[{"id":247975,"date":"2026-10-04T07:34:22","date_gmt":"2026-10-04T12:34:22","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247975"},"modified":"2026-10-04T07:34:22","modified_gmt":"2026-10-04T12:34:22","slug":"miquels-pentagon-theorem","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/04\/miquels-pentagon-theorem\/","title":{"rendered":"Miquel’s pentagon theorem"},"content":{"rendered":"

An earlier post<\/a> presented an elegant plane geometry theorem discovered by the 19th century school teacher Auguste Miquel. This post presents his pentagon theorem.<\/p>\n

Start with a pentagon. It may be irregular, but it needs to be convex.<\/p>\n

Extend each of the sides of the pentagon to form a star, then draw give circles, one through each of the triangles formed by a side of the pentagon and a vertex of the star.<\/p>\n

The five circles intersect in pairs at ten points: the five verices of the pentagon and five new points. The five new points lie on a circle.<\/p>\n

<\/p>\n

The converse of this theorem is known as the five circles theorem.<\/p>\n","protected":false},"excerpt":{"rendered":"

An earlier post presented an elegant plane geometry theorem discovered by the 19th century school teacher Auguste Miquel. This post presents his pentagon theorem. Start with a pentagon. It may be irregular, but it needs to be convex. Extend each of the sides of the pentagon to form a star, then draw give circles, one […]<\/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-247975","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":"Miquel\u2019s pentagon theorem","description":"Miquel's pentagon theorem, an elegant and relatively recent theorem in Euclidean geometry.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/04\/miquels-pentagon-theorem\/","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":"Miquel\u2019s pentagon theorem","og:description":"Miquel's pentagon theorem, an elegant and relatively recent theorem in Euclidean geometry.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/04\/miquels-pentagon-theorem\/","article:published_time":"2026-10-04T12:34:22+00:00","article:modified_time":"2026-10-04T12:34:22+00:00","twitter:card":"summary","twitter:title":"Miquel\u2019s pentagon theorem","twitter:description":"Miquel's pentagon theorem, an elegant and relatively recent theorem in Euclidean geometry.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247975","title":null,"description":"Miquel's pentagon theorem, an elegant and relatively recent theorem in Euclidean geometry.","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":"2026-10-03 23:18:07","updated":"2026-10-04 13:10:00","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":"

\n\t\t\tHome<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tMath<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tMiquel\u2019s pentagon theorem\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":"Miquel’s pentagon theorem","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/04\/miquels-pentagon-theorem\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247975","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=247975"}],"version-history":[{"count":2,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247975\/revisions"}],"predecessor-version":[{"id":247990,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247975\/revisions\/247990"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=247975"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=247975"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=247975"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":247985,"date":"2026-10-04T07:25:19","date_gmt":"2026-10-04T12:25:19","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247985"},"modified":"2026-10-04T07:26:32","modified_gmt":"2026-10-04T12:26:32","slug":"topological-models-of-modal-logic","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/04\/topological-models-of-modal-logic\/","title":{"rendered":"Topological models of modal logic"},"content":{"rendered":"

The previous post<\/a> discussed a superficial connection between modal logic and topology, that both give use the terms\u00a0regular<\/em> and\u00a0normal<\/em> to indicate added sets of axioms. McKinsey and Tarski developed a deeper connection between modal logic and topology that we’ll discuss here.<\/p>\n

Starting with a topological space\u00a0X<\/em> and a proposition p<\/em>, define [[p<\/em>]] as the set of points in\u00a0X<\/em> at which\u00a0p<\/em> is true. Define \u25a1p<\/em> to be true at points in the interior of [[p<\/em>]] and define \u25c7p<\/em> to be true on the closure of [[p<\/em>]].<\/p>\n

You could think of \u25a1p<\/em> as the points where p<\/em> is robustly true. Not only is p<\/em>\u00a0true at x<\/em>, there’s some wiggle room around x<\/em>, i.e. an open set, in which\u00a0p<\/em> remains true.<\/p>\n

You could think of \u25c7p<\/em> as there points where we cannot rule out the possibility of\u00a0p<\/em> being true using open sets. If \u25c7p<\/em> includes\u00a0x<\/em>, any open set containing\u00a0x<\/em> also contains part of \u25c7p<\/em>, though it may also contain points outside of \u25c7p.<\/em><\/p>\n

Regularity<\/h2>\n

For any topology on X<\/em>, the logic constructed above is regular. The axiom<\/p>\n

\u25c7p<\/em> \u21d4 \u00ac (\u25a1 \u00ac p<\/em>)<\/p>\n

holds because the interior of a set is the complement of the interior of its complement [1].<\/p>\n

Note that this is a regularity result for the modal logic, not the topology. The topology could be arbitrary, and not necessarily regular or normal in the topological sense.<\/p>\n

S4<\/h2>\n

The logic constructed above also satisfies a couple more axioms. We have<\/p>\n

\u25a1p<\/em> \u2192\u00a0p<\/em><\/p>\n

because the interior of a set is a subset of the set, and<\/p>\n

\u25a1p<\/em> \u2192 \u25a1\u25a1p<\/em><\/p>\n

because the interior of the interior of a set is simply the interior. This means the modal logic corresponding to a topology satisfies the S4 axioms. You could say S4 is the logic that corresponds to the McKinsey and Tarski logic of all topological spaces.<\/p>\n

More logics and more topologies<\/h2>\n

So S4 is the logic that corresponds to\u00a0all<\/em> topologies. We could look at more restricted topologies and ask what are their corresponding logics. Or we could start with a modal logic and ask whether there’s a topology that models that logic.<\/p>\n

Interesting logics correspond to badly behaved topological spaces. Familiar topological spaces like the real line correspond to S4.<\/p>\n

Trivial modal logic<\/h3>\n

The discrete topology corresponds to the trivial modal logic. All sets are open, and closed, so the any set is the same as its interior and its closure. So \u25a1p<\/em> and \u25c7p<\/em> reduce to just\u00a0p<\/em>.<\/p>\n

S5<\/h3>\n

For the indiscrete topology, \u25a1p<\/em> corresponds to a proposition holding everywhere and \u25c7p<\/em> corresponds to it holding somewhere. If the topological space has infinitely many points, the corresponding modal logic is S5. [2]<\/p>\n

Between S4 and S5<\/h3>\n

The cofinite topology on an infinite set\u00a0X<\/em> defines a set\u00a0U<\/em> to be open if the complement of\u00a0U<\/em> is finite. The McKinsey-Tarski logic of the cofinite topology is somewhere between S4 and S5. You can show that the formula<\/p>\n

p<\/em> \u2227 \u25c7\u25a1p<\/em> \u2192 \u25a1p<\/em><\/p>\n

holds, which doesn’t hold in S4, and the formula<\/p>\n

\u25c7p<\/em> \u2192 \u25a1\u25c7p<\/em><\/p>\n

does not hold, though it must hold in S5.<\/p>\n

Related posts<\/h2>\n