Talk:Semiregular 4-polytope

message: I started this article last October before I recognized uniform polychoron as a wider and more useful category which allows all convex uniform polyhedra cells. Now with other articles fleshed out better, I'm prepared to let this article be deleted.
Opinions and votes?
YES
: Tom Ruen 09:23, 14 March 2006 (UTC)
replies: text: First let's make sure all content is repeated somewhere.  But yes, there's no clear reason for a separate article.  —Tamfang 23:45, 14 March 2006 (UTC)
thread_title:  Votes for deletion 

message: I've been looking at this paper: (Referenced in uniform polychoron history.)
* 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.
This wiki-article includes all of of the forms he enumerated, but he gives ALL for higher polytopes as well. Rather than deleteing this category as being less useful than the uniform polychora, expanding to include his full list seems useful, both to honor his work AND because there's no other work I know that describes ALL the higher semiregular polytopes/tessellations.
I'll look into this expansion and include in talk, and when ready, rename perhaps to something like: List of regular and semregular polytopes and honeycombs. Well, this is CLOSE to the extensive list of regular polytopes article as well, so maybe worth MERGING to that in the end since there's not that many semiregulars added.
=== polytope list ===
To be a bit more clear specifically Gosset lists: (His names in bold, and translating names as I can, and ??? for now otherwise)
==== Regular ====
Regular n-polytopes:
# n-ic pyramid: Simplex {3,...,3}
# n-ic double pyramid: Cross polytope {3,...,3,4}
# n-ic cube: Measure polytope {4,3,...,3}
# (n-1)-ic check: Measure polytope honeycomb {4,3...3,4}
Regular 4-polytopes:
# Octahedric: 24-cell {3,4,3}
# Tetrahedric: 120-cell {5,3,3}
# Dodecahedric: 600-cell {3,3,5}
Regular 5-polytopes: (honeycombs)
# Octahedric-check: {3,3,4,3}
# Double pyramidal check: {3,4,3,3}
==== Semiregular ====
Semiregular n-polytopes (n>2)
# (n-1)-ic semicheck: ??? honeycomb
Semiregular 4-polytopes
# Tetroctahedric: rectified 5-cell = r{3,3,3}
# tetricosahedric: snub 24-cell = r{3,4,3}
# Octicosahedric: rectified 600-cell = r{3,3,5}
# Simple tetroctahedric check: Tetrahedral-octahedral honeycomb
# Complex tetroctahedric check: Gyrated alternated cubic honeycomb
# Tetroctahedric semicheck: ??? honeycomb
Semiregular 5-polytopes
# 5-ic semi-regular: ??? polytope
Semiregular 6-polytopes
# 6-ic semi-regular: ??? polytope
Semiregular 7-polytopes
# 7-ic semi-regular: ??? polytope
Semiregular 8-polytopes
# 8-ic semi-regular: ??? polytope
Semiregular 9-polytopes
# 9-ic semi-regular: ??? honeycomb
The intro claims this list is complete, although no modern references (so far!) to confirm this, alhough Coxeter's 1964 "Regular polytopes" talks about him and this paper.
: On first and second pass I'm rather lost on these higher forms, and won't be extended this quickly! Tom Ruen 01:33, 9 August 2006 (UTC)
thread_title:  Expand and refocus to semiregular polytopes? 

message: To be a bit more clear specifically Gosset lists: (His names in bold, and translating names as I can, and ??? for now otherwise)
==== Regular ====
Regular n-polytopes:
# n-ic pyramid: Simplex {3,...,3}
# n-ic double pyramid: Cross polytope {3,...,3,4}
# n-ic cube: Measure polytope {4,3,...,3}
# (n-1)-ic check: Measure polytope honeycomb {4,3...3,4}
Regular 4-polytopes:
# Octahedric: 24-cell {3,4,3}
# Tetrahedric: 120-cell {5,3,3}
# Dodecahedric: 600-cell {3,3,5}
Regular 5-polytopes: (honeycombs)
# Octahedric-check: {3,3,4,3}
# Double pyramidal check: {3,4,3,3}
==== Semiregular ====
Semiregular n-polytopes (n>2)
# (n-1)-ic semicheck: ??? honeycomb
Semiregular 4-polytopes
# Tetroctahedric: rectified 5-cell = r{3,3,3}
# tetricosahedric: snub 24-cell = r{3,4,3}
# Octicosahedric: rectified 600-cell = r{3,3,5}
# Simple tetroctahedric check: Tetrahedral-octahedral honeycomb
# Complex tetroctahedric check: Gyrated alternated cubic honeycomb
# Tetroctahedric semicheck: ??? honeycomb
Semiregular 5-polytopes
# 5-ic semi-regular: ??? polytope
Semiregular 6-polytopes
# 6-ic semi-regular: ??? polytope
Semiregular 7-polytopes
# 7-ic semi-regular: ??? polytope
Semiregular 8-polytopes
# 8-ic semi-regular: ??? polytope
Semiregular 9-polytopes
# 9-ic semi-regular: ??? honeycomb
The intro claims this list is complete, although no modern references (so far!) to confirm this, alhough Coxeter's 1964 "Regular polytopes" talks about him and this paper.
: On first and second pass I'm rather lost on these higher forms, and won't be extended this quickly! Tom Ruen 01:33, 9 August 2006 (UTC)
thread_title:  polytope list 

message: Semiregular n-polytopes (n>2)
# (n-1)-ic semicheck: ??? honeycomb
Semiregular 4-polytopes
# Tetroctahedric: rectified 5-cell = r{3,3,3}
# tetricosahedric: snub 24-cell = r{3,4,3}
# Octicosahedric: rectified 600-cell = r{3,3,5}
# Simple tetroctahedric check: Tetrahedral-octahedral honeycomb
# Complex tetroctahedric check: Gyrated alternated cubic honeycomb
# Tetroctahedric semicheck: ??? honeycomb
Semiregular 5-polytopes
# 5-ic semi-regular: ??? polytope
Semiregular 6-polytopes
# 6-ic semi-regular: ??? polytope
Semiregular 7-polytopes
# 7-ic semi-regular: ??? polytope
Semiregular 8-polytopes
# 8-ic semi-regular: ??? polytope
Semiregular 9-polytopes
# 9-ic semi-regular: ??? honeycomb
The intro claims this list is complete, although no modern references (so far!) to confirm this, alhough Coxeter's 1964 "Regular polytopes" talks about him and this paper.
: On first and second pass I'm rather lost on these higher forms, and won't be extended this quickly! Tom Ruen 01:33, 9 August 2006 (UTC)
thread_title:  Semiregular