></h3> <div class="wp-block-jetpack-subscriptions__container"> <form action="#" method="post" accept-charset="utf-8" id="subscribe-blog-blog_subscription-2" data-blog="202620863" data-post_access_level="everybody" > <div id="subscribe-text"><p>Enter your email address to subscribe to Feature Column and receive notifications of new posts by email.</p> </div> <p id="subscribe-email"> <label id="jetpack-subscribe-label" class="screen-reader-text" for="subscribe-field-blog_subscription-2"> Email Address </label> <input type="email" name="email" required="required" value="" id="subscribe-field-blog_subscription-2" placeholder="Email Address" /> </p> <p id="subscribe-submit" > <input type="hidden" name="action" value="subscribe"/> <input type="hidden" name="source" value="https://mathvoices.ams.org/featurecolumn/2024/08/01/welcome-to-the-fold/"/> <input type="hidden" name="sub-type" value="widget"/> <input type="hidden" name="redirect_fragment" value="subscribe-blog-blog_subscription-2"/> <input type="hidden" id="_wpnonce" name="_wpnonce" value="c4f43bebc8" /><input type="hidden" name="_wp_http_referer" value="/featurecolumn/2024/08/01/welcome-to-the-fold/" /> <button type="submit" class="wp-block-button__link" name="jetpack_subscriptions_widget" > Subscribe </button> </p> </form> </div> </section><section id="block-3" class="widget widget_block"><div class="wp-block-group"><div class="wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow"><h2 class="wp-block-heading">Recent Posts</h2><ul class="wp-block-latest-posts__list wp-block-latest-posts"><li><a class="wp-block-latest-posts__post-title" href="https://mathvoices.ams.org/featurecolumn/2025/02/01/elliptic-functions/">Elliptic Functions, Doughnuts, and a Secret Integration Technique</a></li> <li><a class="wp-block-latest-posts__post-title" href="https://mathvoices.ams.org/featurecolumn/2025/01/01/visualizing-anti-trans-legislation-using-rstudio/">Visualizing anti-trans legislation using RStudio</a></li> <li><a class="wp-block-latest-posts__post-title" href="https://mathvoices.ams.org/featurecolumn/2024/12/01/capturing-the-invisible-hopping-rabbit/">Capturing the Invisible Hopping Rabbit</a></li> <li><a class="wp-block-latest-posts__post-title" href="https://mathvoices.ams.org/featurecolumn/2024/11/01/strung-out-on-automorphic-forms/">Strung Out on Automorphic Forms</a></li> <li><a class="wp-block-latest-posts__post-title" href="https://mathvoices.ams.org/featurecolumn/2024/10/01/people-and-computers-compared/">People and Computers Compared</a></li> </ul></div></div></section><section id="categories-2" class="widget widget_categories"><h3 class="widget-title"><span>Categories</span></h3> <ul> <li class="cat-item cat-item-2"><a href="https://mathvoices.ams.org/featurecolumn/category/2020/">2020</a> </li> <li class="cat-item cat-item-3"><a href="https://mathvoices.ams.org/featurecolumn/category/2021/">2021</a> </li> <li class="cat-item cat-item-4"><a href="https://mathvoices.ams.org/featurecolumn/category/2022/">2022</a> </li> <li class="cat-item cat-item-112"><a href="https://mathvoices.ams.org/featurecolumn/category/2023/">2023</a> </li> <li class="cat-item cat-item-147"><a href="https://mathvoices.ams.org/featurecolumn/category/2024/">2024</a> </li> <li class="cat-item cat-item-178"><a href="https://mathvoices.ams.org/featurecolumn/category/2025/">2025</a> </li> <li class="cat-item cat-item-5"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/adam-a-smith/">Adam A. Smith</a> </li> <li class="cat-item cat-item-168"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/adriana-salerno/">Adriana Salerno</a> </li> <li class="cat-item cat-item-6"><a href="https://mathvoices.ams.org/featurecolumn/category/algebra-and-number-theory/">Algebra and Number Theory</a> </li> <li class="cat-item cat-item-115"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/allechar-serrano-lopez/">Allechar Serrano López</a> </li> <li class="cat-item cat-item-109"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/anil-venkatesh/">Anil Venkatesh</a> </li> <li class="cat-item cat-item-7"><a href="https://mathvoices.ams.org/featurecolumn/category/archive/">Archive</a> </li> <li class="cat-item cat-item-179"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/bianca-thompson/">Bianca Thompson</a> </li> <li class="cat-item cat-item-8"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/bill-casselman/">Bill Casselman</a> </li> <li class="cat-item cat-item-9"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/colm-mulcahy/">Colm Mulcahy</a> </li> <li class="cat-item cat-item-10"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/courtney-gibbons/">Courtney Gibbons</a> </li> <li class="cat-item cat-item-11"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/david-austin/">David Austin</a> </li> <li class="cat-item cat-item-176"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/deepayan-sarkar/">Deepayan Sarkar</a> </li> <li class="cat-item cat-item-12"><a href="https://mathvoices.ams.org/featurecolumn/category/discrete-math-and-combinatorics/">Discrete Math and Combinatorics</a> </li> <li class="cat-item cat-item-13"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/etienne-ghys/">Étienne Ghys</a> </li> <li class="cat-item cat-item-114"><a href="https://mathvoices.ams.org/featurecolumn/category/geometry-and-topology/">Geometry and Topology</a> </li> <li class="cat-item cat-item-14"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/guillermo-fereyra/">Guillermo Fereyra</a> </li> <li class="cat-item cat-item-15"><a href="https://mathvoices.ams.org/featurecolumn/category/history-of-mathematics/">History of mathematics</a> </li> <li class="cat-item cat-item-172"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/holley-friedlander/">Holley Friedlander</a> </li> <li class="cat-item cat-item-16"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/john-eggers/">John Eggers</a> </li> <li class="cat-item cat-item-17"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/joseph-malkevitch/">Joseph Malkevitch</a> </li> <li class="cat-item cat-item-18"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/josh-leys/">Josh Leys</a> </li> <li class="cat-item cat-item-134"><a href="https://mathvoices.ams.org/featurecolumn/category/math-and-arts-humanities/">Math and Arts/Humanities</a> </li> <li class="cat-item cat-item-137"><a href="https://mathvoices.ams.org/featurecolumn/category/math-and-social-sciences/">Math and Social Sciences</a> </li> <li class="cat-item cat-item-20"><a href="https://mathvoices.ams.org/featurecolumn/category/math-and-technology/">Math and Technology</a> </li> <li class="cat-item cat-item-21"><a href="https://mathvoices.ams.org/featurecolumn/category/math-and-the-sciences/">Math and the Sciences</a> </li> <li class="cat-item cat-item-22"><a href="https://mathvoices.ams.org/featurecolumn/category/mathematics-and-biology/">Mathematics and Biology</a> </li> <li class="cat-item cat-item-23"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/moira-chas/">Moira Chas</a> </li> <li class="cat-item cat-item-100"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/noah-giansiracusa/">Noah Giansiracusa</a> </li> <li class="cat-item cat-item-24"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/patrick-ion/">Patrick Ion</a> </li> <li class="cat-item cat-item-25"><a href="https://mathvoices.ams.org/featurecolumn/category/probability-and-statistics/">Probability and Statistics</a> </li> <li class="cat-item cat-item-26"><a href="https://mathvoices.ams.org/featurecolumn/category/real-and-complex-analysis/">Real and Complex Analysis</a> </li> <li class="cat-item cat-item-169"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/sara-chari/">Sara Chari</a> </li> <li class="cat-item cat-item-77"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/sara-stoudt/">Sara Stoudt</a> </li> <li class="cat-item cat-item-142"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/sarah-wolff/">Sarah Wolff</a> </li> <li class="cat-item cat-item-27"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/steven-h-weintraub/">Steven H. Weintraub</a> </li> <li class="cat-item cat-item-175"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/sunil-chebolu/">Sunil Chebolu</a> </li> <li class="cat-item cat-item-28"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/tamsyn-morrill/">Tamsyn Morrill</a> </li> <li class="cat-item cat-item-29"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/tony-phillips/">Tony Phillips</a> </li> <li class="cat-item cat-item-1"><a href="https://mathvoices.ams.org/featurecolumn/category/uncategorized/">Uncategorized</a> </li> <li class="cat-item cat-item-30"><a href="https://mathvoices.ams.org/featurecolumn/category/columnists/ursula-whitcher/">Ursula Whitcher</a> </li> </ul> </section><section id="tag_cloud-2" class="widget widget_tag_cloud"><h3 class="widget-title"><span>Tags</span></h3><div class="tagcloud"><a href="https://mathvoices.ams.org/featurecolumn/tag/5g/" class="tag-cloud-link tag-link-32 tag-link-position-1" style="font-size: 8pt;" aria-label="5G (1 item)">5G</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/alan-turing/" class="tag-cloud-link tag-link-33 tag-link-position-2" style="font-size: 14.3pt;" aria-label="alan turing (2 items)">alan turing</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/ams/" class="tag-cloud-link tag-link-34 tag-link-position-3" style="font-size: 8pt;" aria-label="ams (1 item)">ams</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/astronauts/" class="tag-cloud-link tag-link-35 tag-link-position-4" style="font-size: 8pt;" aria-label="astronauts (1 item)">astronauts</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/branko-grunbaum/" class="tag-cloud-link tag-link-36 tag-link-position-5" style="font-size: 8pt;" aria-label="Branko Grünbaum (1 item)">Branko Grünbaum</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/cantor-set/" class="tag-cloud-link tag-link-37 tag-link-position-6" style="font-size: 8pt;" aria-label="cantor set (1 item)">cantor set</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/chess/" class="tag-cloud-link tag-link-38 tag-link-position-7" style="font-size: 18.5pt;" aria-label="chess (3 items)">chess</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/claude-shannon/" class="tag-cloud-link tag-link-39 tag-link-position-8" style="font-size: 8pt;" aria-label="Claude Shannon (1 item)">Claude Shannon</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/cryptography/" class="tag-cloud-link tag-link-107 tag-link-position-9" style="font-size: 18.5pt;" aria-label="cryptography (3 items)">cryptography</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/desargues/" class="tag-cloud-link tag-link-41 tag-link-position-10" style="font-size: 8pt;" aria-label="Desargues (1 item)">Desargues</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/elliptic-curves/" class="tag-cloud-link tag-link-154 tag-link-position-11" style="font-size: 18.5pt;" aria-label="elliptic curves (3 items)">elliptic curves</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/epidemic/" class="tag-cloud-link tag-link-42 tag-link-position-12" style="font-size: 14.3pt;" aria-label="epidemic (2 items)">epidemic</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/euclidean-geometry/" class="tag-cloud-link tag-link-43 tag-link-position-13" style="font-size: 18.5pt;" aria-label="Euclidean geometry (3 items)">Euclidean geometry</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/feature-column-history/" class="tag-cloud-link tag-link-44 tag-link-position-14" style="font-size: 8pt;" aria-label="feature column history (1 item)">feature column history</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/games/" class="tag-cloud-link tag-link-84 tag-link-position-15" style="font-size: 14.3pt;" aria-label="games (2 items)">games</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/game-theory/" class="tag-cloud-link tag-link-148 tag-link-position-16" style="font-size: 22pt;" aria-label="game theory (4 items)">game theory</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/generating-functions/" class="tag-cloud-link tag-link-45 tag-link-position-17" style="font-size: 8pt;" aria-label="generating functions (1 item)">generating functions</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/genetics/" class="tag-cloud-link tag-link-46 tag-link-position-18" style="font-size: 8pt;" aria-label="genetics (1 item)">genetics</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/geoffrey-colin-shephard/" class="tag-cloud-link tag-link-47 tag-link-position-19" style="font-size: 8pt;" aria-label="Geoffrey Colin Shephard (1 item)">Geoffrey Colin Shephard</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/herd-immunity/" class="tag-cloud-link tag-link-48 tag-link-position-20" style="font-size: 8pt;" aria-label="herd immunity (1 item)">herd immunity</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/higgs-boson/" class="tag-cloud-link tag-link-49 tag-link-position-21" style="font-size: 8pt;" aria-label="Higgs boson (1 item)">Higgs boson</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/lattices-groups/" class="tag-cloud-link tag-link-97 tag-link-position-22" style="font-size: 18.5pt;" aria-label="lattices (groups) (3 items)">lattices (groups)</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/lie-groups/" class="tag-cloud-link tag-link-111 tag-link-position-23" style="font-size: 14.3pt;" aria-label="Lie groups (2 items)">Lie groups</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/machine-learning/" class="tag-cloud-link tag-link-50 tag-link-position-24" style="font-size: 14.3pt;" aria-label="Machine learning (2 items)">Machine learning</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/measles/" class="tag-cloud-link tag-link-51 tag-link-position-25" style="font-size: 8pt;" aria-label="measles (1 item)">measles</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/medieval/" class="tag-cloud-link tag-link-52 tag-link-position-26" style="font-size: 8pt;" aria-label="medieval (1 item)">medieval</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/mirror-symmetry/" class="tag-cloud-link tag-link-53 tag-link-position-27" style="font-size: 8pt;" aria-label="mirror symmetry (1 item)">mirror symmetry</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/paleontology/" class="tag-cloud-link tag-link-54 tag-link-position-28" style="font-size: 8pt;" aria-label="paleontology (1 item)">paleontology</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/pappuss-theorem/" class="tag-cloud-link tag-link-55 tag-link-position-29" style="font-size: 8pt;" aria-label="Pappus's Theorem (1 item)">Pappus's Theorem</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/partitions/" class="tag-cloud-link tag-link-56 tag-link-position-30" style="font-size: 14.3pt;" aria-label="partitions (2 items)">partitions</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/pca/" class="tag-cloud-link tag-link-57 tag-link-position-31" style="font-size: 8pt;" aria-label="pca (1 item)">pca</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/periods/" class="tag-cloud-link tag-link-129 tag-link-position-32" style="font-size: 14.3pt;" aria-label="periods (2 items)">periods</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/persi-diaconis/" class="tag-cloud-link tag-link-143 tag-link-position-33" style="font-size: 14.3pt;" aria-label="persi diaconis (2 items)">persi diaconis</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/physics/" class="tag-cloud-link tag-link-82 tag-link-position-34" style="font-size: 14.3pt;" aria-label="physics (2 items)">physics</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/poetry/" class="tag-cloud-link tag-link-135 tag-link-position-35" style="font-size: 14.3pt;" aria-label="poetry (2 items)">poetry</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/polar-codes/" class="tag-cloud-link tag-link-58 tag-link-position-36" style="font-size: 8pt;" aria-label="polar codes (1 item)">polar codes</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/polytopes-and-polyhedra/" class="tag-cloud-link tag-link-59 tag-link-position-37" style="font-size: 14.3pt;" aria-label="polytopes and polyhedra (2 items)">polytopes and polyhedra</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/primes/" class="tag-cloud-link tag-link-62 tag-link-position-38" style="font-size: 18.5pt;" aria-label="primes (3 items)">primes</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/probability/" class="tag-cloud-link tag-link-86 tag-link-position-39" style="font-size: 14.3pt;" aria-label="probability (2 items)">probability</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/seir-model/" class="tag-cloud-link tag-link-67 tag-link-position-40" style="font-size: 14.3pt;" aria-label="SEIR model (2 items)">SEIR model</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/statistics/" class="tag-cloud-link tag-link-153 tag-link-position-41" style="font-size: 14.3pt;" aria-label="statistics (2 items)">statistics</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/string-theory/" class="tag-cloud-link tag-link-70 tag-link-position-42" style="font-size: 14.3pt;" aria-label="string theory (2 items)">string theory</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/tic-tac-toe/" class="tag-cloud-link tag-link-165 tag-link-position-43" style="font-size: 14.3pt;" aria-label="tic-tac-toe (2 items)">tic-tac-toe</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/tilings/" class="tag-cloud-link tag-link-72 tag-link-position-44" style="font-size: 14.3pt;" aria-label="tilings (2 items)">tilings</a> <a href="https://mathvoices.ams.org/featurecolumn/tag/voting/" class="tag-cloud-link tag-link-141 tag-link-position-45" style="font-size: 14.3pt;" aria-label="voting (2 items)">voting</a></div> </section><section id="meta-2" class="widget widget_meta"><h3 class="widget-title"><span>Meta</span></h3> <ul> <li><a href="https://mathvoices.ams.org/featurecolumn/wp-login.php">Log in</a></li> <li><a href="https://mathvoices.ams.org/featurecolumn/feed/">Entries feed</a></li> <li><a href="https://mathvoices.ams.org/featurecolumn/comments/feed/">Comments feed</a></li> <li><a href="https://wordpress.org/">WordPress.org</a></li> </ul> </section> </div> </aside> </div> </div> <div id="post-footer" class=" post-footer hgrid-stretch linkstyle"> <div class="hgrid"> <div class="hgrid-span-12"> <p class="credit small"> Designed using <a class="theme-link" href="https://wphoot.com/themes/hoot-business/" title="Hoot Business WordPress Theme">Hoot Business</a>. Powered by <a class="wp-link" href="https://wordpress.org">WordPress</a>. </p> </div> </div> </div> </div> <script type="text/javascript"> window.WPCOM_sharing_counts = {"https:\/\/mathvoices.ams.org\/featurecolumn\/2024\/08\/01\/welcome-to-the-fold\/":2044}; </script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/comment-reply.min.js" id="comment-reply-js" async data-wp-strategy="async"></script> <script src="https://c0.wp.com/c/6.7.2/wp-includes/js/hoverIntent.min.js" id="hoverIntent-js"></script> <script src="https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/js/jquery.superfish.js?ver=1.7.5" id="jquery-superfish-js"></script> <script src="https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/js/jquery.fitvids.js?ver=1.1" id="jquery-fitvids-js"></script> <script src="https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/js/jquery.parallax.js?ver=1.4.2" id="jquery-parallax-js"></script> <script src="https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/js/resizesensor.js?ver=1.7.0" id="resizesensor-js"></script> <script src="https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/js/jquery.theia-sticky-sidebar.js?ver=1.7.0" id="jquery-theia-sticky-sidebar-js"></script> <script src="//cdn.jsdelivr.net/npm/mathjax@2.7.8/MathJax.js?config=TeX-MML-AM_CHTML%2CSafe.js&ver=6.7.2" id="mathjax-js"></script> <script id="sc_ipwd-js-extra"> var sc_ipwd_script = {"sc_ajaxurl":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-admin\/admin-ajax.php"}; </script> <script src="https://mathvoices.ams.org/featurecolumn/wp-content/plugins/simple-comments/js/comments/sc-ipwd-nn.min.js?ver=1.5" id="sc_ipwd-js"></script> <script src="https://mathvoices.ams.org/featurecolumn/wp-content/themes/hoot-business/js/hoot.theme.js?ver=2.11.1" id="hoot-theme-js"></script> <script src="https://stats.wp.com/e-202509.js" id="jetpack-stats-js" data-wp-strategy="defer"></script> <script id="jetpack-stats-js-after"> _stq = window._stq || []; _stq.push([ "view", JSON.parse("{\"v\":\"ext\",\"blog\":\"202620863\",\"post\":\"2044\",\"tz\":\"-5\",\"srv\":\"mathvoices.ams.org\",\"j\":\"1:14.3\"}") ]); _stq.push([ "clickTrackerInit", "202620863", "2044" ]); </script> <script src="https://c0.wp.com/p/jetpack/14.3/_inc/build/likes/queuehandler.min.js" id="jetpack_likes_queuehandler-js"></script> <script id="sharing-js-js-extra"> var sharing_js_options = {"lang":"en","counts":"1","is_stats_active":"1"}; </script> <script src="https://c0.wp.com/p/jetpack/14.3/_inc/build/sharedaddy/sharing.min.js" id="sharing-js-js"></script> <script id="sharing-js-js-after"> var windowOpen; ( function () { function matches( el, sel ) { return !! ( el.matches && el.matches( sel ) || el.msMatchesSelector && el.msMatchesSelector( sel ) ); } document.body.addEventListener( 'click', function ( event ) { if ( ! event.target ) { return; } var el; if ( matches( event.target, 'a.share-facebook' ) ) { el = event.target; } else if ( event.target.parentNode && matches( event.target.parentNode, 'a.share-facebook' ) ) { el = event.target.parentNode; } if ( el ) { event.preventDefault(); // If there's another sharing window open, close it. if ( typeof windowOpen !== 'undefined' ) { windowOpen.close(); } windowOpen = window.open( el.getAttribute( 'href' ), 'wpcomfacebook', 'menubar=1,resizable=1,width=600,height=400' ); return false; } } ); } )(); var windowOpen; ( function () { function matches( el, sel ) { return !! ( el.matches && el.matches( sel ) || el.msMatchesSelector && el.msMatchesSelector( sel ) ); } document.body.addEventListener( 'click', function ( event ) { if ( ! event.target ) { return; } var el; if ( matches( event.target, 'a.share-x' ) ) { el = event.target; } else if ( event.target.parentNode && matches( event.target.parentNode, 'a.share-x' ) ) { el = event.target.parentNode; } if ( el ) { event.preventDefault(); // If there's another sharing window open, close it. if ( typeof windowOpen !== 'undefined' ) { windowOpen.close(); } windowOpen = window.open( el.getAttribute( 'href' ), 'wpcomx', 'menubar=1,resizable=1,width=600,height=350' ); return false; } } ); } )(); </script> <iframe src='https://widgets.wp.com/likes/master.html?ver=20250226#ver=20250226&n=1' scrolling='no' id='likes-master' name='likes-master' style='display:none;'></iframe> <div id='likes-other-gravatars' class='wpl-new-layout' role="dialog" aria-hidden="true" tabindex="-1"><div class="likes-text"><span>%d</span></div><ul class="wpl-avatars sd-like-gravatars"></ul></div> </body> </html>