User:Dc.samizdat/Isoclines
Isoclines in the sequence of regular convex 4-polytopes
[edit | edit source]Isoclinic rotations relate the regular convex 4-polytopes to each other as a sequence of increasingly 3-spherical convex hulls of isoclinic compounds of their predecessors. A general principle of the relationship of a 4-polytope to its adjacent 4-polytopes in the sequence emerges from a comparison of their characteristic isoclines, the circular helical geodesic paths traced by their vertices under isoclinic rotation. A regular 4-polytope's isocline chords are found in the chords of its predecessor's facets, and its isocline vertex figures are the vertex figures of its successor's facets.
Isoclinic compounds in sequence
[edit | edit source]The main sequence of the six regular convex 4-polytopes is evident in their vertex counts,[a] and within that sequence is a subsequence of four with triangular faces representing the major Coxeter symmetry groups: the 5-cell (tetrahedral A4 symmetry), 16-cell (octahedral B4 symmetry), 24-cell (F4 symmetry), and 600-cell (icosahedral H4 symmetry).
Regular convex 4-polytopes | |||||
---|---|---|---|---|---|
5-cell Hyper- |
16-cell Hyper- |
8-cell Hyper- |
24-cell | 600-cell Hyper- |
120-cell Hyper- |
1 {5/2} x 2 | 1 {8/2}=2{4} x 3 | 2 {8/2}=2{4} x 3 | 2 {12/4}=4{3} x 4 | 20 {30/4}=2{15/2} x 6 | 100 {30/4}=2{15/2} x 6 |
Regular convex 4-polytopes
[edit | edit source]Sequence of 6 regular convex 4-polytopes | ||||||
---|---|---|---|---|---|---|
Symmetry group | A4 | B4 | F4 | H4 | ||
Name | 5-cell Hyper-tetrahedron |
16-cell Hyper-octahedron |
8-cell Hyper-cube |
24-cell Hyper-cuboctahedron |
600-cell Hyper-icosahedron |
120-cell Hyper-dodecahedron |
SchlΓ€fli symbol | {3, 3, 3} | {3, 3, 4} | {4, 3, 3} | {3, 4, 3} | {3, 3, 5} | {5, 3, 3} |
Coxeter mirrors | ||||||
Mirror dihedrals | π /3 π /3 π /3 π /2 π /2 π /2 | π /3 π /3 π /4 π /2 π /2 π /2 | π /4 π /3 π /3 π /2 π /2 π /2 | π /3 π /4 π /3 π /2 π /2 π /2 | π /3 π /3 π /5 π /2 π /2 π /2 | π /5 π /3 π /3 π /2 π /2 π /2 |
Graph | ||||||
Vertices[d] | 5 tetrahedral | 8 octahedral | 16 tetrahedral | 24 cubical | 120 icosahedral | 600 tetrahedral |
Edges | 10 triangular | 24 square | 32 triangular | 96 triangular | 720 pentagonal | 1200 triangular |
Faces | 10 triangles | 32 triangles | 24 squares | 96 triangles | 1200 triangles | 720 pentagons |
Cells | 5 {3, 3} | 16 {3, 3} | 8 {4, 3} | 24 {3, 4} | 600 {3, 3} | 120 {5, 3} |
Tori | 5 {3, 3} | 8 {3, 3} x 2 | 4 {4, 3} x 2 | 6 {3, 4} x 4 | 30 {3, 3} x 20 | 10 {5, 3} x 12 |
Inscribed | 120 in 120-cell | 675 in 120-cell | 2 16-cells | 3 8-cells | 25 24-cells | 10 600-cells |
Great polygons | 2 squares x 3[e] | 4 rectangles x 4 | 4 hexagons x 4 | 12 decagons x 6 | 100 irregular hexagons x 4 | |
Petrie polygons | 1 pentagon x 2 | 1 octagon x 3 | 2 octagons x 4 | 2 dodecagons x 4 | 4 30-gons x 6 | 20 30-gons x 4 |
Long radius | ||||||
Edge length[f] | ||||||
Short radius | ||||||
Area | ||||||
Volume | ||||||
4-Content |
Notes
[edit | edit source]- β 1.0 1.1 The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content[1] within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 600-cell is the 120-point 4-polytope: fifth in the ascending sequence that runs from 5-point 4-polytope to 600-point 4-polytope.
- β Cite error: Invalid
<ref>
tag; no text was provided for refs namedisoclinic geodesic
- β Cite error: Invalid
<ref>
tag; no text was provided for refs namedisoclinic compounds
- β Cite error: Invalid
<ref>
tag; no text was provided for refs named4-polytopes ordered by size and complexity
- β Cite error: Invalid
<ref>
tag; no text was provided for refs namedSix orthogonal planes of the Cartesian basis
- β Cite error: Invalid
<ref>
tag; no text was provided for refs namededge length of successor
Citations
[edit | edit source]- β Coxeter 1973, pp. 292-293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.