[{"id":248186,"date":"2026-11-09T11:01:43","date_gmt":"2026-11-09T17:01:43","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=248186"},"modified":"2026-11-09T11:10:54","modified_gmt":"2026-11-09T17:10:54","slug":"steiner-systems","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/11\/09\/steiner-systems\/","title":{"rendered":"Steiner Systems"},"content":{"rendered":"

Steiner triples<\/h2>\n

Several examples from the recent posts on finite geometry are special cases of Steiner triples.<\/p>\n

A Steiner triple system, written STS(n<\/em>), is a set S<\/em> of\u00a0n<\/em> elements and a collection of three-element subsets of\u00a0S<\/em> called blocks, such that every two-element subset of\u00a0S<\/em> belongs to exactly one block.<\/p>\n

STS(7) is the smallest finite projective plane, the Fano plane, and STS(9) is the smallest finite affine plane. In both cases you interpret the elements of S<\/em> as points and the blocks as lines.<\/p>\n

Several questions come up naturally. First, for which values of\u00a0n<\/em> does an STS(n<\/em>) exist? A necessary and sufficient condition for is that n<\/em> = 1 or 3 mod 6.<\/p>\n

For a given\u00a0n<\/em>, can there be more than one STS(n<\/em>), i.e. are there Steiner triple systems of the same size that are not isomorphic? Yes indeed. The number of non-isomorphic instances grows rapidly with\u00a0n<\/em>.<\/p>\n

When does (some instance of) STS(n<\/em>) correspond to a projective or affine space? When n<\/em> = 2k<\/em><\/sup> \u2212 1 for k<\/em> \u2265 3. For n<\/em> \u2265 13 there are multiple isomorphism classes of STS(n<\/em>), so for k<\/em> > 3 there will be an instance of STS(2k<\/em><\/sup> \u2212 1) that corresponds to a projective space and other instances that do not. The instance that does correspond to a projective plane will be isomorphic to a finite field analog of the real projective plane.<\/p>\n

Steiner systems<\/h2>\n

Steiner triple systems are a special case of Steiner systems. A Steiner system\u00a0S<\/em>(t<\/em>,\u00a0k<\/em>,\u00a0n<\/em>) is a set of\u00a0n<\/em> items and a collection of blocks that are\u00a0k<\/em>-element subsets such that every set of\u00a0t<\/em> items belongs to a unique block. Steiner triples are\u00a0S<\/em>(2, 3,\u00a0n<\/em>) systems.<\/p>\n

Steiner systems often correspond to finite geometries with the blocks interpreted as lines. As with the special case of triples, there can be non-isomorphic instances of Steiner systems, and so when I say below that\u00a0S<\/em>(t<\/em>, k<\/em>, n<\/em>) corresponds to a certain geometry, I mean that at least one instance corresonds to that geometry.<\/p>\n

Projective<\/h3>\n

The Steiner system<\/p>\n

S<\/em>(2,\u00a0q<\/em> + 1, (q<\/em>d<\/em> + 1<\/sup> \u2212 1)\/(q<\/em> \u2212 1))<\/p>\n

with q<\/em> a prime power greater than 2 is a d<\/em>-dimensional projective space.<\/p>\n

Affine<\/h3>\n

The space<\/p>\n

S<\/em>(2, q<\/em>, q<\/em>d<\/em><\/sup>)<\/p>\n

is an affine space. Again q<\/em> > 2 is a prime power.<\/p>\n

Hyperbolic<\/h3>\n

The finite hyperbolic plane given as an example in an earlier post corresponds to S<\/em>(2, 2, 6). Many other Steiner systems are hyperbolic, but I’m not aware of a tidy classification similar to the ones above.<\/p>\n

Steiner systems outside geometry<\/h2>\n

Steiner systems were named after Swiss mathematician Jakob Steiner because he published a highly cited article asking questions about what we now call Steiner triples. His interest in these systems was combinatorial. Others developed the connections to geometry later.<\/p>\n

Steiner systems arise in several areas of mathematics outside of geometry, including coding theory, design of experiments, and graph theory.<\/p>\n","protected":false},"excerpt":{"rendered":"

Steiner triples Several examples from the recent posts on finite geometry are special cases of Steiner triples. A Steiner triple system, written STS(n), is a set S of\u00a0n elements and a collection of three-element subsets of\u00a0S called blocks, such that every two-element subset of\u00a0S belongs to exactly one block. STS(7) is the smallest finite projective […]<\/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":[197,224],"class_list":["post-248186","post","type-post","status-publish","format-standard","hentry","category-math","tag-combinatorics","tag-geometry"],"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\t\t\n\n","aioseo_head_json":{"title":"Steiner Systems","description":"Steiner systems are combinatorial structures that often have geometrical interpretations. They appear in several areas of mathematics.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/11\/09\/steiner-systems\/","robots":"max-image-preview:large","keywords":"combinatorics,geometry","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Steiner Systems","og:description":"Steiner systems are combinatorial structures that often have geometrical interpretations. They appear in several areas of mathematics.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/11\/09\/steiner-systems\/","article:published_time":"2026-11-09T17:01:43+00:00","article:modified_time":"2026-11-09T17:10:54+00:00","twitter:card":"summary","twitter:title":"Steiner Systems","twitter:description":"Steiner systems are combinatorial structures that often have geometrical interpretations. They appear in several areas of mathematics.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"248186","title":null,"description":"Steiner systems are combinatorial structures that often have geometrical interpretations. They appear in several areas of mathematics.","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":"2026-11-06 14:58:50","updated":"2026-11-09 18:07:59","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":"

\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\tSteiner Systems\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":"Steiner Systems","link":"https:\/\/www.johndcook.com\/blog\/2026\/11\/09\/steiner-systems\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248186","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=248186"}],"version-history":[{"count":5,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248186\/revisions"}],"predecessor-version":[{"id":248211,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248186\/revisions\/248211"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=248186"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=248186"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=248186"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":248199,"date":"2026-11-09T08:26:09","date_gmt":"2026-11-09T14:26:09","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=248199"},"modified":"2026-11-09T10:28:40","modified_gmt":"2026-11-09T16:28:40","slug":"finite-hyperbolic-planes","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/11\/09\/finite-hyperbolic-planes\/","title":{"rendered":"Finite hyperbolic planes"},"content":{"rendered":"

Euclid’s fifth postulate<\/a> says that given a line \u2113 and a point\u00a0P<\/em> not on \u2113 , there is a unique line through\u00a0P<\/em> parallel to \u2113. In the projective plane there are no parallel lines, and in the hyperbolic plane there are at least two lines through\u00a0P<\/em> parallel to \u2113.<\/p>\n

Recent posts have looked at finite projective and affine planes. This post will look at finite hyperbolic planes.<\/p>\n

Finite projective planes come up far more often in application than finite hyperbolic planes. I’ve seen several resources on finite geometries that don’t mention the hyperbolic case. Finite hyperbolic geometries have less structure than their projective or affine counterparts and so are less useful.<\/p>\n

The definition of a finite hyperbolic plane isn’t entirely standardized. We will use the following axioms for this post.<\/p>\n

    \n
  1. Any two distinct points lie on exactly one line.<\/li>\n
  2. Given a line \u2113 and a point P<\/em> not on \u2113, there exist at least two lines through P<\/em> parallel to \u2113.<\/li>\n
  3. There exist four points, no three collinear.<\/li>\n
  4. Every line contains at least two points.<\/li>\n<\/ol>\n

    An example of a system satisfying these axioms is the complete graph of six vertices. The nodes are the points and the edges are the lines. In set-theoretic terms, the points are the one-element subsets of {1, 2, 3, 4, 5, 6} and the lines are all possible two-element subsets.<\/p>\n

    \"Complete<\/p>\n

    Note that lines in the finite hyperbolic plane are two element sets. The lines {5, 6} and {1, 3}, for example, are parallel because they do not intersect, even though the representations<\/em> of {5, 6} and {1, 3} in the visualization above are not parallel in the Euclidean sense.<\/p>\n

    There are many other examples of system satisfying the hyperbolic plane axioms. Too many in some sense. You could say finite hyperbolic systems are less common in application because they are more common in count.<\/p>\n

    Discrete geometry posts<\/h2>\n