Editing Convex uniform honeycomb (section)

== History ==
* '''1900''': [[Thorold Gosset]] enumerated the list of semiregular convex polytopes with regular cells ([[Platonic solid]]s) in his publication ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', including one regular cubic honeycomb, and two semiregular forms with tetrahedra and octahedra.
* '''1905''': [[Alfredo Andreini]] enumerated 25 of these tessellations.
* '''1991''': [[Norman Johnson (mathematician)|Norman Johnson]]'s manuscript ''Uniform Polytopes'' identified the list of 28.<ref name=OEIS/>
* '''1994''': [[Branko Grünbaum]], in his paper ''Uniform tilings of 3-space'', also independently enumerated all 28, after discovering errors in Andreini's publication. He found the 1905 paper, which listed 25, had 1 wrong, and [[Convex uniform honeycomb#Nonwythoffian forms (gyrated and elongated)|4 being missing]]. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991. He also mentions that [[I. Alexeyev]] of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time.
* '''2006''': George Olshevsky, in his manuscript ''Uniform Panoploid Tetracombs'', along with repeating the derived list of 11 convex uniform tilings, and 28 convex uniform honeycombs, expands a further derived list of 143 convex uniform tetracombs (Honeycombs of [[uniform 4-polytope]]s in 4-space).<ref>George Olshevsky, (2006, ''Uniform Panoploid Tetracombs'', Manuscript ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' [http://bendwavy.org/4HONEYS.pdf]</ref><ref name=OEIS>{{Cite OEIS|A242941|Convex uniform tessellations in dimension ''n''}}</ref>

Only 14 of the convex uniform polyhedra appear in these patterns:
* three of the five [[Platonic solid]]s (the [[tetrahedron]], [[cube]], and [[octahedron]]),
* six of the thirteen [[Archimedean solid]]s (the ones with reflective tetrahedral or octahedral symmetry), and
* five of the infinite family of [[prism (geometry)|prism]]s (the 3-, 4-, 6-, 8-, and 12-gonal ones; the 4-gonal prism duplicates the cube).

The [[icosahedron]], [[snub cube]], and [[square antiprism]] appear in some alternations, but those honeycombs cannot be realised with all edges unit length.

=== Names ===
This set can be called the '''regular and semiregular honeycombs'''. It has been called the '''Archimedean honeycombs''' by analogy with the convex uniform (non-regular) polyhedra, commonly called [[Archimedean solid]]s. Recently [[John Horton Conway|Conway]] has suggested naming the set as the '''Architectonic tessellations''' and the dual honeycombs as the '''[[Catoptric tessellation]]s'''.

The individual honeycombs are listed with names given to them by [[Norman Johnson (mathematician)|Norman Johnson]]. (Some of the terms used below are defined in [[Uniform 4-polytope#Geometric derivations for 46 nonprismatic Wythoffian uniform 4-polytopes]])

For cross-referencing, they are given with list indices from '''A'''ndreini (1-22), '''W'''illiams(1–2,9-19), '''J'''ohnson (11–19, 21–25, 31–34, 41–49, 51–52, 61–65), and '''G'''rünbaum(1-28). Coxeter uses &delta;<sub>4</sub> for a [[cubic honeycomb]], h&delta;<sub>4</sub> for an [[alternated cubic honeycomb]], q&delta;<sub>4</sub> for a [[quarter cubic honeycomb]], with subscripts for other forms based on the ring patterns of the Coxeter diagram.