Editing 8 (section)

=== Geometry ===

A [[polygon]] with eight sides is an [[octagon]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Octagon |url=https://mathworld.wolfram.com/Octagon.html |access-date=2020-08-07 |website=mathworld.wolfram.com |language=en}}</ref> A regular octagon can fill a [[Euclidean tilings by convex regular polygons#Plane-vertex tilings|plane-vertex]] with a regular [[triangle]] and a regular [[icositetragon]], as well as [[tessellation|tessellate]] two-dimensional space alongside squares in the [[truncated square tiling]]. This tiling is one of eight [[Archimedean tiling]]s that are semi-regular, or made of more than one type of regular [[polygon]], and the only tiling that can admit a regular octagon.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Regular Octagon |url=https://mathworld.wolfram.com/RegularOctagon.html|access-date=2022-06-25|website=mathworld.wolfram.com|language=en}}</ref>  The [[Ammann–Beenker tiling]] is a nonperiodic tesselation of [[prototile]]s that feature prominent octagonal ''silver'' eightfold symmetry, that is the two-dimensional [[orthographic projection]] of the four-dimensional [[8-8 duoprism]].<ref>{{Cite book |author =Katz, A |chapter=Matching rules and quasiperiodicity: the octagonal tilings |title=Beyond quasicrystals |publisher=Springer |year=1995 |pages=141–189 |isbn=978-3-540-59251-8 |doi=10.1007/978-3-662-03130-8_6 |editor1-first=F. |editor1-last=Axel |editor2-first=D. |editor2-last=Gratias}}</ref>

An [[octahedron]] is a [[regular polyhedron]] with eight [[equilateral triangle]]s as [[face (geometry)|faces]]. is the [[dual polyhedron]] to the cube and one of eight [[Deltahedron|convex deltahedra]].<ref>{{Citation|last1=Freudenthal|first1=H|last2=van der Waerden|first2=B. L.|authorlink1=Hans Freudenthal | authorlink2=B. L. van der Waerden|title=Over een bewering van Euclides ("On an Assertion of Euclid")|journal=[[Simon Stevin (journal)|Simon Stevin]]|volume=25|pages=115–128|year=1947|language=Dutch}}</ref><ref>{{Cite web|url=http://www.interocitors.com/polyhedra/Deltahedra/Convex |author=Roger Kaufman |title=The Convex Deltahedra And the Allowance of Coplanar Faces |website=The Kaufman Website |access-date=2022-06-25}}</ref> The [[stella octangula]], or ''eight-pointed star'', is the only [[stellation]] with [[octahedral symmetry]]. It has eight triangular faces alongside eight vertices that forms a cubic [[faceting]], composed of two self-dual [[Regular tetrahedron|tetrahedra]] that makes it the simplest of five [[Uniform polyhedron compound|regular compound]]s. The [[cuboctahedron]], on the other hand, is a [[rectification (geometry)|rectified]] cube or rectified octahedron, and one of only two convex [[Quasiregular polyhedron|quasiregular polyhedra]]. It contains eight equilateral triangular faces, whose first [[stellation]] is the [[compound of cube and octahedron|cube-octahedron compound]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Cuboctahedron |url=https://mathworld.wolfram.com/Cuboctahedron.html |access-date=2022-06-25 |website=mathworld.wolfram.com |language=en }}</ref><ref>{{Cite book |last=Coxeter |first=H.S.M. |author-link=Harold Scott MacDonald Coxeter |year=1973 |orig-year=1948 |title=Regular Polytopes |publisher=Dover |place=New York |edition=3rd |pages=18–19 |title-link=Regular Polytopes (book) }}</ref>