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.

See also: Goursat_tetrahedron

The tables below use "x" to indicate a marked node, "o" to indicate an unmarked node, and "s" to indicate a snub node.

Every diagram formed by removing a node and all edges connected to it from a polychoral Coxeter-Dynkin diagram represents a Wythoffian uniform polyhedron, which may all be found at List of uniform polyhedra by Schwarz triangle. When any one of these polyhedral elements are degenerate, so is the resulting polychoron, which is then not given here except as a note "[Grünbaumian]", after Branko Grünbaum who described many polytopes that would traditionally be regarded as degenerate. Nevertheless, degenerate polychora which do not have degenerate elements are included.

Key to Klitzing's linear notation for Coxeter-Dynkin diagrams
ASCII diagrams from Richard Klitzing's website, will be replaced with proper Wikipedia CD graphics soon

Linear
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

 * 1) o3o3o3o =
 * 2) o3o3o3/2o =
 * 3) o3o3/2o3o =
 * 4) o3o3/2o3/2o =
 * 5) o3/2o3o3/2o =
 * 6) o3/2o3/2o3/2o =

Hexadecachoric

 * 1) o3o3o4o =
 * 2) o3/2o3o4o
 * 3) o3o3o4/3o
 * 4) o3o3/2o4o
 * 5) o3/2o3/2o4o
 * 6) o3o3/2o4/3o
 * 7) o3/2o3o4/3o
 * 8) o3/2o3/2o4/3o

Icositetrachoric

 * 1) o3o4o3o =
 * 2) o3o4o3/2o
 * 3) o3o4/3o3o
 * 4) o3o4/3o3/2o
 * 5) o3/2o4o3/2o
 * 6) o3/2o4/3o3/2o

Hexacosichoric

 * 1) o3o3o5o =
 * 2) o3/2o3o5o
 * 3) o3o3o5/4o
 * 4) o3o3/2o5o
 * 5) o3/2o3/2o5o
 * 6) o3o3/2o5/4o
 * 7) o3/2o3o5/4o
 * 8) o3/2o3/2o5/4o
 * 9) o3o3o5/2o
 * 10) o3o3o5/3o
 * 11) o3/2o3o5/2o
 * 12) o3o3/2o5/2o
 * 13) o3o3/2o5/3o
 * 14) o3/2o3o5/3o
 * 15) o3/2o3/2o5/2o
 * 16) o3/2o3/2o5/3o
 * 17) o3o5o5/2o
 * 18) o3o5o5/3o
 * 19) o3/2o5o5/2o
 * 20) o3/2o5o5/3o
 * 21) o3o5/4o5/2o
 * 22) o3o5/4o5/3o
 * 23) o3/2o5/4o5/2o
 * 24) o3/2o5/4o5/3o
 * 25) o3o5/2o5o
 * 26) o3/2o5/2o5o
 * 27) o3o5/3o5o
 * 28) o3/2o5/3o5o
 * 29) o3o5/2o5/4o
 * 30) o3o5/3o5/4o
 * 31) o3/2o5/2o5/4o
 * 32) o3/2o5/3o5/4o
 * 33) o5o3o5/2o
 * 34) o5o3o5/3o
 * 35) o5o3/2o5/2o
 * 36) o5o3/2o5/3o
 * 37) o5/4o3o5/2
 * 38) o5/4o3/2o5/2o
 * 39) o5/4o3o5/3o
 * 40) o5/4o3/2o5/3o
 * 41) o5o5/2o5o
 * 42) o5o5/2o5/4o
 * 43) o5o5/3o5o
 * 44) o5o5/3o5/4o
 * 45) o5/4o5/2o5/4o
 * 46) o5/4o5/3o5/4o
 * 47) o5/2o5o5/2o
 * 48) o5/2o5o5/3o
 * 49) o5/3o5o5/3o
 * 50) o5/2o5/4o5/2o
 * 51) o5/2o5/4o5/3o
 * 52) o5/3o5/4o5/3o

Demitesseractic

 * 1) o3o3o *b3o =
 * 2) o3o3o *b3/2o =
 * 3) o3/2o3/2o *b3o
 * 4) o3/2o3/2o *b3/2o

Loop-and-tail
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Loop
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Two-loop
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Simplicial
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Duoprisms and prismatic prisms

 * 1) on/do om/bo =
 * 2) o o on/do =
 * 3) o o o o =