Euclidean tilings by convex regular polygons: Difference between revisions

Revision as of 02:14, 28 January 2024

Revision as of 16:50, 8 March 2024

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==Plane-vertex tilings==

There are 17 combinations of regular convex polygons that form 21 types of [[Vertex (geometry)#Of a plane tiling|plane-vertex tilings]].<ref>{{citation|title=The Elements of Plane Practical Geometry, Etc|first=Elmslie William|last=Dallas|publisher=John W. Parker & Son|year=1855|page=134|url=https://books.google.com/books?id=y4BaAAAAcAAJ&pg=PA134}}</ref><ref>[[Tilings and Patternspatterns]], Figure 2.1.1, p.60</ref> Polygons in these meet at a point with no gap or overlap. Listing by their [[vertex figure]]s, one has 6 polygons, three have 5 polygons, seven have 4 polygons, and ten have 3 polygons.<ref>[[Tilings and Patternspatterns]], p.58-69</ref>

As detailed in the sections above, three of them can make [[#Regular_tilings|regular tilings]] (6³, 4⁴, 3⁶), and eight more can make [[#Archimedean,_uniform_or_semiregular_tilings|semiregular or archimedean tilings]], (3.12.12, 4.6.12, 4.8.8, (3.6)², 3.4.6.4, 3.3.4.3.4, 3.3.3.4.4, 3.3.3.3.6). Four of them can exist in higher [[#k-uniform_tilings|''k''-uniform tilings]] (3.3.4.12, 3.4.3.12, 3.3.6.6, 3.4.4.6), while six can not be used to completely tile the plane by regular polygons with no gaps or overlaps - they only tessellate space entirely when irregular polygons are included (3.7.42, 3.8.24, 3.9.18, 3.10.15, 4.5.20, 5.5.10).<ref>{{Cite web|url=https://blogs.ams.org/visualinsight/2015/02/01/pentagon-decagon-packing/|title=Pentagon-Decagon Packing|website=American Mathematical Society|publisher=AMS|access-date=2022-03-07}}</ref>

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=== 2-uniform tilings ===

There are twenty (20) '''2-uniform tilings''' of the Euclidean plane. (also called '''2-[[isogonal figure|isogonal]] tilings''' or '''[[demiregular tiling]]s''') {{r|"Critchlow 1969"|p=62-67}} <ref>[[Tilings and Patternspatterns]], Grünbaum and Shephard 1986, pp. 65-67</ref><ref>{{Cite web |url=http://www.math.nus.edu.sg/aslaksen/papers/Demiregular.pdf |title=In Search of Demiregular Tilings |access-date=2015-06-04 |archive-url=https://web.archive.org/web/20160507010400/http://www.math.nus.edu.sg/aslaksen/papers/Demiregular.pdf |archive-date=2016-05-07 |url-status=dead }}</ref> Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2.

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