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Renamed "Types of vertices with regular polygons" to "Plane-vertex tilings," moved section below "Archimedean, uniform or semiregular tilings". Rearranged order of list of plane-vertexes, by order of quantity (top-to-bottom) and relevancy to article. Reworded phrasing and redirected links, as well as noted on the last plane-vertex specific tilings as included for completion |
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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]] possible.<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 Patterns, 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.
As detailed in the sections above, three of them can make [[#Regular_tilings|regular tilings]] (6<sup>3</sup>, 4<sup>4</sup>, 3<sup>6</sup>), 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)<sup>6</sup>, 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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