They are represented by
Coxeter-Dynkin diagrams of three constructive forms: • ... (As an
alternated orthotope) s{21,1,...,1} • ... (As an alternated
hypercube) h{4,3
n−1} • .... (As a demihypercube) {31,
n−3,1}
H.S.M. Coxeter also labeled the third bifurcating diagrams as '
1k
1' representing the lengths of the three branches and led by the ringed branch. An
n-demicube,
n greater than 2, has
n(
n − 1)/2 edges meeting at each vertex. The graphs below show less edges at each vertex due to overlapping edges in the symmetry projection. In general, a demicube's elements can be determined from the original
n-cube: (with C
n,
m =
mth-face count in
n-cube = 2
n−
m n!/(
m!(
n −
m)!)) •
Vertices: D
n,0 = 1/2 C
n,0 = 2
n−1 (Half the
n-cube vertices remain) •
Edges: D
n,1 = C
n,2 = 1/2
n(
n – 1) 2
n−2 (All original edges lost, each square faces create a new edge) •
Faces: D
n,2 = 4 * C
n,3 = 2/3
n(
n − 1)(
n − 2) 2
n−3 (All original faces lost, each cube creates 4 new triangular faces) •
Cells: D
n,3 = C
n,3 + 23 C
n,4 (tetrahedra from original cells plus new ones) •
Hypercells: D
n,4 = C
n,4 + 24 C
n,5 (16-cells and 5-cells respectively) • ... • [For
m = 3, ... ,
n − 1]: D
n,
m = C
n,
m + 2
m C
n,
m+1 (
m-demicubes and
m-simplexes respectively) • ... •
Facets: D
n,
n−1 = 2
n + 2
n−1 ((
n − 1)-demicubes and (
n − 1)-simplices respectively) == Symmetry group ==