{"id":1653,"date":"2025-12-20T07:59:06","date_gmt":"2025-12-20T06:59:06","guid":{"rendered":"https:\/\/control-advent.com\/?p=1653"},"modified":"2025-12-20T07:59:07","modified_gmt":"2025-12-20T06:59:07","slug":"20-how-to-create-a-giant-triangle-of-sweetness","status":"publish","type":"post","link":"https:\/\/control-advent.com\/?p=1653","title":{"rendered":"20 &#8211; How to create a giant triangle of sweetness?"},"content":{"rendered":"\n<p>Santa is testing the <strong>sweetness-control dynamics <\/strong>of his magical Hot Cocoa Machine. To observe its behaviour, the elves run a long enchanted <strong>Sweetness Testing Strip<\/strong>: the machine will push out a strip of paper at a constant speed, and plot on it the sweetness level that the machine produces, so that the sweetness dynamics get drawn as a shape on the strip, where <strong>1cm width corresponds to the sweetness level of one second.<\/strong><\/p>\n\n\n\n<p>For the first test, elf Bushy pulls the brown lever at full strength for <strong>exactly 1 minute<\/strong>. The machine:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>starts sweetening immediately,<\/li>\n\n\n\n<li>slowly ramps up,<\/li>\n\n\n\n<li>peaks right as Bushy lets the lever go,<\/li>\n\n\n\n<li>then gently ramps back down.<\/li>\n<\/ul>\n\n\n\n<p>On the strip, this appears as a <strong>perfect isosceles triangle<\/strong>, <strong>120 seconds wide<\/strong> and <strong>120 sweetness units tall<\/strong>.<\/p>\n\n\n\n<p>Santa beams: \u201cHo-ho-ho! A triangle of sweetness! Now let\u2019s make it <strong>twice as wide in time and twice as tall in sweetness<\/strong>!\u201d<\/p>\n\n\n\n<p>For the second test, two elves \u2014 Bushy and his sister Shinny \u2014 pull the lever together at full strength for <strong>2 minutes<\/strong>.<\/p>\n\n\n\n<p>The machine is known to behave in a magical but engineer-friendly way, so to be:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Linear<\/strong>: two elves \u2192 twice the sweetness.<\/li>\n\n\n\n<li><strong>Time-invariant<\/strong>: doing it later \u2192 same shape, just later.<\/li>\n<\/ul>\n\n\n\n<p>Knowing this, and knowing that Bushy and Shinny are doing such a tes, will Santa get the giant <strong>double-wide, double-tall<\/strong> triangle he imagines?<\/p>\n\n\n\n<p>a. \ud83c\udf6b\ud83d\udccf Not quite\u2026 He gets something only <strong>120 seconds wide<\/strong> but almost <strong>500 sweetness units tall<\/strong> \u2014 the extra sweetness stacks vertically!<br>b. \u2615\u27a1\ufe0f\u2b05\ufe0f Also no\u2026 It becomes almost <strong>500 seconds wide but <\/strong>only <strong>100 sweetness units tall<\/strong>, because the sweetness spreads out over time.<br>c. \ud83c\udf84\ud83d\udcd0 Yes! A perfect <strong>240 seconds \u00d7 240 sweetness units <\/strong>isosceles triangle \u2014 exactly doubled.<br>d. \u2b50\ud83c\udfd4\ufe0f No \u2014 he gets a <strong>trapezoid<\/strong>! The overlapping triangular responses create a flat top.<br>e. \ud83c\udf46 I don\u2019t care, I only drink unsweetened aubergine juice.<\/p>\n\n\n<div class='bootstrap-yop yop-poll-mc'>\n\t\t\t\t\t\t\t<div class=\"basic-yop-poll-container\" style=\"background-color:#b70004; border:0px; border-style:solid; border-color:#b70004; border-radius:0px; padding:10px 10px;\" data-id=\"57\" data-temp=\"basic\" data-skin=\"minimal\" data-cscheme=\"black\" data-cap=\"0\" data-access=\"wordpress\" data-tid=\"\" data-uid=\"c35fa1f0bfe081bf4238c5ed47a16fd6\" data-pid=\"1653\" data-resdet=\"percentages\" data-show-results-to=\"\" data-show-results-moment=\"never\" data-show-results-only=\"false\" data-show-message=\"true\" data-show-results-as=\"bar\" data-sort-results-by=\"as-defined\" data-sort-results-rule=\"asc\"data-is-ended=\"0\" data-gdpr=\"no\" data-gdpr-sol=\"consent\" data-css=\".basic-yop-poll-container[data-uid] .basic-vote { text-align: center; }\" data-counter=\"0\" data-load-with=\"1\" data-notification-section=\"top\"><div class=\"row\"><div class=\"col-md-12\"><div class=\"basic-inner\"><div class=\"basic-message hide\" style=\"border-left: 10px solid #008000; padding: 0px 10px;\" data-error=\"#ff0000\" data-success=\"#008000\"><p class=\"basic-message-text\" style=\"color:#000000; font-size:14px; font-weight:normal;\"><\/p><\/div><div class=\"basic-overlay hide\"><div class=\"basic-vote-options\"><\/div><div class=\"basic-preloader\"><div class=\"basic-windows8\"><div class=\"basic-wBall basic-wBall_1\"><div class=\"basic-wInnerBall\"><\/div><\/div><div class=\"basic-wBall basic-wBall_2\"><div class=\"basic-wInnerBall\"><\/div><\/div><div class=\"basic-wBall basic-wBall_3\"><div class=\"basic-wInnerBall\"><\/div><\/div><div class=\"basic-wBall basic-wBall_4\"><div class=\"basic-wInnerBall\"><\/div><\/div><div class=\"basic-wBall basic-wBall_5\"><div class=\"basic-wInnerBall\"><\/div><\/div><\/div><\/div><\/div><form class=\"basic-form\"><input type=\"hidden\" name=\"_token\" value=\"ae68b26c6e\" autocomplete=\"off\"><div class=\"basic-elements\"><div class=\"basic-element basic-question basic-question-text-horizontal\" data-id=\"57\" data-uid=\"ee377634ce56bd369fd000af3a88286f\" data-type=\"question\" data-question-type=\"text\" data-allow-multiple=\"no\" data-min=\"1\" data-max=\"1\" data-display=\"horizontal\" data-colnum=\"\" data-display-others=\"no\" data-others-color=\"\" data-others=\"\"><div role=\"heading\" aria-level=\"5\" class=\"basic-question-title\" style=\"color:#ffffff; font-size:14px; font-weight:normal; text-align:center;\">Choose an option:<\/div><ul class=\"basic-answers basic-h-answers\"><li class=\"basic-answer col-xs-6 col-sm-6 col-md-2\" style=\"padding:0px 0px;\" data-id=\"596\" data-type=\"text\" data-vn=\"3\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[596]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[596]\" name=\"answer[57]\" value=\"596\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">a<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 col-md-2\" style=\"padding:0px 0px;\" data-id=\"597\" data-type=\"text\" data-vn=\"0\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[597]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[597]\" name=\"answer[57]\" value=\"597\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">b<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 col-md-2\" style=\"padding:0px 0px;\" data-id=\"598\" data-type=\"text\" data-vn=\"9\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[598]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[598]\" name=\"answer[57]\" value=\"598\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">c<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 col-md-2\" style=\"padding:0px 0px;\" data-id=\"602\" data-type=\"text\" data-vn=\"15\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[602]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[602]\" name=\"answer[57]\" value=\"602\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">d<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 col-md-2\" style=\"padding:0px 0px;\" data-id=\"613\" data-type=\"text\" data-vn=\"0\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[613]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[613]\" name=\"answer[57]\" value=\"613\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">e<\/span><\/label><\/div><\/li><li class=\"basic-answer col-xs-6 col-sm-6 col-md-2\" style=\"padding:0px 0px;\" data-id=\"614\" data-type=\"text\" data-vn=\"0\" data-color=\"#000000\" data-make-link=\"no\" data-link=\"\"><div class=\"basic-answer-content basic-text-horizontal\"><label for=\"answer[614]\" class=\"basic-answer-label\"><input type=\"radio\" id=\"answer[614]\" name=\"answer[57]\" value=\"614\" autocomplete=\"off\"><span class=\"basic-text\" style=\"color: #ffffff; font-size: 14px; font-weight: normal;\" tabindex=\"0\">e<\/span><\/label><\/div><\/li><\/ul><\/div><div class=\"clearfix\"><\/div><\/div><div class=\"basic-vote\"><a href=\"#\" class=\"button basic-vote-button\" role=\"button\" style=\"background:#ffffff; border:1px; border-style: solid; border-color:#000000; border-radius:0px; padding:5px 10px; color:#000000; font-size:14px; font-weight:normal;\">&gt;&gt;&gt;<\/a><\/div><\/form><div class=\"yop-poll-modal\"><div class=\"yop-poll-modal-content\"><div class=\"yop-poll-modal-header\"><span class=\"yop-poll-modal-close\">&times;<\/span><\/div><div class=\"yop-poll-modal-body\"><div class=\"yop-poll-form-group\"><label class=\"yop-poll-user-or-email-label\">Username or Email Address<\/label><input type=\"text\" class=\"yop-poll-user-or-email-input\" autocomplete=\"off\"><\/div><div class=\"yop-poll-form-group\"><label class=\"yop-poll-password-label\">Password<\/label><input type=\"password\" class=\"yop-poll-password-input\" autocomplete=\"off\"><\/div><div class=\"yop-poll-form-group submit\"><button class=\"button button-primary button-large yop-poll-login-button\">Log In<\/button><\/div><div class=\"yop-poll-form-group yop-poll-section-login-options\"><p><a href=\"https:\/\/control-advent.com\/wp-login.php?action=register\" target=\"_blank\">No account? Register here<\/a><\/p><p><a href=\"https:\/\/control-advent.com\/wp-login.php?action=lostpassword\" target=\"_blank\">Forgot password<\/a><\/p><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div>\n\t\t\t\t\t\t<\/div>\n\n\n\n<p><strong><strong>Related control theory topic<\/strong>: linear, time invariant systems<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Santa is testing the sweetness-control dynamics of his magical Hot Cocoa Machine. To observe its behaviour, the elves run a long enchanted Sweetness Testing Strip: the machine will push out a strip of paper at a constant speed, and plot &hellip; <span class=\"more-link\"><a href=\"https:\/\/control-advent.com\/?p=1653\" class=\"readmore\">Continue reading<span class=\"screen-reader-text\">20 &#8211; How to create a giant triangle of sweetness?<\/span><\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":1657,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[22],"tags":[111],"class_list":["post-1653","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-english","tag-2025-en"],"_links":{"self":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/1653","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1653"}],"version-history":[{"count":2,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/1653\/revisions"}],"predecessor-version":[{"id":1659,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/1653\/revisions\/1659"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/media\/1657"}],"wp:attachment":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1653"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1653"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1653"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}