cos(c<\/em>) = cos(a<\/em>) cos(b<\/em>) + sin(a<\/em>) sin(b<\/em>) cos(C<\/em>).<\/p>\n If you know the (angular) lengths of sides\u00a0a<\/em> and\u00a0b<\/em>, and (tangential) angle\u00a0C<\/em> between the two sides, you can solve for\u00a0c<\/em> by taking the inverse cosine of the right hand side above.<\/p>\n Now suppose you want to solve this big triangle because you’re a navigator<\/strong> on a ship a couple centuries ago, doing calculations by looking up trig functions and inverse trig functions in a table. You’re interested in triangles that are so big that you have to account for the fact that you’re living on a sphere. But at the same time, you’re triangles are still fairly small relative to the size of the globe.<\/p>\n The numbers\u00a0a<\/em> and\u00a0b<\/em> will often be fairly small, and so their cosines will be near 1 and their sines are near zero. So the calculation<\/p>\n cos(a<\/em>) cos(b<\/em>) + sin(a<\/em>) sin(b<\/em>) cos(C<\/em>)<\/p>\n will add a number near 1 and a number near zero. That’s a problem.<\/p>\n Say you’re working with five decimal place arithmetic. Then if the second term above is less than 10\u22125<\/sup>, its contribution to the sum gets completely lost in the addition to the first term. If the second term is larger than 10\u22125<\/sup> but still small, its contribution to the sum will be partially lost.<\/p>\n Enter the haversine<\/a>, defined by<\/p>\n hav(\u03b8) = (1 \u2212 cos(\u03b8))\/2.<\/p>\n The expression 1 \u2212 cos \u03b8 was called the versine, and so half of the versine is the haversine.<\/p>\n In terms of the haversine, the law of cosines above becomes the law of haversines:<\/p>\n hav(c<\/em>) = hav(a<\/em> \u2212\u00a0b<\/em>) + sin(a<\/em>) sin(b<\/em>) hav(C<\/em>).<\/p>\n Now suppose you have a table of haversines and inverse haversines. The law of haversines requires a little less work: you have one less table lookup, and you trade a product for a subtraction.<\/p>\n But the primary advantage is numerical accuracy: the terms on the right side have roughly the same size.<\/p>\n Note that we’re assuming the values in your table of haversines have been calculated correctly to the given precision. If you calculated your own values of haversines from the definition above, you’d lose precision in the subtraction 1 \u2212 cos \u03b8, defeating the advantage of the law of haversines [1].<\/p>\n According to Wikipedia<\/a>.<\/p>\n The first table of haversines in English was published by James Andrew in 1805, but Florian Cajori credits an earlier use by Jos\u00e9 de Mendoza y R\u00edos in 1801. The term haversine<\/em> was coined in 1835 by James Inman.<\/p><\/blockquote>\n I ran some experiments that carried out arithmetic in float16 (11 bits of precision) to approximate what someone might have done by hand. When the difference between a<\/em> and\u00a0b<\/em> was on the order of 1\u00b0 or 0.1\u00b0, the law of cosine method often overflowed: the right-hand side evaluated to something larger than 1 even though theoretically it should be less than 1. The haversine method never overflowed.<\/p>\n The median error for the haversine method was a couple orders of magnitude less than that of the cosine method.<\/p>\n [1] hav(\u03b8) = (1 \u2212 cos(\u03b8))\/2 = sin\u00b2(\u03b8\/2). If you calculated hav \u03b8 by looking up sin(\u03b8\/2) and squaring it, you’d be doing extra work, but you wouldn’t have numerical problems.<\/p>\n","protected":false},"excerpt":{"rendered":" Suppose you want to solve a triangle. You know two sides and the angle between them. Then you can solve for the third side using the law of cosines. Now suppose you want to solve a\u00a0big triangle, a triangle on the surface of the earth so large that the curvature of the earth matters. You […]<\/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":[224,312],"class_list":["post-247940","post","type-post","status-publish","format-standard","hentry","category-math","tag-geometry","tag-navigation"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Haversine law","description":"The haversine law was used to solve triangles in navigation at sea. It gives more accurate results in practice than the law of cosines.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/21\/haversine-law\/","robots":"max-image-preview:large","keywords":"geometry,navigation","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Haversine law","og:description":"The haversine law was used to solve triangles in navigation at sea. It gives more accurate results in practice than the law of cosines.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/21\/haversine-law\/","article:published_time":"2026-09-21T23:13:22+00:00","article:modified_time":"2026-09-21T23:13:22+00:00","twitter:card":"summary","twitter:title":"Haversine law","twitter:description":"The haversine law was used to solve triangles in navigation at sea. It gives more accurate results in practice than the law of cosines.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247940","title":null,"description":"The haversine law was used to solve triangles in navigation at sea. It gives more accurate results in practice than the law of cosines.","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":"2026-09-21 16:29:49","updated":"2026-09-21 23:13:22","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":" Nothing grows exponentially forever. What appears to be an exponential curve often turns out to be some sort of S curve, such as a logistic curve. Suppose you’re collecting data on the left side of the curve. If there’s even a small amount of error in your data, you won’t be able to predict the asymptotic value with any accuracy. But if you have data on both sides of the inflection point, you can make a good prediction of the limiting value.<\/p>\n I’ve written about this before<\/a>, explaining that the problem is hard, but I didn’t say\u00a0why<\/em> it’s hard. Here I’d like to give an idea why it’s hard.<\/p>\n Suppose you want to fit a logistic equation<\/p>\n to three distinct values of t<\/em> and the corresponding values of y<\/em>. There is a unique solution, but in general you cannot find a solution in closed form. However, if the values of t<\/em> are evenly spaced<\/p>\n there is a method [1] to solve for the parameters L<\/em>, k<\/em>, and t<\/em>0<\/sub>. For this post we’re only interested in the limiting value L<\/em>, and it can be found by<\/p>\n independent of\u00a0h<\/em>.<\/p>\n To find out how small changes in the\u00a0y<\/em>‘s change the estimate of\u00a0L<\/em>, we take the partial derivatives of\u00a0L<\/em> with respect to the\u00a0y<\/em>‘s and find<\/p>\n and<\/p>\n If the function\u00a0y<\/em>(t<\/em>) were an exponential, this expression would be exactly zero [2]. The function y<\/em>(t<\/em>) is not exactly exponential, but it is approximately<\/em> exponential when the t<\/em>‘s are in the left or right tail of the logistic curve. The further out in either tail the t<\/em>‘s are, the closer the expression is to zero.<\/p>\n So when all the t<\/em>‘s come from the same side of the inflection point, y<\/em>(t<\/em>) is nearly exponential the partial derivatives are huge and so the fitted value of L<\/em> is extremely sensitive to changes in the y<\/em>‘s.<\/p>\n As a concrete example, set\u00a0L<\/em> =\u00a0k<\/em> = 1 and t<\/em>0<\/sub> = 0. Evaluate\u00a0y<\/em>(t<\/em>) at \u22122, \u22121.5, and \u22121. Then the values of\u00a0y<\/em> are<\/p>\n y<\/em>0<\/sub> = 0.11920292 If you forecast L<\/em> using exactly these three values you’ll get L<\/em> = 1.<\/p>\n But if you change y<\/em>0<\/sub> to 0.12374097, the forecasted value of L<\/em> is infinite. Values of y<\/em>0<\/sub> in the interval [0.11920292, 0.12374097] predict values of K<\/em> in [1, \u221e].<\/p>\n [1] Raymond Pearl and Lowell J. Reed. On the Rate of Growth of the Population of the United States Since 1790 and its Mathematical Representation. Proceedings of the National Academy of Sciences of the United States of America, Vol. 6, No. 6 (Jun. 15, 1920), pp. 275-288<\/p>\n [2] exp(x<\/em> +\u00a0h<\/em>)\u00b2 = exp(x<\/em>)\u00b2 exp(h<\/em>)\u00b2 = exp(x<\/em>) exp(x<\/em> + 2h<\/em>)<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" Nothing grows exponentially forever. What appears to be an exponential curve often turns out to be some sort of S curve, such as a logistic curve. Suppose you’re collecting data on the left side of the curve. If there’s even a small amount of error in your data, you won’t be able to predict the […]<\/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":[106],"class_list":["post-247934","post","type-post","status-publish","format-standard","hentry","category-math","tag-probability-and-statistics"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Why fitting a logistic is nearly impossible from early data","description":"Why fitting a logistic to data is very sensitive when all the data are on one side of the inflection point.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/18\/logistic-fit-sensitivity\/","robots":"max-image-preview:large","keywords":"probability and statistics","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Why fitting a logistic is nearly impossible from early data","og:description":"Why fitting a logistic to data is very sensitive when all the data are on one side of the inflection point.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/18\/logistic-fit-sensitivity\/","article:published_time":"2026-09-19T01:17:36+00:00","article:modified_time":"2026-09-19T14:30:16+00:00","twitter:card":"summary","twitter:title":"Why fitting a logistic is nearly impossible from early data","twitter:description":"Why fitting a logistic to data is very sensitive when all the data are on one side of the inflection point.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247934","title":null,"description":"Why fitting a logistic to data is very sensitive when all the data are on one side of the inflection point.","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":"2026-09-18 23:53:59","updated":"2026-09-21 16:29:59","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":" There’s a common saying in discussion of fractals that the length of a coastline depends on how small a device you use to measure it. I thought this was a hypothetical, say as applied to the steps in the construction of the Koch snowflake. But the saying has its roots in actually surveying.<\/p>\n Lewis Fry Richardson (1881\u20131953) noticed that the length of the coast of Scotland depended on the size of segments used to measure it. More specifically, he found that the length followed a power law, i.e. that there’s a linear relation between the log of the coastline length and the log of the ruler length.<\/p>\n Here’s a reproduction of Richardson’s plot, taken from [1].<\/p>\n Mandelbrot built on Richardson’s observation and defined the idea of fractal dimension.<\/p>\n I was under the impression that fractals were invented as mathematical novelties that researchers later found applications for. But as is often the case, the applications came first. Or at least some<\/em> applications came first.<\/p>\n Ideally there’s always a feedback cycle where applications lead to theory and theory leads to applications. As Donald Knuth put it, “The best theory is inspired by practice. The best practice is inspired by theory.”<\/p>\n [1] Eoghan Bradley and Mark McCartney. Four hundred years of the fractal coastline of Scotland. The Mathematical Gazette, November 2019, Vol. 103, No. 558 (November 2019), pp. 518-521<\/p>\n","protected":false},"excerpt":{"rendered":" There’s a common saying in discussion of fractals that the length of a coastline depends on how small a device you use to measure it. I thought this was a hypothetical, say as applied to the steps in the construction of the Koch snowflake. But the saying has its roots in actually surveying. Lewis Fry […]<\/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-247924","post","type-post","status-publish","format-standard","hentry","category-math","tag-geometry"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Empirical fractal","description":"Richardson's measurements of the coast of Scotland were a precursor to the theory of fractals.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/17\/empirical-fractal\/","robots":"max-image-preview:large","keywords":"geometry","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Empirical fractal","og:description":"Richardson's measurements of the coast of Scotland were a precursor to the theory of fractals.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/09\/17\/empirical-fractal\/","article:published_time":"2026-09-17T16:48:14+00:00","article:modified_time":"2026-09-18T12:10:00+00:00","twitter:card":"summary","twitter:title":"Empirical fractal","twitter:description":"Richardson's measurements of the coast of Scotland were a precursor to the theory of fractals.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247924","title":null,"description":"Richardson's measurements of the coast of Scotland were a precursor to the theory of fractals.","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":"2026-09-17 16:30:32","updated":"2026-09-18 12:10:01","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":" 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<\/a>.<\/p>\n One of the cards in the book is for Eugene Frank, President of B & F Instruments. His card lists his phone number as<\/p>\n (215) MErcury 9-7100<\/p>\n At first glance I thought the “E” in “MErcury” had been accidentally capitalized. Then I realized the intention was that someone would dial ME (i.e. 63) and ingore “rcury”. So the phone number would be (215) 639-7100.<\/p>\n This card was from 1968, the height of the space race. Maybe the card was alluding to the Project Mercury or the planet Mercury, or both. [1]<\/p>\n The telephone keypad mapping (ITU E.161 standard) is a poor attempt at making phone numbers more memorable. For starters, there’s no way to encode 0 or 1 [2]. It’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 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’s not much advantage. Still, I wanted to play around with it for fun. I found 109 words [3] containing the letters from a telephone encoding of 4228646. (I’m using the file Here are some of the more interesting hits.<\/p>\n There are over 30,000 words containing an encoding of the area code 832. One of these is traditional<\/em>, and so I could write my phone number as<\/p>\n Another choice for 832 is intercosmic<\/em>, so<\/p>\n is another possibility.<\/p>\n 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 intercosmic<\/em> fits with the later sense.<\/p>\n I got greedy and tried to find a word containing the full phone number, 8324228646, but didn’t find anything.<\/p>\n Here’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 Now suppose you remembered “gladiatorism” but not which letters were capitalized. Then you’d have to try up to 792, i.e. 12 choose 7, possible numbers, so this really isn’t a practical mnemonic. If you remembered “traditional semicatholicism” without capitalization it would be worse, with over a million possibilities (11 choose 3 times 15 choose 7). Some possibilities are counted twice, since different ways of selecting letters can lead to the same phone number, but still there are too many possibilities to try.<\/p>\n [1] Thanks to Andrew for pointing out in his comment that it was common at one time to encode the first two numbers of the exchange (the second triplet of numbers in a phone number) as letters, and assign a word to those letters. Sometimes this was standardized, such as Pennsylvania 6 for 736, an example made famous by Glenn Miller. But from what I can tell, not all exchanges had standard names, and proposed standards weren’t always adopted in practice.<\/p>\n In the example above, I don’t know whether it was common to encode 639 as Mercury 9, or even ME 9, or whether Mr. Frank chose this. It was common chose\u00a0some<\/em> encoding for the first two numbers of the exchange, though that practice was going away by 1968. Perhaps Mr. Frank was an older man who retained a habit he acquired when it was more common. None of the other cards in the book spelled out the exchange.<\/p>\n Update<\/strong>: Thanks to Chuck for pointing out this list<\/a> of recommended words for exchanges. Note that there are multiple suggestions for most exchanges, including six for 63X.<\/p>\n [2] 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 [3] 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":" 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 & F Instruments. His card lists his phone number […]<\/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\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\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":"2026-09-18T14:15:24+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 letters.","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":"2026-09-17 11:34:21","updated":"2026-09-18 23:55:03","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":" I’ve written three posts on cosine similarity lately. The first<\/a> looked at interpreting cosine similarity. The second<\/a> looked at an approximation related to the first. The third<\/a> looked at how ranking according to cosine similarity works better than cosine similarity itself.<\/p>\n 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>\nThe problem with the law of cosines<\/h2>\n
Law of haversines<\/h2>\n
Tables<\/h2>\n
History<\/h2>\n
Experiments<\/h2>\n
Related posts<\/h2>\n
\n
![\"logistic](\"https:\/\/www.johndcook.com\/logistic_tangents.png\")
<\/p>\n![\"y(t)](\"https:\/\/www.johndcook.com\/logistic_fit1.svg\")
<\/p>\n![\"y_2](\"https:\/\/www.johndcook.com\/logistic_fit2.svg\")
<\/p>\n![\"L](\"https:\/\/www.johndcook.com\/logistic_fit3.svg\")
<\/p>\n![\"\\frac{\\partial](\"https:\/\/www.johndcook.com\/logistic_fit4.svg\")
<\/p>\n![\"\\frac{\\partial](\"https:\/\/www.johndcook.com\/logistic_fit5.svg\")
\nAll three derivatives have the same expression in the denominator: y<\/em>1<\/sub>\u00b2 \u2212 y<\/em>0<\/sub> y<\/em>2<\/sub>.<\/p>\n
\ny<\/em>1<\/sub> = 0.18242552
\ny<\/em>2<\/sub> = 0.26894142<\/p>\n![](\"https:\/\/www.johndcook.com\/scottish_coast.png\")
<\/p>\nRelated posts<\/h2>\n
\n
\/usr\/share\/dict\/words<\/code> on my laptop as my list of words.)<\/p>\n\n
TraDitionAl semICAThOlIcisM<\/code>.<\/p>\ninTErCosmic GAlaCTOGeNetic<\/code><\/p>\n![](\"https:\/\/www.johndcook.com\/vintage_card.png\")
<\/p>\nRelated posts<\/h2>\n
\n
![](\"https:\/\/www.johndcook.com\/concentration_ratio.png\")
<\/p>\n