{"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 class=\"wp-block-paragraph\">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 class=\"wp-block-paragraph\">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 class=\"wp-block-paragraph\">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 class=\"wp-block-paragraph\">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 class=\"wp-block-paragraph\">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 class=\"wp-block-paragraph\">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 class=\"wp-block-paragraph\">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 class=\"wp-block-paragraph\">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=\"yop-poll-container\" data-yop-poll-id=\"57\"><script type=\"application\/json\" data-yop-poll-init>{\"poll\":{\"id\":\"57\",\"name\":\"20-Dec-2025\",\"template\":\"3\",\"template_base\":\"classic\",\"skin_base\":\"red-v1\",\"status\":\"published\",\"meta_data\":{\"style\":{\"poll\":{\"backgroundColor\":\"#b70004\",\"borderSize\":0,\"borderStyle\":\"none\",\"borderColor\":\"#b70004\",\"borderRadius\":0,\"paddingLeftRight\":10,\"paddingTopBottom\":10},\"questions\":{\"paddingLeftRight\":0,\"paddingTopBottom\":10,\"textColor\":\"#ffffff\",\"textSize\":14,\"textWeight\":\"normal\",\"textAlign\":\"center\"},\"answers\":{\"paddingLeftRight\":0,\"paddingTopBottom\":0,\"textColor\":\"#ffffff\",\"textSize\":14,\"textWeight\":\"normal\",\"skin\":\"minimal\",\"colorScheme\":\"black\"},\"buttons\":{\"backgroundColor\":\"#ffffff\",\"borderSize\":1,\"borderStyle\":\"solid\",\"borderColor\":\"#000000\",\"borderRadius\":0,\"paddingLeftRight\":10,\"paddingTopBottom\":5,\"textColor\":\"#000000\",\"textSize\":14,\"textWeight\":\"normal\",\"align\":\"center\"},\"errors\":{\"borderLeftColorForSuccess\":\"#008000\",\"borderLeftColorForError\":\"#ff0000\",\"borderLeftSize\":10,\"paddingTopBottom\":0,\"textColor\":\"#000000\",\"textSize\":14,\"textWeight\":\"normal\"},\"captcha\":[],\"custom\":{\"css\":\"\"}},\"options\":{\"poll\":{\"voteButtonLabel\":\"&gt;&gt;&gt;\",\"showResultsLink\":\"no\",\"resultsLabelText\":\"Results\",\"showTotalVotes\":\"no\",\"showTotalAnswers\":\"no\",\"startDateOption\":\"now\",\"startDateCustom\":\"\",\"endDateOption\":\"never\",\"endDateCustom\":\"\",\"redirectAfterVote\":\"no\",\"redirectUrl\":\"\",\"redirectAfter\":\"\",\"pageLink\":\"\",\"useCaptcha\":\"no\",\"enableGdpr\":\"no\",\"gdprSolution\":\"ask_consent\",\"gdprConsentText\":\"\",\"loadWithAjax\":\"no\",\"notificationMessageLocation\":\"top\"},\"results\":{\"showResultsMoment\":[\"never\"],\"customDateResults\":\"\",\"showResultsTo\":[],\"resultsDetails\":[\"percentages\"],\"backToVoteOption\":\"no\",\"backToVoteCaption\":\"Back to vote\",\"sortResults\":\"as-defined\",\"sortResultsRule\":\"asc\",\"displayResultsAs\":\"bar\"},\"access\":{\"votePermissions\":[\"wordpress\"]}}}},\"elements\":[{\"id\":\"57\",\"poll_id\":\"57\",\"etext\":\"Choose an option:\",\"etype\":\"question-text\",\"status\":\"active\",\"sorder\":\"1\",\"meta_data\":{\"allowOtherAnswers\":\"no\",\"otherAnswersLabel\":\"Other\",\"addOtherAnswers\":\"no\",\"displayOtherAnswersInResults\":\"no\",\"resultsColorForOtherAnswers\":\"#000000\",\"allowMultipleAnswers\":\"no\",\"multipleAnswersMinim\":\"1\",\"multipleAnswersMaxim\":\"1\",\"answersDisplay\":\"horizontal\",\"answersColumns\":\"\",\"answersSort\":\"as-defined\"},\"subelements\":[{\"id\":\"596\",\"poll_id\":\"57\",\"element_id\":\"57\",\"stext\":\"a\",\"stype\":\"text\",\"status\":\"active\",\"sorder\":\"1\",\"meta_data\":{\"makeDefault\":\"0\",\"makeLink\":\"0\",\"link\":\"\",\"resultsColor\":\"#000000\"}},{\"id\":\"597\",\"poll_id\":\"57\",\"element_id\":\"57\",\"stext\":\"b\",\"stype\":\"text\",\"status\":\"active\",\"sorder\":\"2\",\"meta_data\":{\"makeDefault\":\"0\",\"makeLink\":\"0\",\"link\":\"\",\"resultsColor\":\"#000000\"}},{\"id\":\"598\",\"poll_id\":\"57\",\"element_id\":\"57\",\"stext\":\"c\",\"stype\":\"text\",\"status\":\"active\",\"sorder\":\"3\",\"meta_data\":{\"makeDefault\":\"0\",\"makeLink\":\"0\",\"link\":\"\",\"resultsColor\":\"#000000\"}},{\"id\":\"602\",\"poll_id\":\"57\",\"element_id\":\"57\",\"stext\":\"d\",\"stype\":\"text\",\"status\":\"active\",\"sorder\":\"4\",\"meta_data\":{\"makeDefault\":\"0\",\"makeLink\":\"0\",\"link\":\"\",\"resultsColor\":\"#000000\"}},{\"id\":\"613\",\"poll_id\":\"57\",\"element_id\":\"57\",\"stext\":\"e\",\"stype\":\"text\",\"status\":\"active\",\"sorder\":\"5\",\"meta_data\":{\"makeDefault\":\"0\",\"makeLink\":\"0\",\"link\":\"\",\"resultsColor\":\"#000000\"}},{\"id\":\"614\",\"poll_id\":\"57\",\"element_id\":\"57\",\"stext\":\"e\",\"stype\":\"text\",\"status\":\"active\",\"sorder\":\"5\",\"meta_data\":{\"makeDefault\":\"0\",\"makeLink\":\"0\",\"link\":\"\",\"resultsColor\":\"#000000\"}}]}],\"messages\":{\"captcha\":{\"accessibility-alt\":\"Sound icon\",\"accessibility-title\":\"Accessibility option: listen to a question and answer it!\",\"accessibility-description\":\"Type below the [STRONG]answer[\\\/STRONG] to what you hear. Numbers or words:\",\"explanation\":\"Click or touch the [STRONG]ANSWER[\\\/STRONG]\",\"refresh-alt\":\"Refresh\\\/reload icon\",\"refresh-title\":\"Refresh\\\/reload: get new images and accessibility option!\"},\"buttons\":{\"anonymous\":\"Anonymous Vote\",\"wordpress\":\"Sign in with WordPress\",\"facebook\":\"Sign in with Facebook\",\"google\":\"Sign in with Google\"},\"voting\":{\"poll-ended\":\"This poll is no longer accepting votes\",\"poll-not-started\":\"This poll is not accepting votes yet\",\"already-voted-on-poll\":\"Thank you for your vote\",\"invalid-poll\":\"Invalid Poll\",\"no-answers-selected\":\"No answer selected\",\"min-answers-required\":\"At least {min_answers_allowed} answer(s) required\",\"max-answers-required\":\"A max of {max_answers_allowed} answer(s) accepted\",\"no-answer-for-other\":\"No other answer entered\",\"no-value-for-custom-field\":\"{custom_field_name} is required\",\"consent-not-checked\":\"You must agree to our terms and conditions\",\"no-captcha-selected\":\"Captcha is required\",\"not-allowed-by-ban\":\"Vote not allowed\",\"not-allowed-by-block\":\"Vote not allowed\",\"not-allowed-by-limit\":\"Vote not allowed\",\"thank-you\":\"Thank you for your vote\",\"too-many-chars-for-custom-field\":\"Text for {custom_field_name} is too long\"},\"results\":{\"single-vote\":\"vote\",\"multiple-votes\":\"votes\",\"single-answer\":\"answer\",\"multiple-answers\":\"answers\"}},\"date_format\":\"F j, Y\",\"nonce\":\"6cfccd22a0\",\"page_id\":0,\"tracking_id\":\"https:\\\/\\\/control-advent.com\"}<\/script><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><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}]}}