In geometry , the 5-CELL is a four-dimensional object bounded by 5
tetrahedral cells . It is also known as a C5, PENTACHORON, PENTATOPE,
PENTAHEDROID, or TETRAHEDRAL PYRAMID . It is a 4-SIMPLEX , the
simplest possible convex regular 4-polytope (four-dimensional analogue
of a
The REGULAR 5-CELL is bounded by regular tetrahedra , and is one of the six regular convex 4-polytopes , represented by Schläfli symbol {3,3,3}. CONTENTS * 1 Alternative names * 2.1 Construction
* 2.2
* 3 Irregular
ALTERNATIVE NAMES * Pentachoron * 4-simplex * Pentatope * Pentahedroid (Henry Parker Manning) * Pen (Jonathan Bowers: for pentachoron) * Hyperpyramid, tetrahedral pyramid GEOMETRY The
CONSTRUCTION The
The simplest set of coordinates is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (τ,τ,τ,τ), with edge length 2√2, where τ is the golden ratio . The
Another set of origin-centered coordinates in 4-space can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2√2: ( 1 , 1 , 1 , 1 / 5 ) {displaystyle left(1,1,1,-1/{sqrt {5}}right)} ( 1 , 1 , 1 , 1 / 5 ) {displaystyle left(1,-1,-1,-1/{sqrt {5}}right)} ( 1 , 1 , 1 , 1 / 5 ) {displaystyle left(-1,1,-1,-1/{sqrt {5}}right)} ( 1 , 1 , 1 , 1 / 5 ) {displaystyle left(-1,-1,1,-1/{sqrt {5}}right)} ( 0 , 0 , 0 , 5 1 / 5 ) {displaystyle left(0,0,0,{sqrt {5}}-1/{sqrt {5}}right)} The vertices of a 4-simplex (with edge √2) can be more simply
constructed on a hyperplane in 5-space, as (distinct) permutations of
(0,0,0,0,1) or (0,1,1,1,1); in these positions it is a facet of,
respectively, the
BOERDIJK–COXETER HELIX A
PROJECTIONS The A4
orthographic projections
Ak
GRAPH DIHEDRAL SYMMETRY PROJECTIONS TO 3 DIMENSIONS
A 3D projection of a
The vertex-first projection of the
The face-first projection of the
IRREGULAR 5-CELL There are many lower symmetry forms, including these found in uniform polytope vertex figures : SYMMETRY Order 120 Order 24 Order 12 Order 6 + Order 10 NAME Regular 5-cell Tetrahedral pyramid Triangular-pyramidal pyramid Pentagonal hyperdisphenoid SCHLäFLI SYMBOL {3,3,3} {3,3} ∨ ( ) {3} ∨ { } Example
Vertex
figure
The TETRAHEDRAL PYRAMID is a special case of a 5-CELL, a polyhedral pyramid , constructed as a regular tetrahedron base in a 3-space hyperplane , and an apex point above the hyperplane. The four sides of the pyramid are made of tetrahedron cells. Many uniform 5-polytopes have TETRAHEDRAL PYRAMID vertex figures : Symmetry , order 24 Schlegel diagram Name Coxeter diagram { }×{3,3,3} { }×{4,3,3} { }×{5,3,3} t{3,3,3,3} t{4,3,3,3} t{3,4,3,3} Other uniform 5-polytopes have irregular
SYMMETRY , ORDER 12 , ORDER 6 , ORDER 8 , ORDER 4 Schlegel diagram Name Coxeter diagram t12α5 t12γ5 t012α5 t012γ5 t123α5 t123γ5 SYMMETRY , ORDER 2 , ORDER 2 +, ORDER 1 Schlegel diagram Name Coxeter diagram t0123α5 t0123γ5 t0123β5 t01234α5 t01234γ5 COMPOUND The compound of two 5-cells in dual configurations can be seen in
this A5
RELATED POLYTOPES AND HONEYCOMB The pentachoron (5-cell) is the simplest of 9 uniform polychora
constructed from the
SCHLäFLI {3,3,3} T{3,3,3} R{3,3,3} RR{3,3,3} 2T{3,3,3} TR{3,3,3} T0,3{3,3,3} T0,1,3{3,3,3} T0,1,2,3{3,3,3} COXETER SCHLEGEL 1K2 FIGURES IN N DIMENSIONS SPACE FINITE EUCLIDEAN HYPERBOLIC N 3 4 5 6 7 8 9 10 Coxeter group E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = E 8 {displaystyle {tilde {E}}_{8}} = E8+ E10 = T 8 {displaystyle {bar {T}}_{8}} = E8++ Coxeter diagram Symmetry (order) ] ORDER 12 120 192 103,680 2,903,040 696,729,600 ∞ GRAPH - - NAME 1−1,2 102 112 122 132 142 152 162 2K1 FIGURES IN N DIMENSIONS SPACE FINITE EUCLIDEAN HYPERBOLIC N 3 4 5 6 7 8 9 10 Coxeter group E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = E 8 {displaystyle {tilde {E}}_{8}} = E8+ E10 = T 8 {displaystyle {bar {T}}_{8}} = E8++ Coxeter diagram SYMMETRY ] ORDER 12 120 384 51,840 2,903,040 696,729,600 ∞ GRAPH - - NAME 2−1,1 201 211 221 231 241 251 261 It is in the sequence of regular polychora : the tesseract {4,3,3},
{P,3,3} POLYTOPES SPACE S3 H3 FORM FINITE PARACOMPACT NONCOMPACT NAME {3,3,3} {4,3,3} {5,3,3} {6,3,3} {7,3,3} {8,3,3} ...{∞,3,3} IMAGE Cells {p,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} It is similar to three regular polychora : the tesseract {4,3,3},
{3,3,P} POLYTOPES SPACE S3 H3 FORM FINITE PARACOMPACT NONCOMPACT NAME {3,3,3} {3,3,4} {3,3,5} {3,3,6} {3,3,7} {3,3,8} ... {3,3,∞} IMAGE Vertex figure {3,3} {3,4} {3,5} {3,6} {3,7} {3,8} {3,∞} {3,P,3} POLYTOPES SPACE S3 H3 FORM FINITE COMPACT PARACOMPACT NONCOMPACT {3,P,3} {3,3,3} {3,4,3} {3,5,3} {3,6,3} {3,7,3} {3,8,3} ... {3,∞,3} IMAGE CELLS {3,3} {3,4} {3,5} {3,6} {3,7} {3,8} {3,∞} Vertex figure {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {P,3,P} REGULAR HONEYCOMBS SPACE S3 EUCLIDEAN E3 H3 FORM FINITE AFFINE COMPACT PARACOMPACT NONCOMPACT NAME {3,3,3} {4,3,4} {5,3,5} {6,3,6} {7,3,7} {8,3,8} ...{∞,3,∞} IMAGE CELLS {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} Vertex figure {3,3} {3,4} {3,5} {3,6} {3,7} {3,8} {3,∞} REFERENCES * ^ Matila Ghyka, The geometry of Art and Life (1977), p.68
* ^
* T. Gosset : On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 * H.S.M. Coxeter : * Coxeter, Regular Polytopes , (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) * H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) * 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, * (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, * (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, * John H. Conway , Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1) * Norman Johnson Uniform Polytopes, Manuscript (1991) * N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) EXTERNAL LINKS * Weisstein, Eric W. "Pentatope".
* |