[{"id":247915,"date":"2026-09-17T09:03:51","date_gmt":"2026-09-17T14:03:51","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247915"},"modified":"2026-09-17T10:42:45","modified_gmt":"2026-09-17T15:42:45","slug":"phone-words","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/09\/17\/phone-words\/","title":{"rendered":"Phone words"},"content":{"rendered":"<p>I recently bought a copy of Los Alamos Rolodex, a book displaying business cards from Los Alamos Nation Labs from 1967 to 1978. You can find some examples of the cards <a href=\"https:\/\/clui.org\/collections\/los-alamos-business-cards\/selection-cards\">here<\/a>.<\/p>\n<p>One of the cards in the book is for Eugene Frank, President of B &amp; F Instruments. His card lists his phone number as<\/p>\n<p style=\"padding-left: 40px;\">(215) MErcury 9-7100<\/p>\n<p>At first glance I thought the &#8220;E&#8221; in &#8220;MErcury&#8221; had been accidentally capitalized. Then I realized the intention was that someone would dial ME (i.e. 63) and ingore &#8220;rcury&#8221;. So the phone number would be (215) 639-7100.<\/p>\n<p>This card was from 1968, the height of the space race. Presumably the card was alluding to the Project Mercury or the planet Mercury, or both.<\/p>\n<p>The telephone keypad mapping (ITU E.161 standard) is a poor attempt at making phone numbers more memorable. For starters, there&#8217;s no way to encode 0 or 1 [1]. It&#8217;s unlikely a phone number will correspond to anything memorable unless you come up with the word first and then try to obtain the phone number, such as 800 FLOWERS.<\/p>\n<p>Inserting extra letters, as Mr. Frank did, greatly increases the chances of encoding a phone number as a word. But then you need to denote which letters count and which ones are filler, so there&#8217;s not much advantage. Still, I wanted to play around with it for fun. I found 109 words [1] containing the letters from a telephone encoding of 4228646. (I&#8217;m using the file <code>\/usr\/share\/dict\/words<\/code> on my laptop as my list of words.)<\/p>\n<p>Here are some of the more interesting hits.<\/p>\n<ul>\n<li>semicatholicism<\/li>\n<li>heartburning<\/li>\n<li>gladiatorism<\/li>\n<li>diabetogenic<\/li>\n<li>galactogenetic<\/li>\n<li>xanthocreatinine<\/li>\n<\/ul>\n<p>There are over 30,000 words containing an encoding of the area code 832. One of these is <em>traditional<\/em>, and so I could write my phone number as<\/p>\n<p style=\"padding-left: 40px;\"><code>TraDitionAl semICAThOlIcisM<\/code>.<\/p>\n<p>Another choice for 832 is <em>intercosmic<\/em>, so<\/p>\n<p style=\"padding-left: 40px;\"><code>inTErCosmic GAlaCTOGeNetic<\/code><\/p>\n<p>is another possibility.<\/p>\n<p><em>Galactogentic<\/em> can refer to the production of milk by the mammary glands or to the formation of galaxies (e.g. the Milky Way). Here <em>intercosmic<\/em> fits with the later sense.<\/p>\n<p>I got greedy and tried to find a word containing the full phone number, 8324228646, but didn&#8217;t find anything.<\/p>\n<p>Here&#8217;s my business card in the style of the Los Alamos Rolodex cards, created by Grok, using (832) GlAdiATOrIsM as the phone number.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium\" src=\"https:\/\/www.johndcook.com\/vintage_card.png\" width=\"600\" height=\"366\" \/><\/p>\n<h2>Related posts<\/h2>\n<ul>\n<li class='link'><a href=\"https:\/\/www.johndcook.com\/blog\/2021\/07\/26\/major-memory-keypad\/\">Major memory system telephone keypad<\/a><\/li>\n<li class='link'><a href='https:\/\/www.johndcook.com\/blog\/2023\/11\/17\/phone-number-intel\/'>What can you learn from a phone number?<\/a><\/li>\n<li class='link'><a href='https:\/\/www.johndcook.com\/blog\/2022\/03\/14\/phone-tones-in-musical-notation\/'>Phone tones inn musical notation<\/a><\/li>\n<\/ul>\n<p>[1] Not only are there no letters for 0 and 1, the letters O and I represent digits. At one point in time the first digit of an exchange (the middle three digits) could not be a 0 or 1, but these digits could appear anywhere else.<\/p>\n<p>[2] I initially found a list of 185 words, but some of these were duplicates: a word can represent a phone number in more than one way.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I recently bought a copy of Los Alamos Rolodex, a book displaying business cards from Los Alamos Nation Labs from 1967 to 1978. You can find some examples of the cards here. One of the cards in the book is for Eugene Frank, President of B &amp; F Instruments. His card lists his phone number [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-247915","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"Vintage business cards and phone number encodings adding extra letters.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"John\"\/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/17\/phone-words\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.1.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"John D. Cook | Applied Mathematics Consulting\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"Phone words\" \/>\n\t\t<meta property=\"og:description\" content=\"Vintage business cards and phone number encodings adding extra letters.\" \/>\n\t\t<meta property=\"og:url\" content=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/17\/phone-words\/\" \/>\n\t\t<meta property=\"article:published_time\" content=\"2026-09-17T14:03:51+00:00\" \/>\n\t\t<meta property=\"article:modified_time\" content=\"2026-09-17T15:42:45+00:00\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary\" \/>\n\t\t<meta name=\"twitter:title\" content=\"Phone words\" \/>\n\t\t<meta name=\"twitter:description\" content=\"Vintage business cards and phone number encodings adding extra letters.\" \/>\n\t\t<meta name=\"twitter:image\" content=\"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png\" \/>\n\t\t<!-- All in One SEO -->\n\n","aioseo_head_json":{"title":"Phone words","description":"Vintage business cards and phone number encodings adding extra letters.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/17\/phone-words\/","robots":"max-image-preview:large","keywords":"","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Phone words","og:description":"Vintage business cards and phone number encodings adding extra letters.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/17\/phone-words\/","article:published_time":"2026-09-17T14:03:51+00:00","article:modified_time":"2026-09-17T15:42:45+00:00","twitter:card":"summary","twitter:title":"Phone words","twitter:description":"Vintage business cards and phone number encodings adding extra letters.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247915","title":null,"description":"Vintage business cards and phone number encodings adding extra 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11:34:21","updated":"2026-09-17 15:42:45","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\/category\/uncategorized\/\" title=\"Uncategorized\">Uncategorized<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tPhone words\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog"},{"label":"Uncategorized","link":"https:\/\/www.johndcook.com\/blog\/category\/uncategorized\/"},{"label":"Phone words","link":"https:\/\/www.johndcook.com\/blog\/2026\/09\/17\/phone-words\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247915","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=247915"}],"version-history":[{"count":6,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247915\/revisions"}],"predecessor-version":[{"id":247923,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247915\/revisions\/247923"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=247915"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=247915"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=247915"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":247909,"date":"2026-09-16T11:05:11","date_gmt":"2026-09-16T16:05:11","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247909"},"modified":"2026-09-16T11:05:11","modified_gmt":"2026-09-16T16:05:11","slug":"concentration-ratio","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/09\/16\/concentration-ratio\/","title":{"rendered":"Converting between cosine similarity and concentration ratio"},"content":{"rendered":"<p>I&#8217;ve written three posts on cosine similarity lately. The <a href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/cosine-similarity\/\">first<\/a> looked at interpreting cosine similarity. The <a href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/simple-approximation-for-spherical-cap-area\/\">second<\/a> looked at an approximation related to the first. The <a href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/16\/coffee-milk-latte\/\">third<\/a> looked at how ranking according to cosine similarity works better than cosine similarity itself.<\/p>\n<p>Normalized word vectors are points on a high dimensional sphere, and geometry in high dimensions is counterintuitive. See the first post in this series for an explanation.<\/p>\n<p>The set of points within a given angular distance of a point on a hypersphere is called a <a href=\"https:\/\/www.johndcook.com\/blog\/2023\/08\/09\/hypersphere-cap\/\">spherical cap<\/a>. The ratio of the area of this spherical cap to that of the whole sphere is called <strong>cap fraction<\/strong> or <strong>concentration ratio<\/strong>. Concentration ratio explains why a modest cosine similarity value corresponds to a tiny portion of the area of the sphere and should be interpreted as a close match.<\/p>\n<p>For this post, I wanted to share a plot of concentration ratio as a function of cosine similarity.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium\" src=\"https:\/\/www.johndcook.com\/concentration_ratio.png\" width=\"480\" height=\"360\" \/><\/p>\n<p>This shows that moderate values of cosine similarity correspond to infinitesimal concentration ratios. And yet, as the third post linked at the top showed, word vectors are very unevenly distributed, and even extremely small regions of the sphere can contain multiple word vectors.<\/p>\n<p>I only included cosine similarity values up to 0.8 because the function plotted above takes a nosedive for larger values, even on a logarithmic scale.<\/p>\n<p>Here&#8217;s the Python code to make the plot, using the function <code>cap_fraction<\/code> from <a href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/simple-approximation-for-spherical-cap-area\/\">here<\/a>.<\/p>\n<pre>\r\ns = np.linspace(0, 0.8, 500)\r\nplt.plot(s, cap_fraction(np.acos(s), 200))\r\nplt.yscale(\"log\")\r\nplt.xlabel(\"cosine similarity\")\r\nplt.ylabel(\"concentration ratio\")\r\nplt.show()\r\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>I&#8217;ve written three posts on cosine similarity lately. The first looked at interpreting cosine similarity. The second looked at an approximation related to the first. The third looked at how ranking according to cosine similarity works better than cosine similarity itself. Normalized word vectors are points on a high dimensional sphere, and geometry in high [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[224,146],"class_list":["post-247909","post","type-post","status-publish","format-standard","hentry","category-math","tag-geometry","tag-machine-learning"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"Concentration ratio may be easier to understand than cosine similarity, but it&#039;s still a little tricky to understand.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"John\"\/>\n\t<meta name=\"keywords\" content=\"geometry,machine learning\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/16\/concentration-ratio\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.1.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"John D. Cook | Applied Mathematics Consulting\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"Converting between cosine similarity and concentration ratio\" \/>\n\t\t<meta property=\"og:description\" content=\"Concentration ratio may be easier to understand than cosine similarity, but it&#039;s still a little tricky to understand.\" \/>\n\t\t<meta property=\"og:url\" content=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/16\/concentration-ratio\/\" \/>\n\t\t<meta property=\"article:published_time\" content=\"2026-09-16T16:05:11+00:00\" \/>\n\t\t<meta property=\"article:modified_time\" content=\"2026-09-16T16:05:11+00:00\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary\" \/>\n\t\t<meta name=\"twitter:title\" content=\"Converting between cosine similarity and concentration ratio\" \/>\n\t\t<meta name=\"twitter:description\" content=\"Concentration ratio may be easier to understand than cosine similarity, but it&#039;s still a little tricky to understand.\" \/>\n\t\t<meta name=\"twitter:image\" content=\"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png\" \/>\n\t\t<!-- All in One SEO -->\n\n","aioseo_head_json":{"title":"Converting between cosine similarity and concentration ratio","description":"Concentration ratio may be easier to understand than cosine similarity, but it's still a little tricky to understand.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/16\/concentration-ratio\/","robots":"max-image-preview:large","keywords":"geometry,machine learning","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Converting between cosine similarity and concentration ratio","og:description":"Concentration ratio may be easier to understand than cosine similarity, but it's still a little tricky to understand.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/16\/concentration-ratio\/","article:published_time":"2026-09-16T16:05:11+00:00","article:modified_time":"2026-09-16T16:05:11+00:00","twitter:card":"summary","twitter:title":"Converting between cosine similarity and concentration ratio","twitter:description":"Concentration ratio may be easier to understand than cosine similarity, but it's still a little tricky to understand.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247909","title":null,"description":"Concentration ratio may be easier to understand than cosine similarity, but it's still a little tricky to 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15:28:04","updated":"2026-09-16 18:00:59","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\/category\/math\/\" title=\"Math\">Math<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tConverting between cosine similarity and concentration 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+ milk \u2260 latte"},"content":{"rendered":"<p><a href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/cosine-similarity\/\">Yesterday<\/a> I wrote about the canonical example of how vector embeddings of words add:<\/p>\n<p style=\"padding-left: 40px;\">\u201cking\u201d \u2212 \u201cman\u201d + \u201cwoman\u201d \u2248 \u201cqueen\u201d<\/p>\n<p>This should be interpreted as saying that the word vector for\u00a0<em>king<\/em>, minus the word vector for\u00a0<em>man<\/em>, plus the word vector for\u00a0<em>woman<\/em>, is in some sense close to the word vector for\u00a0<em>queen<\/em>.<\/p>\n<p>This post will look at another example. Is the expression<\/p>\n<p style=\"padding-left: 40px;\">&#8220;coffee&#8221; + &#8220;milk&#8221; \u2248 &#8220;latte&#8221;<\/p>\n<p>true in some sense?<\/p>\n<h2>Notation<\/h2>\n<p>In this post I will use &#8220;foo&#8221; to mean the vector embedding of the word\u00a0<em>foo<\/em>.<\/p>\n<h2>Coffee + milk<\/h2>\n<p>The cosine similarity between &#8220;coffee&#8221; + &#8220;milk&#8221; and &#8220;latte&#8221; is about 0.63. And for reasons given in the previous post, this is a large value of cosine similarity. But there are 11 words that are more similar to &#8220;milk&#8221; + &#8220;coffee&#8221; than &#8220;latte&#8221;. Here are the top 12 matches in order.<\/p>\n<ol>\n<li>coffee<\/li>\n<li>milk<\/li>\n<li>tea<\/li>\n<li>drink<\/li>\n<li>chocolate<\/li>\n<li>cream<\/li>\n<li>breakfast<\/li>\n<li>ice<\/li>\n<li>beer<\/li>\n<li>vanilla<\/li>\n<li>starbucks<\/li>\n<li>latte<\/li>\n<\/ol>\n<p>There are two questions to resolve. First, why isn&#8217;t\u00a0<em>latte<\/em> one of the closest words? Second, why is the cosine similarity large even though\u00a0<em>latte<\/em> is not one of the best matches?<\/p>\n<h2>Concept arithmetic<\/h2>\n<p>When word vector arithmetic works, as in the king and queen example, the vectors combine <em>concepts<\/em>. If you replace the male gender component of <em>king<\/em> with a female component, you get a vector close to the vector for\u00a0<em>queen<\/em>.<\/p>\n<p>But when you add the vectors for\u00a0<em>milk<\/em> and\u00a0<em>coffee<\/em>, you&#8217;re not adding concepts, you&#8217;re adding ingredients.<\/p>\n<p>The concepts of\u00a0<em>milk<\/em> and\u00a0<em>coffee<\/em> are similar in that they&#8217;re both common beverages, as are tea and even beer. A latte is a beverage, but it&#8217;s not as common as milk, coffee, tea, or beer.<\/p>\n<h2>Extremely uneven distribution<\/h2>\n<p>If you divide word vectors by their norm, you get a point on a high-dimensional sphere. In the case of the glove-twitter-200 vector embedding, you get a point on a sphere in 200 dimensions. As explained in the earlier post, a fairly large cosine similarity corresponds to a tiny portion of the sphere&#8217;s surface area.<\/p>\n<p>In the example of \u201cking\u201d \u2212 \u201cman\u201d + \u201cwoman\u201d, the vector &#8220;queen&#8221; is the closest match (except for &#8220;king&#8221; itself).<\/p>\n<p>But there are a lot of words whose vectors are within a tiny region around &#8220;coffee&#8221; + &#8220;milk&#8221;. And by tiny, I mean a region that accounts for a proportion of the sphere on the order of 10<sup>\u221223<\/sup>.<\/p>\n<p>The glove-twitter-200 vector list contains vectors for 1.2 million words. If these vectors were roughly evenly distributed on the sphere when normalized, you&#8217;d expect each patch representing 10<sup>\u22126<\/sup> of the sphere to contain about a word or two. You wouldn&#8217;t expect a patch taking up 10<sup>\u221212 <\/sup>of the sphere to contain more than one word, and you certainly wouldn&#8217;t expect a patch taking up 10<sup>\u221223 <\/sup>of the sphere to contain 12 words [1].<\/p>\n<h2>Rank order<\/h2>\n<p>Rank order based on cosine similarity is more robust than cosine similarity itself. This is an example of a phenomenon that occurs regularly: a metric whose values are dubious might still rank things well. Naive Bayes is another example. It naively computes probabilities in a way that is blatantly wrong, and yet ranking things by these spurious probabilities works well in some cases.<\/p>\n<p>The cosine similarity between \u201cking\u201d \u2212 \u201cman\u201d + \u201cwoman\u201d and &#8220;queen&#8221; is roughly the same as the cosine similarity between &#8220;coffee&#8221; + &#8220;milk&#8221; and &#8220;latte.&#8221; But in the former example, rank order picks out\u00a0<em>queen<\/em> as the best match; rank order works like you&#8217;d expect, because you&#8217;re working with attributes that can be decomposed.<\/p>\n<h2>Dog + infant = puppy?<\/h2>\n<p>I wouldn&#8217;t be surprised if the Anglo-Saxon word for\u00a0<em>puppy<\/em> was something like\u00a0<em>dogchild<\/em>. The language was full of colorful compound words, such as <em>hronrad<\/em> (&#8220;whale-road&#8221;) for the sea and <em>nosethyrl<\/em> (&#8220;nose-hole&#8221;) for nostril.<\/p>\n<p>Here are the top ten matches for &#8220;dog&#8221; + &#8220;infant&#8221; along with their cosine similarities.<\/p>\n<ol>\n<li>dog, 0.819<\/li>\n<li>infant, 0.809<\/li>\n<li>toddler, 0.734<\/li>\n<li>dogs, 0.697<\/li>\n<li>puppy, 0.688<\/li>\n<li>cat, 0.682<\/li>\n<li>pet, 0.676<\/li>\n<li>child, 0.671<\/li>\n<li>newborn, 0.670<\/li>\n<li>baby, 0.650<\/li>\n<\/ol>\n<p>This shows that &#8220;puppy&#8221; is close to &#8220;dog&#8221; + &#8220;infant&#8221;, both in terms of cosine similarity and rank order, though it&#8217;s not the closet.<\/p>\n<p>This also shows that you have to take the addition of word vectors with a grain of salt. It&#8217;s no surprise that\u00a0<em>puppy<\/em> was a good match, but it&#8217;s surprising that\u00a0<em>cat<\/em> is nearly as good.<\/p>\n<p>[1] I poked around a little to get an idea just how unevenly words are distributed. The closest pair of words is <em>jajaja<\/em> and <em>jajajaja<\/em> with a cosine similarity of 0.993. The most isolated word, meaning the word whose nearest neighbor is furthest away, the the Thai word <span lang=\"th\">\u0e40\u0e04\u0e22\u0e44\u0e2b\u0e21<\/span>. It&#8217;s nearest neighbor is the Russian word <span lang=\"ru\">\u0431\u043e\u043b\u044c<\/span> with a cosine similarity of 0.283.<\/p>\n<p>The glove-twitter-200 vectors were created from a corpus that is about half English and about other languages and strings of symbols that are not words in any language. Presumably <span lang=\"th\">\u0e40\u0e04\u0e22\u0e44\u0e2b\u0e21<\/span> would have a much closer neighbor in a corpus containing more Thai words.<\/p>\n<p>I didn&#8217;t search the entire corpus, only the 50,000 most frequently occurring vectors, because a full search would require running an <em>O<\/em>(<em>N<\/em>\u00b2) search with <em>N<\/em> = 1,200,000.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Yesterday I wrote about the canonical example of how vector embeddings of words add: \u201cking\u201d \u2212 \u201cman\u201d + \u201cwoman\u201d \u2248 \u201cqueen\u201d This should be interpreted as saying that the word vector for\u00a0king, minus the word vector for\u00a0man, plus the word vector for\u00a0woman, is in some sense close to the word vector for\u00a0queen. This post will [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[17],"tags":[146],"class_list":["post-247906","post","type-post","status-publish","format-standard","hentry","category-statistics","tag-machine-learning"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"Subtle aspects of adding word vectors. Does coffee + milk = latte? 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We have<\/p>\n<p style=\"padding-left: 40px;\"><em>F<\/em><sub><em>n<\/em><\/sub> <em>F<\/em><sub><em>n<\/em>+1<\/sub> <em>F<\/em><sub><em>n<\/em>+2<\/sub> <em>F<\/em><sub><em>n<\/em>+3<\/sub> = <em>m<\/em>(<em>m<\/em> + 1)<\/p>\n<p>where <em>m<\/em> equals<\/p>\n<p style=\"padding-left: 40px;\"><em>F<\/em><sub><em>n<\/em>+1<\/sub> <em>F<\/em><sub><em>n<\/em>+2<\/sub><\/p>\n<p>if <em>n<\/em> is odd and<\/p>\n<p style=\"padding-left: 40px;\"><em>F<\/em><sub><em>n<\/em><\/sub> <em>F<\/em><sub><em>n<\/em>+3<\/sub><\/p>\n<p>if <em>n<\/em> is even.<\/p>\n<p>In the example at the top, 3 is the 4th Fibonacci number, so <em>n<\/em> = 4. Since 4 is even,\u00a0<em>m<\/em> is the product of the 4th and 7th Fibonacci numbers, i.e.\u00a0<em>m<\/em> = 3 \u00d7 13 = 39.<\/p>\n<h2>More Fibonacci posts<\/h2>\n<ul>\n<li class='link'><a href='https:\/\/www.johndcook.com\/blog\/2018\/07\/13\/fibonacci-meets-pythagoras\/'>Fibonacci meets Pythagoras<\/a><\/li>\n<li class='link'><a href='https:\/\/www.johndcook.com\/blog\/2026\/02\/05\/fibonacci-certificate\/'>Certified Fibonacci numbers<\/a><\/li>\n<li class='link'><a href='https:\/\/www.johndcook.com\/blog\/2025\/10\/17\/trig-fibonacci\/'>Turning trig identities into Fibonacci identities<\/a><\/li>\n<\/ul>\n<p>[1] K. B. Subramaniam. On a link between Triangular and Fibonacci numbers. The Mathematical Gazette, Vol. 103, No. 558 (November 2019), p. 489.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The product of four consecutive Fibonacci numbers equals the product of two consecutive integers. For example, 3 \u00d7 5 \u00d7 8 \u00d7 13 = 39 \u00d7 40. I ran across this theorem in a note [1] that says &#8220;The product of any four consecutive Fibonacci numbers is twice a triangular number.&#8221; Since triangular numbers have [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[94],"class_list":["post-247901","post","type-post","status-publish","format-standard","hentry","category-math","tag-number-theory"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"The product of four consecutive Fibonacci numbers equals the product of two consecutive integers.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"John\"\/>\n\t<meta name=\"keywords\" content=\"number theory\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/16\/fibonacci-product\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.1.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"John D. 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Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Fibonacci product","og:description":"The product of four consecutive Fibonacci numbers equals the product of two consecutive integers.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/16\/fibonacci-product\/","article:published_time":"2026-09-16T12:04:09+00:00","article:modified_time":"2026-09-16T12:08:27+00:00","twitter:card":"summary","twitter:title":"Fibonacci product","twitter:description":"The product of four consecutive Fibonacci numbers equals the product of two consecutive integers.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247901","title":null,"description":"The product of four consecutive Fibonacci numbers equals the product of two consecutive 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11:43:26","updated":"2026-09-16 13:39:53","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\/category\/math\/\" title=\"Math\">Math<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tFibonacci product\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog"},{"label":"Math","link":"https:\/\/www.johndcook.com\/blog\/category\/math\/"},{"label":"Fibonacci 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approximation for spherical cap area"},"content":{"rendered":"<p>The <a href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/cosine-similarity\/\">previous post<\/a> looked at how to interpret cosine similarity, or equivalently angles between word vectors. In a high-dimensional space, randomly chosen vectors are likely nearly perpendicular, and so relatively large angles, such as 50\u00b0, indicate very closely related words.<\/p>\n<p>Another way to look at this, as explained in the previous post, is that in high dimensions, a spherical cap of angular radius \u03b8 represents a small portion of a sphere, even for moderately large \u03b8.<\/p>\n<p>The proportion of the area inside the spherical cap, given <a href=\"https:\/\/www.johndcook.com\/blog\/2023\/08\/09\/hypersphere-cap\/\">here<\/a>, involves the &#8220;regularized incomplete beta function&#8221; and so it&#8217;s hard to have an intuition for the value.<\/p>\n<p>For large dimension <em>n<\/em>, the approximation<\/p>\n<p style=\"padding-left: 40px;\"><em>n<\/em><sup>\u22121\/2<\/sup> sin<sup><em>n<\/em> \u2212 1<\/sup>(\u03b8)<\/p>\n<p>gives the proportion of the area inside the cap to within an order of magnitude. It&#8217;s easy to see that this function goes to zero quickly as\u00a0<em>n<\/em> increases, provided |\u03b8| &lt; \u03c0\/2.<\/p>\n<p>If you have the cosine similarity\u00a0<em>c<\/em> = cos \u03b8 rather than \u03b8 itself, the approximation becomes<\/p>\n<p style=\"padding-left: 40px;\"><em>n<\/em><sup>\u22121\/2<\/sup> (1 \u2212 <em>c<\/em>\u00b2)<sup>(<em>n<\/em> \u2212 1)\/2<\/sup>.<\/p>\n<h2>Python script<\/h2>\n<p>Let&#8217;s try it on the example from the previous post, in which\u00a0<em>n<\/em> = 200 and \u03b8 = 49\u00b0.<\/p>\n<pre>import numpy as np\r\nfrom scipy.special import betainc\r\n\r\n# Fraction of S^{n-1} inside a spherical cap of angular radius theta\r\n# theta is measured from the pole\r\n# Assume 0 &lt; theta &lt; pi\/2\r\n\r\ndef cap_fraction(theta, n):\r\n    x = np.sin(theta) ** 2\r\n    return 0.5 * betainc(0.5 * (n - 1), 0.5, x)\r\n\r\ndef cap_fraction_approx(theta, n):\r\n    return n**(-0.5) * np.sin(theta)**(n-1)\r\n\r\ntheta = np.deg2rad(49)\r\nprint(cap_fraction(theta, 200)) \r\nprint(cap_fraction_approx(theta, 200)) \r\n<\/pre>\n<p>This prints 2.03e-26 and 3.37e-26. The order of magnitude is correct as advertised.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The previous post looked at how to interpret cosine similarity, or equivalently angles between word vectors. In a high-dimensional space, randomly chosen vectors are likely nearly perpendicular, and so relatively large angles, such as 50\u00b0, indicate very closely related words. Another way to look at this, as explained in the previous post, is that in [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[224],"class_list":["post-247896","post","type-post","status-publish","format-standard","hentry","category-math","tag-geometry"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"A simple expression for roughly calculating the proportion of area inside a spherical cap of a high-dimensional sphere, useful for interpreting cosine similarity.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"John\"\/>\n\t<meta name=\"keywords\" content=\"geometry\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/simple-approximation-for-spherical-cap-area\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.1.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"John D. 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Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Simple approximation for spherical cap area","og:description":"A simple expression for roughly calculating the proportion of area inside a spherical cap of a high-dimensional sphere, useful for interpreting cosine similarity.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/simple-approximation-for-spherical-cap-area\/","article:published_time":"2026-09-15T22:00:44+00:00","article:modified_time":"2026-09-16T15:34:56+00:00","twitter:card":"summary","twitter:title":"Simple approximation for spherical cap area","twitter:description":"A simple expression for roughly calculating the proportion of area inside a spherical cap of a high-dimensional sphere, useful for interpreting cosine similarity.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247896","title":null,"description":"A simple expression for roughly calculating the 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16:46:24","updated":"2026-09-16 15:44:00","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\/category\/math\/\" title=\"Math\">Math<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tSimple approximation for spherical cap 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counts as a large cosine similarity?"},"content":{"rendered":"<p>Machine learning represents words as vectors and measures the similarity of words by the angles between the vectors.<\/p>\n<p>For vectors\u00a0<strong>x<\/strong>\u00a0and\u00a0<strong>y<\/strong>,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium\" style=\"background-color: white;\" src=\"https:\/\/www.johndcook.com\/dotproduct3.svg\" alt=\"\\mathbf{x} \\cdot \\mathbf{y} = ||\\mathbf{x} || \\,||\\mathbf{y} || \\, \\cos(\\theta)\" width=\"173\" height=\"18\" \/><\/p>\n<p>where \u03b8 is the angle between the vectors, and so<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium\" style=\"background-color: white;\" src=\"https:\/\/www.johndcook.com\/dotproduct4.svg\" alt=\"\\cos(\\theta) = \\frac{\\mathbf{x} \\cdot \\mathbf{y}}{ ||\\mathbf{x} || \\,||\\mathbf{y} || }\" width=\"136\" height=\"37\" \/><\/p>\n<p>This is the cosine similarity between the words represented by <strong>x<\/strong> and <strong>y<\/strong>.<\/p>\n<p>Small angles have large cosines, and so words with larger cosine similarities are closer together than words with smaller cosine similarities. The cosine similarity between a word and itself equals 1, and we&#8217;d expect unrelated words to have a cosine similarity near 0.<\/p>\n<p>You can do a sort of arithmetic with vector embeddings of words. The canonical example is that<\/p>\n<p style=\"padding-left: 40px;\">&#8220;king&#8221; \u2212 &#8220;man&#8221; + &#8220;woman&#8221; \u2248 &#8220;queen&#8221;<\/p>\n<p>Implicit in this equation is that we&#8217;re really adding vector representations of the words. Let\u00a0<strong>a<\/strong>,\u00a0<strong>b<\/strong>,\u00a0<strong>c<\/strong>, and\u00a0<strong>d<\/strong> be the vector embeddings of the words\u00a0<em>king<\/em>,\u00a0<em>man<\/em>,\u00a0<em>woman<\/em>, and\u00a0<em>queen<\/em>. What we&#8217;re really asserting is that<\/p>\n<p style=\"padding-left: 40px;\"><strong>a<\/strong> \u2212\u00a0<strong>b<\/strong> +\u00a0<strong>c<\/strong> \u2248\u00a0<strong>d<\/strong>,<\/p>\n<p>except that&#8217;s not true! Or at least it&#8217;s not true unless you view it in the right context.<\/p>\n<p>The angle between <strong>a<\/strong> \u2212\u00a0<strong>b<\/strong> +\u00a0<strong>c<\/strong> and\u00a0<strong>d<\/strong> is about 49\u00b0, which corresponds to a cosine similarity of 0.656. Here I&#8217;m using the gensim glove-twitter-200 embedding that represents words as 200-dimensional vectors.<\/p>\n<p>The way to interpret the equation above is not that a 49\u00b0 degree angle is approximately 0, or that a similarity of 0.656 is approximately 1.<\/p>\n<p>In high dimensions, such as 200-dimensional word embeddings, nearly all vectors are nearly perpendicular. I wrote a post about this <a href=\"https:\/\/www.johndcook.com\/blog\/2023\/08\/09\/random-points-hypersphere-orthant\/\">here<\/a>. So the angle between randomly selected words will usually be close to 90\u00b0, and so in that context an angle of 49\u00b0 is relatively small. For example, the angle between the vector representations of\u00a0<em>king<\/em> and\u00a0<em>fireplace<\/em> is 89.25\u00b0.<\/p>\n<p>If you divide word vectors by their norm, you can think of each vector as a point on a high-dimensional sphere, in our case a sphere in 200 dimensions. The proportion of vectors within 49\u00b0 of a given point is surprisingly small in high dimensions.<\/p>\n<p>Let&#8217;s say our point of interest is the north pole of an <em>n<\/em>-dimensional sphere. We&#8217;d like to calculate the proportion of the area of the sphere that is within an angle \u03b8 of the pole. I go through the calculations <a href=\"https:\/\/www.johndcook.com\/blog\/2023\/08\/09\/hypersphere-cap\/\">here<\/a>. (Update: I give an approximation <a href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/simple-approximation-for-spherical-cap-area\/\">here<\/a> that&#8217;s easier to work with than the exact formula.)<\/p>\n<p>When <em>n<\/em> = 3, 17% of the area is with 49 degrees of the pole. But when <em>n<\/em> = 200, the proportion is on the order of 10<sup>\u221226<\/sup>, essentially zero.<\/p>\n<p>The vector <strong>d<\/strong> above representing <em>queen<\/em> is within a relatively tiny region around the vector <strong>a<\/strong> \u2212\u00a0<strong>b<\/strong> +\u00a0<strong>c<\/strong>.<\/p>\n<p>In terms of cosine similarity, 0.656 is a large similarity. Words with a cosine similarity in this range are quite close, even though we wouldn&#8217;t normally think of 0.656 being close to 1. In this context, 0.656\u00a0<em>is<\/em> close to 1.<\/p>\n<h2>Related posts<\/h2>\n<ul>\n<li class=\"link\"><a href=\"https:\/\/www.johndcook.com\/blog\/2023\/08\/08\/angles-between-words\/\">Angles between words<\/a><\/li>\n<li class=\"link\"><a href=\"https:\/\/www.johndcook.com\/blog\/2023\/08\/09\/hypersphere-cap\/\">Area and volume of a hypersphere cap<\/a><\/li>\n<li class=\"link\"><a href=\"https:\/\/www.johndcook.com\/blog\/2023\/08\/09\/cosine-similarity-not-a-metric\/\">Cosine similarity does not satisfy the triangle inequality<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Machine learning represents words as vectors and measures the similarity of words by the angles between the vectors. For vectors\u00a0x\u00a0and\u00a0y, where \u03b8 is the angle between the vectors, and so This is the cosine similarity between the words represented by x and y. Small angles have large cosines, and so words with larger cosine similarities [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[260],"tags":[48],"class_list":["post-247894","post","type-post","status-publish","format-standard","hentry","category-ai","tag-differential-geometry"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"Vector embeddings let you conclude things like &quot;king&quot; \u2212 &quot;man&quot; + &quot;woman&quot; approximately equals &quot;queen&quot;. But this might not seem true when you look at the numbers.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"John\"\/>\n\t<meta name=\"keywords\" content=\"differential geometry\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/cosine-similarity\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.1.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"John D. Cook | Applied Mathematics Consulting\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"What counts as a large cosine similarity?\" \/>\n\t\t<meta property=\"og:description\" content=\"Vector embeddings let you conclude things like &quot;king&quot; \u2212 &quot;man&quot; + &quot;woman&quot; approximately equals &quot;queen&quot;. But this might not seem true when you look at the numbers.\" \/>\n\t\t<meta property=\"og:url\" content=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/cosine-similarity\/\" \/>\n\t\t<meta property=\"article:published_time\" content=\"2026-09-15T16:06:02+00:00\" \/>\n\t\t<meta property=\"article:modified_time\" content=\"2026-09-15T22:10:49+00:00\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary\" \/>\n\t\t<meta name=\"twitter:title\" content=\"What counts as a large cosine similarity?\" \/>\n\t\t<meta name=\"twitter:description\" content=\"Vector embeddings let you conclude things like &quot;king&quot; \u2212 &quot;man&quot; + &quot;woman&quot; approximately equals &quot;queen&quot;. But this might not seem true when you look at the numbers.\" \/>\n\t\t<meta name=\"twitter:image\" content=\"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png\" \/>\n\t\t<!-- All in One SEO -->\n\n","aioseo_head_json":{"title":"What counts as a large cosine similarity?","description":"Vector embeddings let you conclude things like \"king\" \u2212 \"man\" + \"woman\" approximately equals \"queen\". But this might not seem true when you look at the numbers.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/cosine-similarity\/","robots":"max-image-preview:large","keywords":"differential geometry","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"What counts as a large cosine similarity?","og:description":"Vector embeddings let you conclude things like &quot;king&quot; \u2212 &quot;man&quot; + &quot;woman&quot; approximately equals &quot;queen&quot;. But this might not seem true when you look at the numbers.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/cosine-similarity\/","article:published_time":"2026-09-15T16:06:02+00:00","article:modified_time":"2026-09-15T22:10:49+00:00","twitter:card":"summary","twitter:title":"What counts as a large cosine similarity?","twitter:description":"Vector embeddings let you conclude things like &quot;king&quot; \u2212 &quot;man&quot; + &quot;woman&quot; approximately equals &quot;queen&quot;. But this might not seem true when you look at the numbers.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247894","title":null,"description":"Vector embeddings let you conclude things like \"king\" \u2212 \"man\" + \"woman\" approximately equals \"queen\". But this might not seem true when you look at the numbers.","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-09-15 14:48:31","updated":"2026-09-16 01:15:55","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\/category\/ai\/\" title=\"AI\">AI<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tWhat counts as a large cosine similarity?\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog"},{"label":"AI","link":"https:\/\/www.johndcook.com\/blog\/category\/ai\/"},{"label":"What counts as a large cosine similarity?","link":"https:\/\/www.johndcook.com\/blog\/2026\/09\/15\/cosine-similarity\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247894","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=247894"}],"version-history":[{"count":2,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247894\/revisions"}],"predecessor-version":[{"id":247899,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247894\/revisions\/247899"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=247894"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=247894"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=247894"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":247892,"date":"2026-09-14T05:44:45","date_gmt":"2026-09-14T10:44:45","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247892"},"modified":"2026-09-14T05:44:45","modified_gmt":"2026-09-14T10:44:45","slug":"guessing-the-meaning-of-a-number","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/09\/14\/guessing-the-meaning-of-a-number\/","title":{"rendered":"Guessing the meaning of a number"},"content":{"rendered":"<p>Suppose I give you an\u00a0<em>n<\/em>-digit number and ask you what it represents. This seems impossible, and in theory it\u00a0<em>is<\/em> impossible. But in practice it&#8217;s often possible.<\/p>\n<p>Apps on a phone may automatically interpret a 10-digit number as a phone number or a 16-digit number as a package tracking number. And very often these interpretations are correct, given the kinds of things most people use their phones for.<\/p>\n<p>It&#8217;s not surprising that a 10-digit number\u00a0<em>on a phone<\/em> is a\u00a0<em>phone number<\/em>. It&#8217;s more interesting that a 16-digit number is likely a tracking number. It could be other things, such as a credit card number. But people don&#8217;t usually write out credit card numbers in a text note; credit card numbers likely saved in some more opaque way.<\/p>\n<p>I run into a variation of this problem routinely, trying to infer what a number represents inside medical notes.<\/p>\n<p>A five-digit number could be a US postal code, or it could be a <a href=\"https:\/\/www.johndcook.com\/blog\/2022\/09\/23\/hcpcs-codes\/\">medical procedure code<\/a>.<\/p>\n<p>A six-digit number could be a date in MMDDYY format, or it could be a medical record number.<\/p>\n<p>A ten-digit number could be a phone number, or it could be an <a href=\"https:\/\/www.johndcook.com\/blog\/2024\/06\/26\/npi-number\/\">NPI<\/a> (National Provider Identifier) number.<\/p>\n<p>It&#8217;s interesting that it&#8217;s possible make a good guess at what a number means inside unstructured text. Context has been lost, but not all context: you know you&#8217;re looking at medical notes. And that meager bit of context can be surprisingly useful.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Suppose I give you an\u00a0n-digit number and ask you what it represents. This seems impossible, and in theory it\u00a0is impossible. But in practice it&#8217;s often possible. Apps on a phone may automatically interpret a 10-digit number as a phone number or a 16-digit number as a package tracking number. And very often these interpretations are [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-247892","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"If all you know about a number is its length, how could you possibly tell what it means? Given a little context, it might be possible to make a good guess.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"John\"\/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/14\/guessing-the-meaning-of-a-number\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.1.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"John D. Cook | Applied Mathematics Consulting\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"Guessing the meaning of a number\" \/>\n\t\t<meta property=\"og:description\" content=\"If all you know about a number is its length, how could you possibly tell what it means? 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Given a little context, it might be possible to make a good guess.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/14\/guessing-the-meaning-of-a-number\/","article:published_time":"2026-09-14T10:44:45+00:00","article:modified_time":"2026-09-14T10:44:45+00:00","twitter:card":"summary","twitter:title":"Guessing the meaning of a number","twitter:description":"If all you know about a number is its length, how could you possibly tell what it means? Given a little context, it might be possible to make a good guess.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247892","title":null,"description":"If all you know about a number is its length, how could you possibly tell what it means? 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It could calculate some kind of distance between between the pixel pattern of the character and the pixel patterns of beta and eszett. But that would be discarding context.<\/p>\n<p>If you&#8217;re scanning a Greek document and run into a beta-like symbol, it&#8217;s very likely a beta. If you&#8217;re scanning a German document and run into a beta-like symbol, it\u00a0<em>could<\/em> be a beta. For example, it could be a scientific paper that mentions beta particles or beta carotene. But most likely the symbol is an eszett.<\/p>\n<p>The previous paragraph is saying you should compute the\u00a0<em>conditional<\/em> probability of a set of pixels representing a character\u00a0<em>given<\/em> the language of the document. You could be more sophisticated and look at the position of the symbol in a word as well. For example, if you see a symbol at the end of a Greek word that could either be \u03bf (omicron) or \u03c3 (sigma), it&#8217;s likely an omicron because Greek has a different symbol \u03c2 for final sigma.<\/p>\n<p>This post is a follow-on to my <a href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/07\/ngram-error-rate\/\">earlier post<\/a> on the error rate in Google&#8217;s Ngram database. OCR errors are fairly common in that database, so why don&#8217;t they &#8220;just&#8221; fix the errors by using some sort of Bayesian method? OCR software probably does use some sort of Bayesian method, but it&#8217;s not that simple.<\/p>\n<p>In that post I looked at the use of the word\u00a0<em>grok<\/em> in English. The Ngram database shows the word being used before it was coined in 1961 due to OCR errors. Why didn&#8217;t Google compute the probability of a word being &#8220;grok&#8221; conditional on the publication date? That would be circular. We happen to know exactly when <em>grok<\/em> was coined, but in general we might try to determine when a word was coined by looking at a large set of scanned books, like the Ngram database!<\/p>\n<p>Now we could compute the probable value of an ambiguously scanned word by conditioning on the language of the surrounding text. That would be a reasonable thing to do in general, but it could also lead to exactly the kind of errors we see in the Ngram data for\u00a0<em>grok<\/em>.<\/p>\n<p>Suppose you see an ambiguously scanned word in a book written in English. There is a higher prior probability that the word is an English word than a German word. Now suppose you see &#8220;gro?&#8221; where ? could be \u03b2, \u00df, or k. Without any context, perhaps the probability of the symbol being a <em>k<\/em> is small. But\u00a0<em>grok<\/em> is an English word and gro\u00df is a German word which may lead\u00a0you to conclude &#8220;?&#8221; is a\u00a0<em>k<\/em> and the ambiguous word is\u00a0<em>grok<\/em>.<\/p>\n<p>Assigning higher prior probability to English words in English texts is the best thing to do\u00a0<em>on average<\/em>, but in particular instances it will lead to errors. That&#8217;s life.<\/p>\n<p>The Ngram database includes millions of scanned books. Google had to use OCR algorithms that work well on average. A linguist with a special interest in a particular word can be more careful and create a more sophisticated probability model (explicit or implicit) customized for their interests. Google did what they could operating at such a large scale.<\/p>\n<h2>Related posts<\/h2>\n<ul>\n<li class=\"link\"><a href=\"https:\/\/www.johndcook.com\/blog\/2025\/08\/14\/uppercase-eszett\/\">Uppercase eszett<\/a><\/li>\n<li class=\"link\"><a href=\"https:\/\/www.johndcook.com\/blog\/2011\/09\/27\/bayesian-amazon\/\">A Bayesian view of Amazon resellers<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The Greek letter \u03b2 (beta) and the German letter \u00df (eszett) look similar, especially in some fonts. Now suppose an OCR program sees some character that could be a beta or could be an eszett. It could calculate some kind of distance between between the pixel pattern of the character and the pixel patterns of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[25,172],"class_list":["post-247884","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-bayesian","tag-typography"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"The error rate in Google&#039;s Ngram database is understandable given its scale. More specialized probability models could do better in particular instances.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"John\"\/>\n\t<meta name=\"keywords\" content=\"bayesian,typography\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/10\/bayesian-ocr\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.1.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"John D. Cook | Applied Mathematics Consulting\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"Bayesian OCR\" \/>\n\t\t<meta property=\"og:description\" content=\"The error rate in Google&#039;s Ngram database is understandable given its scale. 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More specialized probability models could do better in particular instances.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/10\/bayesian-ocr\/","article:published_time":"2026-09-10T12:32:31+00:00","article:modified_time":"2026-09-10T13:17:36+00:00","twitter:card":"summary","twitter:title":"Bayesian OCR","twitter:description":"The error rate in Google's Ngram database is understandable given its scale. More specialized probability models could do better in particular instances.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247884","title":null,"description":"The error rate in Google's Ngram database is understandable given its scale. 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The first major computer-assisted proof was published in 1976, the proof of the four color theorem by Kenneth Appel and Wolfgang Haken. The authors reduced the proof of the four color theorem to verifying calculations on 1,834 configurations, each checked by a computer program.<\/p>\n<p>The proof was simplified over the years, and formalized in Coq in 2005. Everyone is satisfied that the theorem is true, but there has never been a satisfying proof, one that a human could read and say &#8220;I see now why any map can be colored using only four colors.&#8221; And there may never be one, but see <a href=\"https:\/\/www.johndcook.com\/blog\/2013\/09\/04\/homework-problems-for-2090\/\">this post<\/a> for a contrary prediction.<\/p>\n<p>The IBM mainframe that ran the calculations completing the proof of the four color theorem did not generate the proof. It simply executed the FORTRAN program that Haken and Appel (and Koch [1]) gave it.<\/p>\n<p>I don&#8217;t see the recent proof of finite-time blowup for solutions to the Navier-Stokes equations as entirely different. Computers did higher-level tasks for the OpenAI team than the mainframe did for Haken and Appel, and these tasks were not as directly programmed as the tasks that were given to the mainframe, but still machines do what they are told to do.<\/p>\n<h2>Related posts<\/h2>\n<ul>\n<li class=\"link\"><a href=\"https:\/\/www.johndcook.com\/blog\/2013\/07\/19\/the-seven-color-map-theorem\/\">The seven color map theorem<\/a><\/li>\n<li class=\"link\"><a href=\"https:\/\/www.johndcook.com\/blog\/2019\/09\/12\/detecting-typos\/\">Detecting errors with the four color theorem<\/a><\/li>\n<\/ul>\n<p>[1] John A. Koch was a programmer who worked on the four color proof with Haken and Appel. I don&#8217;t know how much credit he deserves, but I suspect it may be more than he was given.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The idea of using computers to assist with proofs is not new. The first major computer-assisted proof was published in 1976, the proof of the four color theorem by Kenneth Appel and Wolfgang Haken. The authors reduced the proof of the four color theorem to verifying calculations on 1,834 configurations, each checked by a computer [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[5],"tags":[],"class_list":["post-247877","post","type-post","status-publish","format-standard","hentry","category-computing"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"Computer-assisted proofs are older than you may imagine. The first major theorem proved with the assistance of a computer was over 50 years ago.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"John\"\/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/09\/four-colors\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.1.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"John D. Cook | Applied Mathematics Consulting\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"A 50-year-old computer-assisted proof\" \/>\n\t\t<meta property=\"og:description\" content=\"Computer-assisted proofs are older than you may imagine. 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The first major theorem proved with the assistance of a computer was over 50 years ago.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/09\/four-colors\/","article:published_time":"2026-09-09T14:59:48+00:00","article:modified_time":"2026-09-09T14:59:48+00:00","twitter:card":"summary","twitter:title":"A 50-year-old computer-assisted proof","twitter:description":"Computer-assisted proofs are older than you may imagine. The first major theorem proved with the assistance of a computer was over 50 years ago.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247877","title":null,"description":"Computer-assisted proofs are older than you may imagine. The first major theorem proved with the assistance of a computer was over 50 years ago.","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-09-09 14:24:10","updated":"2026-09-09 15:13:34","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/www.johndcook.com\/blog\/category\/computing\/\" title=\"Computing\">Computing<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tA 50-year-old computer-assisted proof\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog"},{"label":"Computing","link":"https:\/\/www.johndcook.com\/blog\/category\/computing\/"},{"label":"A 50-year-old computer-assisted proof","link":"https:\/\/www.johndcook.com\/blog\/2026\/09\/09\/four-colors\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247877","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=247877"}],"version-history":[{"count":2,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247877\/revisions"}],"predecessor-version":[{"id":247879,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247877\/revisions\/247879"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=247877"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=247877"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=247877"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":247875,"date":"2026-09-09T09:17:20","date_gmt":"2026-09-09T14:17:20","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247875"},"modified":"2026-09-09T10:18:22","modified_gmt":"2026-09-09T15:18:22","slug":"ai-multiplier","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/09\/09\/ai-multiplier\/","title":{"rendered":"AI is an intelligence multiplier"},"content":{"rendered":"<p>A rising tide may lift all boats, but the AI tide lifts some boats much more than others.<\/p>\n<p>By all accounts, the best programmers have had the biggest productivity boost from AI. And top tier mathematicians are using AI to settle long-standing mathematical conjectures. AI is a powerful tool, but tools don&#8217;t come to life and make things on their own.<\/p>\n<p>I routinely have naive amateurs [1] send me proofs of open conjectures, and naturally more recent such proofs involve AI. I&#8217;ll get an email saying something like &#8220;I&#8217;ve solved the Collatz conjecture using ChatGPT, but I&#8217;m not a mathematician so I need some help verifying the proof.&#8221; And of course the supposed proof is rubbish.<\/p>\n<p>The recent Navier-Stokes proof is impressive, but AI didn&#8217;t initiate the proof any more than LaTeX did. Nor did a child steer AI into proving the conjecture. Professional mathematicians were able to use AI to pursue their ideas at superhuman speed. But someone without an understanding of the Navier-Stokes problem, and familiarity with recent ideas for approaching the problem, could not have directed AI to produce a proof.<\/p>\n<p>Computer scientists have been saying &#8220;garbage in, garbage out&#8221; from the beginning. A variation on this aphorism for the age of AI would be &#8220;mediocrity in, mediocrity out.&#8221;<\/p>\n<p style=\"text-align: center;\">***<\/p>\n<p>[1] Amateurs can and do make contributions to mathematics. For example, in 2022 David Smith, a retired print technician, discovered a single shape that can be used to create an aperiodic tiling of the plane. By &#8220;naive amateurs&#8221; I mean people who literally do not know what they are talking about.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A rising tide may lift all boats, but the AI tide lifts some boats much more than others. By all accounts, the best programmers have had the biggest productivity boost from AI. And top tier mathematicians are using AI to settle long-standing mathematical conjectures. AI is a powerful tool, but tools don&#8217;t come to life [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[260],"tags":[],"class_list":["post-247875","post","type-post","status-publish","format-standard","hentry","category-ai"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"The most productive, most creative, and most intelligent people will benefit the most from AI.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"John\"\/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.johndcook.com\/blog\/2026\/09\/09\/ai-multiplier\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.1.1\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"John D. 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