Regular 4-polytope

In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogs of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

Regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century, although the full set were not discovered until later.

There are six convex and ten star regular 4-polytopes, giving a total of sixteen.

History

The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures.

Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell). He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F  E + V = 2). That excludes cells and vertex figures as {5,5/2} and {5/2,5}.

Edmund Hess (18431903) published the complete list in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder.

Construction

The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which form its cells and a dihedral angle constraint

to ensure that the cells meet to form a closed 3-surface.

The six convex and ten star polytopes described are the only solutions to these constraints.

There are four nonconvex Schläfli symbols {p,q,r} that have valid cells {p,q} and vertex figures {q,r}, and pass the dihedral test, but fail to produce finite figures: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}.

Regular convex 4-polytopes

The regular convex 4-polytopes are the four-dimensional analogs of the Platonic solids in three dimensions and the convex regular polygons in two dimensions.

Five of them may be thought of as close analogs of the Platonic solids. One additional figure, the 24-cell, has no close 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.

Properties

The following tables lists some properties of the six convex regular 4-polytopes. The symmetry groups of these 4-polytopes 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.

NamesImageFamilySchläfli
Coxeter
VEFCVert.
fig.
Dual Symmetry group
5-cell
pentachoron
pentatope
4-simplex
n-simplex
(An family)
{3,3,3}
51010
{3}
5
{3,3}
{3,3}(self-dual)A4
[3,3,3]
120
8-cell
octachoron
tesseract
4-cube
n-cube
(Bn family)
{4,3,3}
163224
{4}
8
{4,3}
{3,3}16-cellB4
[4,3,3]
384
16-cell
hexadecachoron
4-orthoplex
n-orthoplex
(Bn family)
{3,3,4}
82432
{3}
16
{3,3}
{3,4}8-cellB4
[4,3,3]
384
24-cell
icositetrachoron
octaplex
polyoctahedron (pO)
Fn family{3,4,3}
249696
{3}
24
{3,4}
{4,3}(self-dual)F4
[3,4,3]
1152
120-cell
hecatonicosachoron
dodecacontachoron
dodecaplex
polydodecahedron (pD)
n-pentagonal polytope
(Hn family)
{5,3,3}
6001200720
{5}
120
{5,3}
{3,3}600-cellH4
[5,3,3]
14400
600-cell
hexacosichoron
tetraplex
polytetrahedron (pT)
n-pentagonal polytope
(Hn family)
{3,3,5}
1207201200
{3}
600
{3,3}
{3,5}120-cellH4
[5,3,3]
14400

John Conway advocates the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), dodecaplex or polydodecahedron (pD), and tetraplex or polytetrahedron (pT).[1]

Norman Johnson advocates the names n-cell, or pentachoron, tesseract or octachoron, hexadecachoron, icositetrachoron, hecatonicosachoron (or dodecacontachoron), and hexacosichoron, coining the term polychoron being a 4D analogy to the 3D polyhedron, and 2D polygon, expressed from the Greek roots poly ("many") and choros ("room" or "space").[2][3]

The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analog of Euler's polyhedral formula:

where Nk denotes the number of k-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).

The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.[4]

As configurations

A regular 4-polytope can be completely described as a configuration matrix containing counts of its component elements. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers (upper left to lower right) say how many of each element occur in the whole 4-polytope. The non-diagonal numbers say how many of the column's element occur in or at the row's element. For example, there are 2 vertices in each edge (each edge has 2 vertices), and 2 cells meet at each face (each face belongs to 2 cells), in any regular 4-polytope. Notice that the configuration for the dual polytope can be obtained by rotating the matrix by 180 degrees.[5][6]

5-cell
{3,3,3}
16-cell
{3,3,4}
tesseract
{4,3,3}
24-cell
{3,4,3}
600-cell
{3,3,5}
120-cell
{5,3,3}

Visualization

The following table shows some 2-dimensional projections of these 4-polytopes. Various other visualizations can be found in the external links below. The Coxeter-Dynkin diagram graphs are also given below the Schläfli symbol.

A4 = [3,3,3]BC4 = [4,3,3]F4 = [3,4,3]H4 = [5,3,3]
5-cell8-cell16-cell24-cell120-cell600-cell
{3,3,3}{4,3,3}{3,3,4}{3,4,3}{5,3,3}{3,3,5}
Solid 3D orthographic projections

Tetrahedral
envelope

(cell/vertex-centered)

Cubic envelope
(cell-centered)

cubic envelope
(cell-centered)

Cuboctahedral
envelope

(cell-centered)

Truncated rhombic
triacontahedron
envelope

(cell-centered)

pentakis icosidodecahedral
envelope

(vertex-centered)
Wireframe Schlegel diagrams (Perspective projection)

Cell-centered

Cell-centered

Cell-centered

Cell-centered

Cell-centered

Vertex-centered
Wireframe stereographic projections (3-sphere)

Regular star (Schläfli–Hess) 4-polytopes

The SchläfliHess 4-polytopes are the complete set of 10 regular self-intersecting star polychora (four-dimensional polytopes).[8] They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. Each is represented by a Schläfli symbol {p,q,r} in which one of the numbers is 5/2. They are thus analogous to the regular nonconvex KeplerPoinsot polyhedra, which are in turn analogous to the pentagram.

Names

Their names given here were given by John Conway, extending Cayley's names for the Kepler–Poinsot polyhedra: along with stellated and great, he adds a grand modifier. Conway offered these operational definitions:

  1. stellation – replaces edges by longer edges in same lines. (Example: a pentagon stellates into a pentagram)
  2. greatening – replaces the faces by large ones in same planes. (Example: an icosahedron greatens into a great icosahedron)
  3. aggrandizement – replaces the cells by large ones in same 3-spaces. (Example: a 600-cell aggrandizes into a grand 600-cell)

John Conway names the 10 forms from 3 regular celled 4-polytopes: pT=polytetrahedron {3,3,5} (a tetrahedral 600-cell), pI=polyicoshedron {3,5,5/2} (an icosahedral 120-cell), and pD=polydodecahedron {5,3,3} (a dodecahedral 120-cell), with prefix modifiers: g, a, and s for great, (ag)grand, and stellated. The final stellation, the great grand stellated polydodecahedron contains them all as gaspD.

Symmetry

All ten polychora have [3,3,5] (H4) hexacosichoric symmetry. They are generated from 6 related Goursat tetrahedra rational-order symmetry groups: [3,5,5/2], [5,5/2,5], [5,3,5/2], [5/2,5,5/2], [5,5/2,3], and [3,3,5/2].

Each group has 2 regular star-polychora, except for two groups which are self-dual, having only one. So there are 4 dual-pairs and 2 self-dual forms among the ten regular star polychora.

Properties

Note:

The cells (polyhedra), their faces (polygons), the polygonal edge figures and polyhedral vertex figures are identified by their Schläfli symbols.

Name
Conway (abbrev.)
Orthogonal
projection
Schläfli
Coxeter
C
{p, q}
F
{p}
E
{r}
V
{q, r}
Dens. χ
Icosahedral 120-cell
polyicosahedron (pI)
{3,5,5/2}
120
{3,5}
1200
{3}
720
{5/2}
120
{5,5/2}
4 480
Small stellated 120-cell
stellated polydodecahedron (spD)
{5/2,5,3}
120
{5/2,5}
720
{5/2}
1200
{3}
120
{5,3}
4 480
Great 120-cell
great polydodecahedron (gpD)
{5,5/2,5}
120
{5,5/2}
720
{5}
720
{5}
120
{5/2,5}
6 0
Grand 120-cell
grand polydodecahedron (apD)
{5,3,5/2}
120
{5,3}
720
{5}
720
{5/2}
120
{3,5/2}
20 0
Great stellated 120-cell
great stellated polydodecahedron (gspD)
{5/2,3,5}
120
{5/2,3}
720
{5/2}
720
{5}
120
{3,5}
20 0
Grand stellated 120-cell
grand stellated polydodecahedron (aspD)
{5/2,5,5/2}
120
{5/2,5}
720
{5/2}
720
{5/2}
120
{5,5/2}
66 0
Great grand 120-cell
great grand polydodecahedron (gapD)
{5,5/2,3}
120
{5,5/2}
720
{5}
1200
{3}
120
{5/2,3}
76 480
Great icosahedral 120-cell
great polyicosahedron (gpI)
{3,5/2,5}
120
{3,5/2}
1200
{3}
720
{5}
120
{5/2,5}
76 480
Grand 600-cell
grand polytetrahedron (apT)
{3,3,5/2}
600
{3,3}
1200
{3}
720
{5/2}
120
{3,5/2}
191 0
Great grand stellated 120-cell
great grand stellated polydodecahedron (gaspD)
{5/2,3,3}
120
{5/2,3}
720
{5/2}
1200
{3}
600
{3,3}
191 0

See also

References

Citations

  1. Conway, 2008, Chapter 26, Higher Still
  2. "Convex and abstract polytopes", Programme and abstracts, MIT, 2005
  3. Johnson (2015), Chapter 11, Section 11.5 Spherical Coxeter groups
  4. Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
  5. Coxeter, Regular Polytopes, sec 1.8 Configurations
  6. Coxeter, Complex Regular Polytopes, p.117
  7. The Symmetries of Things, John Conway, (2008), p. 406, Fig 26.2
  8. Coxeter, Star polytopes and the Schläfli function f{α,β,γ) p. 122 2. The Schläfli-Hess polytopes

Bibliography

  • 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.
  • D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26, Regular Star-polytopes, pp. 404–408)
  • Edmund Hess, (1883) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder .
  • Edmund Hess Uber die regulären Polytope höherer Art, Sitzungsber Gesells Beförderung gesammten Naturwiss Marburg, 1885, 31-57
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
    • (Paper 10) H.S.M. Coxeter, Star Polytopes and the Schlafli Function f(α,β,γ) [Elemente der Mathematik 44 (2) (1989) 2536]
  • H. S. M. Coxeter, Regular Complex Polytopes, 2nd. ed., Cambridge University Press 1991. ISBN 978-0-521-39490-1.
  • Peter McMullen and Egon Schulte, Abstract Regular Polytopes, 2002, PDF
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