{"id":2071,"date":"2026-10-11T10:34:33","date_gmt":"2026-10-11T10:34:33","guid":{"rendered":"https:\/\/stadiaorg.com\/?page_id=2071"},"modified":"2026-10-11T10:34:35","modified_gmt":"2026-10-11T10:34:35","slug":"stadia-numbers-the-universe","status":"publish","type":"page","link":"https:\/\/stadiaorg.com\/fr\/stadia-numbers-the-universe\/","title":{"rendered":"NOMBRE DE STADES &amp; UNIVERSE"},"content":{"rendered":"<style>.snu{--ink:#211b40;--muted:#625c75;--purple:#6650bd;max-width:1260px;margin:38px auto 60px;padding:0 20px;font-family:system-ui,-apple-system,\"Segoe UI\",sans-serif;color:var(--ink);line-height:1.75}.snu,.snu *{box-sizing:border-box}.snu [hidden]{display:none!important}.snu p{margin:0 0 16px}.snu h1,.snu h2,.snu h3{font-family:inherit;line-height:1.18;letter-spacing:-.035em;color:var(--ink);margin:0 0 18px}.snu h2{font-size:clamp(28px,3vw,39px)}.snu h3{font-size:22px}.snu a{color:#5841a8}.snu 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.snu-readouts span{display:block;font-size:10px;color:var(--muted);margin-bottom:6px}.snu .snu-readouts strong{font-size:14px;overflow-wrap:anywhere;color:#49347e}.snu .snu-question{font-size:13px;color:#65577a;border-top:1px solid #e2d9ef;padding-top:16px;margin-top:20px}.snu .snu-practice{background:#f3eefb;padding:34px;border-radius:22px;margin-top:48px;scroll-margin-top:120px}.snu .snu-practice>.snu-kicker{color:var(--purple)}.snu .snu-practice-grid{display:grid;grid-template-columns:repeat(3,minmax(0,1fr));gap:24px;margin-top:26px}.snu .snu-practice h3{font-size:20px}.snu .snu-practice-grid p{font-size:14px;color:var(--muted)}.snu .snu-footer{margin-top:30px;padding-top:25px;border-top:1px solid #e4daef;font-size:12px;color:var(--muted)}.snu .snu-footer h3{font-size:18px}.snu .snu-footer a{display:block;margin:8px 0}.snu .snu-footer p{margin-top:18px}.snu :is(a,summary,input):focus-visible{outline:3px solid #bca0ec;outline-offset:4px}\r\n@media(max-width:850px){.snu 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                  <img decoding=\"async\" src=\"https:\/\/stadiaorg.com\/wp-content\/plugins\/translatepress-multilingual\/assets\/flags\/4x3\/de_DE_formal.svg\" class=\"trp-flag-image\" alt=\"\" role=\"presentation\" loading=\"lazy\" \/>                                                            <span class=\"trp-language-item-name\" data-no-translation>German<\/span>\r\n                                                                                <\/a>\r\n                                            <a class=\"trp-language-item\" href=\"https:\/\/stadiaorg.com\/fr\/wp-json\/wp\/v2\/pages\/2071#TRPLINKPROCESSED\" title=\"Italian\">\r\n                            <img decoding=\"async\" src=\"https:\/\/stadiaorg.com\/wp-content\/plugins\/translatepress-multilingual\/assets\/flags\/4x3\/it_IT.svg\" class=\"trp-flag-image\" alt=\"\" role=\"presentation\" loading=\"lazy\" \/>                                                            <span class=\"trp-language-item-name\" data-no-translation>Italian<\/span>\r\n                                                                                <\/a>\r\n                                            <a class=\"trp-language-item\" href=\"https:\/\/stadiaorg.com\/fr\/wp-json\/wp\/v2\/pages\/2071#TRPLINKPROCESSED\" title=\"Spanish\">\r\n                            <img decoding=\"async\" src=\"https:\/\/stadiaorg.com\/wp-content\/plugins\/translatepress-multilingual\/assets\/flags\/4x3\/es_ES.svg\" class=\"trp-flag-image\" alt=\"\" role=\"presentation\" loading=\"lazy\" \/>                                                            <span class=\"trp-language-item-name\" data-no-translation>Spanish<\/span>\r\n                                                                                <\/a>\r\n                                                <\/div>\r\n        <\/div>\r\n\r\n    <\/div>\r\n<\/div>\r\n<header class=\"snu-hero\"><div><p class=\"snu-kicker\">STADIA \/ NUMBERS &amp; THE UNIVERSE<\/p><h1 id=\"snu-1title\">Simple rules.<br><em>Surprising forms.<\/em><\/h1><p class=\"snu-lead\">Explore Fibonacci, repeating patterns and fractals. Learn to understand mathematics by looking, experimenting and explaining one step at a time.<\/p><a class=\"snu-button\" href=\"#snu-1method\">Start with the method \u2193<\/a><a class=\"snu-quiet\" href=\"#snu-1fibonacci\">Try the first experiment \u2192<\/a><div class=\"snu-stats\"><div><strong>3<\/strong><span>Interactive experiments<\/span><\/div><div><strong>5<\/strong><span>Steps to understanding<\/span><\/div><\/div><\/div><figure><svg viewBox=\"0 0 500 500\" role=\"img\" aria-label=\"A mathematical pattern of points placed at a constant angle\"><rect width=\"500\" height=\"500\" fill=\"#211848\"\/><g><circle cx=\"240.414\" cy=\"258.781\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"251.607\" cy=\"231.686\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"263.7\" cy=\"267.869\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"224.397\" cy=\"245.471\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"274.527\" cy=\"234.398\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"241.733\" cy=\"280.752\" r=\"4.1\" 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cx=\"368.061\" cy=\"92.103\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"269.646\" cy=\"446.604\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"102.391\" cy=\"118.017\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"448.422\" cy=\"247.607\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"104.996\" cy=\"386.088\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"265.027\" cy=\"51.281\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"373.414\" cy=\"407.013\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"52.521\" cy=\"217.52\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"417.908\" cy=\"140.324\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"200.17\" cy=\"444.7\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"155.027\" cy=\"72.405\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"440.39\" cy=\"316.939\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"64.011\" cy=\"329.412\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"333.673\" cy=\"65.431\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"313.107\" cy=\"443.012\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"72.727\" cy=\"150.103\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"448.599\" cy=\"203.821\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"134.519\" cy=\"418.544\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"221.244\" cy=\"47.305\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"408.44\" cy=\"380.298\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"44.745\" cy=\"260.969\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"394.227\" cy=\"102.971\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"242.954\" cy=\"456.248\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"115.611\" cy=\"92.849\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"455.654\" cy=\"275.149\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"81.037\" cy=\"370.61\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"293.202\" cy=\"46.536\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"355.789\" cy=\"429.56\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"50.316\" cy=\"188.939\" r=\"4.1\" fill=\"#d7bb79\"\/><circle cx=\"438.852\" cy=\"159.966\" r=\"4.1\" fill=\"#b3a0f1\"\/><circle cx=\"171.412\" cy=\"444.33\" r=\"4.1\" fill=\"#b3a0f1\"\/><\/g><\/svg><figcaption>One placement rule. Hundreds of points. A mathematical model, not a photograph of nature.<\/figcaption><\/figure><\/header>\r\n<nav class=\"snu-nav\" aria-label=\"Explore the mathematics\"><a href=\"#snu-1method\">How to understand<\/a><a href=\"#snu-1fibonacci\">Fibonacci<\/a><a href=\"#snu-1patterns\">Repeating patterns<\/a><a href=\"#snu-1fractals\">Fractals<\/a><a href=\"#snu-1practice\">Practise the method<\/a><\/nav>\r\n<div class=\"snu-intro\"><div><p class=\"snu-kicker\">MATHEMATICS FOR READING THE WORLD<\/p><h2>Begin with a question.<br>Give the symbols meaning.<\/h2><\/div><div><p>A formula expresses a relationship. Understanding it means knowing what its symbols stand for, how its parts work together and when the relationship applies.<\/p><p>This first chapter starts with patterns you can build yourself. Experiments help you form a conjecture; explanations and proofs help you decide whether it is true.<\/p><\/div><\/div>\r\n<section class=\"snu-section\" id=\"snu-1method\"><p class=\"snu-kicker\">THE METHOD \/ FIVE SMALL STEPS<\/p><h2>How to understand mathematics<\/h2><div class=\"snu-method\">\r\n<article><span>01<\/span><h3>Observe<\/h3><p>Describe what changes and what stays the same. Start with words and a drawing.<\/p><\/article>\r\n<article><span>02<\/span><h3>Try a small case<\/h3><p>Use a few numbers or shapes. Write down each step so you can check it.<\/p><\/article>\r\n<article><span>03<\/span><h3>Find a rule<\/h3><p>Ask which operation connects the examples. A pattern suggests a rule; it does not prove one.<\/p><\/article>\r\n<article><span>04<\/span><h3>Use symbols<\/h3><p>Give each quantity a name. Read the formula aloud and connect it to your example.<\/p><\/article>\r\n<article><span>05<\/span><h3>Explain and test<\/h3><p>Explain why the rule works. Try to find a counterexample and state its conditions.<\/p><\/article><\/div><\/section>\r\n<section class=\"snu-section\" id=\"snu-1fibonacci\"><div class=\"snu-section-head\"><span class=\"snu-number\">01<\/span><div><p class=\"snu-kicker\">FIBONACCI \/ BUILDING A SEQUENCE<\/p><h2>Two numbers make the next.<\/h2><p>Start with 0 and 1. Add the last two numbers to obtain the next: 0, 1, 1, 2, 3, 5, 8\u2026<\/p><\/div><\/div><div class=\"snu-lab\"><div class=\"snu-lab-copy\"><h3>Read the rule<\/h3><div class=\"snu-formula\" translate=\"no\" data-no-translation data-no-dynamic-translation>F\u2080 = 0, F\u2081 = 1<br>F\u2099 = F\u2099\u208b\u2081 + F\u2099\u208b\u2082 &nbsp; (n \u2265 2)<\/div><p>The small index below F tells you the position in the sequence. F\u2099 means \u201cthe Fibonacci number at position n\u201d, not F multiplied by n.<\/p><details><summary>Why do the ratios approach the golden ratio?<\/summary><p>For positive consecutive terms, divide the recurrence by the previous term. If their ratios approach a positive limit r, then r = 1 + 1\/r. This gives r\u00b2 = r + 1, whose positive solution is (1 + \u221a5)\/2. A full proof must also establish that the ratios converge.<\/p><\/details><p class=\"snu-note\">These ratios approach the golden ratio; a finite ratio of integer Fibonacci numbers is not exactly equal to it. Not every spiral in nature follows Fibonacci.<\/p><\/div><div class=\"snu-experiment\"><div class=\"snu-controls\" hidden data-snu-controls><label for=\"snu-1terms\">How many Fibonacci numbers to display <output data-snu-terms-value>10<\/output><\/label><input id=\"snu-1terms\" data-snu-terms type=\"range\" min=\"4\" max=\"16\" value=\"10\"><\/div><div class=\"snu-sequence\" data-snu-sequence translate=\"no\" data-no-translation data-no-dynamic-translation>0 \u00b7 1 \u00b7 1 \u00b7 2 \u00b7 3 \u00b7 5 \u00b7 8 \u00b7 13 \u00b7 21 \u00b7 34<\/div><svg data-snu-bars viewBox=\"0 0 600 280\" role=\"img\" aria-label=\"Bars comparing Fibonacci numbers\"><rect x=\"40\" y=\"230\" width=\"70\" height=\"20\" fill=\"#bbaae9\"\/><rect x=\"130\" y=\"210\" width=\"70\" height=\"40\" fill=\"#a08bdb\"\/><rect x=\"220\" y=\"185\" width=\"70\" height=\"65\" fill=\"#8971ce\"\/><rect x=\"310\" y=\"145\" width=\"70\" height=\"105\" fill=\"#7158bd\"\/><rect x=\"400\" y=\"80\" width=\"70\" height=\"170\" fill=\"#6048ad\"\/><\/svg><div class=\"snu-readouts\"><div><span>Add the two previous numbers<\/span><strong data-snu-addition translate=\"no\" data-no-translation data-no-dynamic-translation>13 + 21 = 34<\/strong><\/div><div><span>Ratio of consecutive Fibonacci numbers<\/span><strong data-snu-ratio translate=\"no\" data-no-translation data-no-dynamic-translation>34 \/ 21 \u2248 1.619048<\/strong><\/div><\/div><p class=\"snu-question\">Before moving the slider: what will the next term be? Explain how you found it.<\/p><\/div><\/div><\/section>\r\n<section class=\"snu-section\" id=\"snu-1patterns\"><div class=\"snu-section-head\"><span class=\"snu-number\">02<\/span><div><p class=\"snu-kicker\">REPETITION \/ SAME RULE, NEW POSITION<\/p><h2>Build a pattern from one tile.<\/h2><p>A periodic pattern repeats after a fixed shift. Move the same tile horizontally and vertically to create a grid.<\/p><\/div><\/div><div class=\"snu-lab\"><div class=\"snu-lab-copy\"><h3>Separate the tile from the rule.<\/h3><p>The tile is the object being repeated. In geometry, a translation moves every point by the same displacement. The tile keeps its orientation and size.<\/p><div class=\"snu-formula\" translate=\"no\" data-no-translation data-no-dynamic-translation>(x, y) \u2192 (x + a, y + b)<\/div><p>Here a and b are the horizontal and vertical shifts. Rotating alternate tiles adds a second rule and can change which symmetries the pattern has.<\/p><details><summary>Try it with paper<\/summary><p>Draw an asymmetric tile, such as a diagonal with a dot at one end. Copy it into four squares. Rotate alternate copies by 180\u00b0. Which movements make the whole pattern match itself?<\/p><\/details><p class=\"snu-note\">A repeating grid and a fractal use different kinds of repetition: a grid repeats across positions; an exactly self-similar fractal repeats across scales.<\/p><\/div><div class=\"snu-experiment\"><div class=\"snu-controls\" hidden data-snu-controls><label for=\"snu-1tiles\">Number of tiles in each row <output data-snu-tiles-value>6<\/output><\/label><input id=\"snu-1tiles\" data-snu-tiles type=\"range\" min=\"3\" max=\"10\" value=\"6\"><label class=\"snu-check\"><input type=\"checkbox\" data-snu-rotate> Rotate alternate tiles by 180\u00b0<\/label><\/div><svg data-snu-pattern viewBox=\"0 0 480 480\" role=\"img\" aria-label=\"A grid of repeated diagonal tiles\"><g transform=\"translate(0 0)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(80 0)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(160 0)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(240 0)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(320 0)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(400 0)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(0 80)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(80 80)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(160 80)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(240 80)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(320 80)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(400 80)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(0 160)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(80 160)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(160 160)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(240 160)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(320 160)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(400 160)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(0 240)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(80 240)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(160 240)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(240 240)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(320 240)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(400 240)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(0 320)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(80 320)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(160 320)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(240 320)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(320 320)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(400 320)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(0 400)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(80 400)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(160 400)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(240 400)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(320 400)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><g transform=\"translate(400 400)\"><rect width=\"80\" height=\"80\" fill=\"#f1ebff\" stroke=\"#ddd0f0\"\/><path d=\"M12 68 L68 12\" stroke=\"#7255bd\" stroke-width=\"7\"\/><circle cx=\"68\" cy=\"12\" r=\"9\" fill=\"#c88b4c\"\/><\/g><\/svg><p><span>Total tiles:<\/span> <strong data-snu-tile-count translate=\"no\" data-no-translation data-no-dynamic-translation>36<\/strong><\/p><p class=\"snu-question\">Does doubling the number of tiles per side double the total, or multiply it by four? Explain using rows and columns.<\/p><\/div><\/div><\/section>\r\n<section class=\"snu-section\" id=\"snu-1fractals\"><div class=\"snu-section-head\"><span class=\"snu-number\">03<\/span><div><p class=\"snu-kicker\">FRACTALS \/ REPEAT AT A SMALLER SCALE<\/p><h2>One triangle becomes three.<\/h2><p>Start with a filled equilateral triangle. Join the midpoints of its sides, remove the central triangle, and repeat on the three remaining triangles.<\/p><\/div><\/div><div class=\"snu-lab\"><div class=\"snu-lab-copy\"><h3>The Sierpi\u0144ski construction<\/h3><p>Each step leaves three copies with half the previous side length. After n steps, there are 3\u207f small triangles, and the remaining area is (3\/4)\u207f of the starting area.<\/p><div class=\"snu-formula\" translate=\"no\" data-no-translation data-no-dynamic-translation>N\u2099 = 3\u207f<br>A\u2099 \/ A\u2080 = (3\/4)\u207f<\/div><p>The screen shows a finite stage. The mathematical fractal is the limit of this construction as the number of steps grows without bound.<\/p><details><summary>Why does 75% of the area remain after each step?<\/summary><p>The midpoints divide a triangle into four congruent triangles. Removing one leaves three of the four equal areas. Repeating multiplies the remaining area by 3\/4 again at each step.<\/p><\/details><p class=\"snu-note\">Natural branching and irregular surfaces can resemble fractals over limited scales. They do not contain infinitely small copies of themselves.<\/p><\/div><div class=\"snu-experiment\"><div class=\"snu-controls\" hidden data-snu-controls><label for=\"snu-1depth\">Construction step <output data-snu-depth-value>0<\/output><\/label><input id=\"snu-1depth\" data-snu-depth type=\"range\" min=\"0\" max=\"6\" value=\"0\"><\/div><svg data-snu-fractal viewBox=\"0 0 600 530\" role=\"img\" aria-label=\"A finite stage of the Sierpi\u0144ski triangle\"><polygon points=\"300,20 20,505 580,505\" fill=\"#7051be\"\/><\/svg><div class=\"snu-readouts\"><div><span>Number of small triangles<\/span><strong data-snu-triangles translate=\"no\" data-no-translation data-no-dynamic-translation>1<\/strong><\/div><div><span>Percentage of the original area retained<\/span><strong data-snu-area translate=\"no\" data-no-translation data-no-dynamic-translation>100%<\/strong><\/div><\/div><p class=\"snu-question\">Predict the number of triangles at the next step. Why can their count grow while their total area decreases?<\/p><\/div><\/div><\/section>\r\n<section class=\"snu-practice\" id=\"snu-1practice\"><p class=\"snu-kicker\">PRACTISE \/ EXPLAIN THE IDEA YOURSELF<\/p><h2>Understanding grows through practice.<\/h2><div class=\"snu-practice-grid\"><article><h3>Change representation<\/h3><p>Write a rule as a sentence, a table, a drawing and a formula. What does each representation make easier to see?<\/p><\/article><article><h3>Look for a counterexample<\/h3><p>\u201cAll increasing sequences double at each step.\u201d Fibonacci already gives a counterexample: 2 is followed by 3. One counterexample disproves a universal claim.<\/p><\/article><article><h3>Distinguish seeing from proving<\/h3><p>A thousand matching examples can support a conjecture. A proof explains why a statement holds for every case covered by its assumptions.<\/p><\/article><\/div><details><summary>A short exercise with an explanation<\/summary><p>At step 2, the Sierpi\u0144ski construction has 9 triangles. At step 3 it has 27. The remaining area changes from 9\/16 to 27\/64 of the original. Each surviving triangle is replaced by three smaller ones, while one quarter of its area is removed.<\/p><\/details><\/section>\r\n<footer class=\"snu-footer\"><h3>Further reading<\/h3><a href=\"https:\/\/mathworld.wolfram.com\/FibonacciNumber.html\" target=\"_blank\" rel=\"noopener noreferrer\">Fibonacci numbers \u00b7 Wolfram MathWorld \u2197<\/a><a href=\"https:\/\/mathworld.wolfram.com\/GoldenRatio.html\" target=\"_blank\" rel=\"noopener noreferrer\">Golden ratio \u00b7 Wolfram MathWorld \u2197<\/a><a href=\"https:\/\/mathworld.wolfram.com\/SierpinskiSieve.html\" target=\"_blank\" rel=\"noopener noreferrer\">Sierpi\u0144ski construction \u00b7 Wolfram MathWorld \u2197<\/a><p>This chapter offers selected mathematical models and exercises. It does not claim that every natural form follows the same rule.<\/p><\/footer>\r\n<noscript><p>The explanations remain available. Enable JavaScript to change the experimental controls.<\/p><\/noscript>\r\n<\/section>\r\n\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":44,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2071","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/stadiaorg.com\/fr\/wp-json\/wp\/v2\/pages\/2071","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/stadiaorg.com\/fr\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/stadiaorg.com\/fr\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/stadiaorg.com\/fr\/wp-json\/wp\/v2\/users\/44"}],"replies":[{"embeddable":true,"href":"https:\/\/stadiaorg.com\/fr\/wp-json\/wp\/v2\/comments?post=2071"}],"version-history":[{"count":1,"href":"https:\/\/stadiaorg.com\/fr\/wp-json\/wp\/v2\/pages\/2071\/revisions"}],"predecessor-version":[{"id":2072,"href":"https:\/\/stadiaorg.com\/fr\/wp-json\/wp\/v2\/pages\/2071\/revisions\/2072"}],"wp:attachment":[{"href":"https:\/\/stadiaorg.com\/fr\/wp-json\/wp\/v2\/media?parent=2071"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}