User:Double sharp/List of uniform polychora by Goursat tetrahedron

There are many relationships among the polychora. The Wythoff construction is able to construct some of the uniform polychora from the Goursat tetrahedra. The numbers that can be used for the sides of a non-dihedral Goursat tetrahedron that does not necessarily lead to only degenerate uniform polychora are 2, 3, 3/2, 4, 4/3, 5, 5/2, 5/3, and 5/4 (but numbers with numerator 4 and those with numerator 5 may not occur together). (4/2 can also be used, but only leads to degenerate uniform polychora as 4 and 2 have a common factor.) There are ??? such Schwarz triangles (? with X symmetry) which, together with the infinite family of dihedral Goursat tetrahedra, can form almost all of the uniform polyhedra.

There are many non-Wythoffian uniform polyhedra, which no Goursat tetrahedra can generate: some are given at the bottom.

Key to Klitzing's linear notation for Coxeter-Dynkin diagrams[edit]

ASCII diagrams from Richard Klitzing's website, will be replaced with proper Wikipedia CD graphics soon

Linear	Tridental	Loop-and-tail	Loop	Two-loop	Simplicial
oPoQoRo	oPoQo *bRo	oPoQoRoS *b	oPoQoRoS *a	oPoQoRoS a T c	oPoQoRoS a T c b U d
o-P-o-Q-o-R-o	o_ -P_ >o---R---o _Q- o-	o_ \| -Q_ R >o---P---o \| _S- o-	o---P---o \| \| S Q \| \| o---R---o	_o_ _P- \| -S_ o< T >o -Q_ \| _R- -o-	o / T \ P _o_ S /_Q R_\ o-----U-----o

Linear[edit]

Only three numbers, p, q, and r, are needed to define a linear Coxeter-Dynkin diagram for a polychoron symmetry. The notation [p q r] is used to distinguish them from the spherical Schwarz triangles for polyhedra, which are notated as (p q r).

Pentachoric[edit]

o3o3o3o =
o3o3o3/2o =
o3o3/2o3o =
o3o3/2o3/2o =
o3/2o3o3/2o =
o3/2o3/2o3/2o =

#	[p q r]	xooo Base	oxoo Rectified	ooxo Birectified	ooox Trirectified (Dual)	xxoo Truncated	xoxo Cantellated	xoox Runcinated	oxxo Bitruncated	oxox Bicantellated	ooxx Tritruncated	xxxo Cantitruncated	xxox Runcitruncated	xoxx Runcicantellated	oxxx Bicantitruncated	xxxx Runcicantitruncated (Omnitruncated)
1	[3 3 3] (μ=1)	pen	rap	rap	pen	tip	srip	spid	deca	srip	tip	grip	prip	prip	grip	gippid
2	[3 3 3/2] (μ=4)	pen	rap	rap	pen	tip	srip	[2firp]	deca	[pinnip+5(2thah)]	[Grünbaumian]	grip	[pirpop+5(2thah)]	[Grünbaumian]	[Grünbaumian]	[Grünbaumian]