CONVEX REGULAR 4-POLYTOPE

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In mathematics, a 'convex regular 4-polytope' (or 'polychoron') is 4-dimensional polytope which is both regular and convex. These are the four-dimensional analogs of the Platonic solids (in three dimensions) and the regular polygons (in two dimensions).
These polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. Schläfli discovered that there are precisely six such figures. Five of these may be thought of as higher dimensional analogs of the Platonic solids. There is one additional figure (the 24-cell) which has no three-dimensional equivalent.
Each convex regular 4-polytope is bounded by a set of 3-dimensional ''cells'' which are all Platonic solids of the same type and size. These are fitted together along their respective faces in a regular fashion.

Contents

Properties

Visualizations

See also

References

External links

Properties

The following tables lists some properties of the six convex regular polychora. The symmetry groups of these polychora are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group.
{| class="wikitable" style="margin: auto; text-align: center;"
! Name || Family || Schläfli
symbol || Vertices || Edges || Faces || Cells || Vertex figures || Dual polytope
!colspan=2 | Symmetry group
|-
| pentachoron || simplex || {3,3,3} || 5 || 10 || 10
triangles || 5
tetrahedra || tetrahedra || (self-dual) || ''A''₄ || 120
|-
| tesseract || hypercube || {4,3,3} || 16 || 32 || 24
squares || 8
cubes || tetrahedra || 16-cell || ''B''₄ || 384
|-
| 16-cell || cross-polytope || {3,3,4} || 8 || 24 || 32
triangles || 16
tetrahedra || octahedra || tesseract || ''B''₄ || 384
|-
| 24-cell || || {3,4,3} || 24 || 96 || 96
triangles || 24
octahedra || cubes || (self-dual) || ''F''₄ || 1152
|-
| 120-cell || || {5,3,3} || 600 || 1200 || 720
pentagons || 120
dodecahedra || tetrahedra || 600-cell || ''H''₄ || 14400
|-
| 600-cell || || {3,3,5} || 120 || 720 || 1200
triangles || 600
tetrahedra || icosahedra || 120-cell || ''H''₄ || 14400
|}
Since the boundaries of each of these figures is topologically equivalent to a 3-sphere, whose Euler characteristic is zero, we have the 4-dimensional analog of Euler's polyhedral formula:
:$N\_0\; -\; N\_1\; +\; N\_2\; -\; N\_3\; =\; 0,$
where ''N''_''k'' denotes the number of ''k''-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).

Visualizations

The following table shows some 2 dimensional projections of these polytopes. Various other visualizations can be found in the external links below.
{| class="wikitable" style="margin: auto; text-align: center;"
|-
! {3,3,3} || {4,3,3} || {3,3,4} || {3,4,3} || {5,3,3} || {3,3,5}
|-
|colspan=6|Wireframe orthographic projections
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|colspan=6|Solid orthographic projections (cell-centered)
|-
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tetrahedral
envelope
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cubic envelope
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octahedral
envelope
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cuboctahedral
envelope
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truncated rhombic
triacontahedron
envelope
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pentakis dodecahedral
envelope
|-
|colspan=6|Wireframe Schlegel diagrams (Perspective projection)
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(Cell-centered)
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(Cell-centered)
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(Cell-centered)
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(Cell-centered)
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(Cell-centered)
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(Vertex-centered)
|-
|colspan=6|Wireframe stereographic projections (Hyperspherical)
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|}

See also

★ Uniform polychoron Includes families of uniform polychora created from these 6 regular forms.

★ Schläfli-Hess polychoron - 10 nonconvex regular polychora

★ Regular polytope

★ List of regular polytopes

★ Platonic solid

References

★ H. S. M. Coxeter, ''Introduction to Geometry, 2nd ed.'', John Wiley & Sons Inc., 1969. ISBN 0-471-50458-0.

★ H. S. M. Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.

External links

★ Jonathan Bowers, 16 regular polychora

★ Regular 4D Polytope Foldouts

★ Catalog of Polytope Images A collection of stereographic projections of 4-polytopes.

★ [1]