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B4 polytope

{{DISPLAYTITLE:B4 polytope}}

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|+ Orthographic projections in the B4 Coxeter plane

align=center

|120px
Tesseract
{{CDD|node_1|4|node|3|node|3|node}}

|120px
16-cell
{{CDD|node_1|3|node|3|node|4|node}}

In 4-dimensional geometry, there are 15 uniform 4-polytopes with B4 symmetry. There are two regular forms, the tesseract and 16-cell, with 16 and 8 vertices respectively.

Visualizations

They can be visualized as symmetric orthographic projections in Coxeter planes of the B5 Coxeter group, and other subgroups.

Symmetric orthographic projections of these 32 polytopes can be made in the B5, B4, B3, B2, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.

These 32 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

The pictures are drawn as Schlegel diagram perspective projections, centered on the cell at pos. 3, with a consistent orientation, and the 16 cells at position 0 are shown solid, alternately colored.

class="wikitable"

!rowspan=2|#

!rowspan=2|Name

!colspan=4| Coxeter plane projections

!colspan=2| Schlegel
diagrams

!rowspan=2|Net

B4
[8]

!B3
[6]

!B2
[4]

!A3
[4]

!Cube
centered

!Tetrahedron
centered

BGCOLOR="#f0e0e0"

!1

8-cell or tesseract
{{CDD|node_1|4|node|3|node|3|node}} = {4,3,3}

|80px

|80px

|80px

|80px

|80px

|

|70px

BGCOLOR="#f0e0e0"

!2

rectified 8-cell
{{CDD|node|4|node_1|3|node|3|node}} = r{4,3,3}

|80px

|80px

|80px

|80px

|70px

|

|70px

BGCOLOR="#e0e0f0"

!3

16-cell
{{CDD|node|4|node|3|node|3|node_1}} = {3,3,4}

|80px

|80px

|80px

|80px

|

|80px

|70px

BGCOLOR="#f0e0e0"

!4

truncated 8-cell
{{CDD|node_1|4|node_1|3|node|3|node}} = t{4,3,3}

|80px

|80px

|80px

|80px

|80px

|

|70px

BGCOLOR="#f0e0e0"

!5

cantellated 8-cell
{{CDD|node_1|4|node|3|node_1|3|node}} = rr{4,3,3}

|80px

|80px

|80px

|80px

|80px

|

|70px

BGCOLOR="#e0f0e0"

!6

runcinated 8-cell
(also runcinated 16-cell)
{{CDD|node_1|4|node|3|node|3|node_1}} = t03{4,3,3}

|80px

|80px

|80px

|80px

|80px

|80px

|70px

BGCOLOR="#e0f0e0"

!7

bitruncated 8-cell
(also bitruncated 16-cell)
{{CDD|node|4|node_1|3|node_1|3|node}} = 2t{4,3,3}

|80px

|80px

|80px

|80px

| 80px

|80px

|70px

BGCOLOR="#e0e0f0"

!8

truncated 16-cell
{{CDD|node|4|node|3|node_1|3|node_1}} = t{3,3,4}

|80px

|80px

|80px

|80px

|

|80px

|70px

BGCOLOR="#f0e0e0"

!9

cantitruncated 8-cell
{{CDD|node|4|node_1|3|node_1|3|node_1}} = tr{3,3,4}

|80px

|80px

|80px

|80px

| 80px

|

|70px

BGCOLOR="#f0e0e0"

!10

runcitruncated 8-cell
{{CDD|node_1|4|node_1|3|node|3|node_1}} = t013{4,3,3}

|80px

|80px

|80px

|80px

| 80px

|

|70px

BGCOLOR="#e0e0f0"

!11

runcitruncated 16-cell
{{CDD|node_1|4|node|3|node_1|3|node_1}} = t013{3,3,4}

|80px

|80px

|80px

|80px

|

| 80px

|70px

BGCOLOR="#e0f0e0"

!12

omnitruncated 8-cell
(also omnitruncated 16-cell)
{{CDD|node_1|4|node_1|3|node_1|3|node_1}} = t0123{4,3,3}

|80px

|80px

|80px

|80px

|80px

|80px

|70px

class="wikitable"

!rowspan=2|#

!rowspan=2|Name

!colspan=5| Coxeter plane projections

!colspan=2| Schlegel
diagrams

!rowspan=2|Net

F4
[12]

!B4
[8]

!B3
[6]

!B2
[4]

!A3
[4]

!Cube
centered

!Tetrahedron
centered

BGCOLOR="#e0e0f0"

!13

|*rectified 16-cell
(Same as 24-cell)
{{CDD|node|4|node|3|node_1|3|node}} = {{CDD|node_1|3|node|4|node|3|node}}
r{3,3,4} = {3,4,3}

|80px

|80px

|80px

|80px

|80px

|

|80px

|70px

BGCOLOR="#e0e0f0"

!14

*cantellated 16-cell
(Same as rectified 24-cell)
{{CDD|node|4|node_1|3|node|3|node_1}} = {{CDD|node|3|node_1|4|node|3|node}}
rr{3,3,4} = r{3,4,3}

|80px

|80px

|80px

|80px

|80px

|

| 80px

| 70px

BGCOLOR="#e0e0f0"

!15

*cantitruncated 16-cell
(Same as truncated 24-cell)
{{CDD|node|4|node_1|3|node_1|3|node_1}} = {{CDD|node_1|3|node_1|4|node|3|node}}
tr{3,3,4} = t{3,4,3}

|80px

|80px

|80px

|80px

|80px

|

|80px

|70px

class="wikitable"

!rowspan=2|#

!rowspan=2|Name

!colspan=5| Coxeter plane projections

!colspan=2| Schlegel
diagrams

!rowspan=2|Net

F4
[12]

!B4
[8]

!B3
[6]

!B2
[4]

!A3
[4]

!Cube
centered

!Tetrahedron
centered

BGCOLOR="#d0f0f0"

!16

alternated cantitruncated 16-cell
(Same as the snub 24-cell)
{{CDD|node|4|node_h|3|node_h|3|node_h}} = {{CDD|node_h|3|node_h|4|node|3|node}}
sr{3,3,4} = s{3,4,3}

|80px

|80px

|80px

|80px

|

|

|80px

|70px

Coordinates

The tesseractic family of 4-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform 4-polytopes. All coordinates correspond with uniform 4-polytopes of edge length 2.

class="wikitable"

|+Coordinates for uniform 4-polytopes in Tesseract/16-cell family

#

!Base point

! Name

!Coxeter diagram

!colspan=2|Vertices

BGCOLOR="#f0e0e0"

!3

|(0,0,0,1){{radic|2}}

|16-cell

|{{CDD|node|4|node|3|node|3|node_1}}

|8

24-34!/3!
BGCOLOR="#e0e0f0"

!1

|(1,1,1,1)

|Tesseract

|{{CDD|node_1|4|node|3|node|3|node}}

|16

244!/4!
BGCOLOR="#f0e0e0"

!13

|(0,0,1,1){{radic|2}}

|Rectified 16-cell (24-cell)

|{{CDD|node|4|node|3|node_1|3|node}}

|24

24-24!/(2!2!)
BGCOLOR="#e0e0f0"

!2

|(0,1,1,1){{radic|2}}

|Rectified tesseract

|{{CDD|node|4|node_1|3|node|3|node}}

|32

244!/(3!2!)
BGCOLOR="#f0e0e0"

!8

|(0,0,1,2){{radic|2}}

|Truncated 16-cell

|{{CDD|node|4|node|3|node_1|3|node_1}}

|48

24-24!/2!
BGCOLOR="#e0f0e0"

!6

|(1,1,1,1) + (0,0,0,1){{radic|2}}

|Runcinated tesseract

|{{CDD|node_1|4|node|3|node|3|node_1}}

|64

244!/3!
BGCOLOR="#e0e0f0"

!4

|(1,1,1,1) + (0,1,1,1){{radic|2}}

|Truncated tesseract

|{{CDD|node_1|4|node_1|3|node|3|node}}

|64

244!/3!
BGCOLOR="#f0e0e0"

!14

|(0,1,1,2){{radic|2}}

|Cantellated 16-cell (rectified 24-cell)

|{{CDD|node|4|node_1|3|node|3|node_1}}

|96

244!/(2!2!)
BGCOLOR="#e0f0e0"

!7

|(0,1,2,2){{radic|2}}

|Bitruncated 16-cell

|{{CDD|node|4|node_1|3|node_1|3|node}}

|96

244!/(2!2!)
BGCOLOR="#e0e0f0"

!5

|(1,1,1,1) + (0,0,1,1){{radic|2}}

|Cantellated tesseract

|{{CDD|node_1|4|node|3|node_1|3|node}}

|96

244!/(2!2!)
BGCOLOR="#f0e0e0"

!15

|(0,1,2,3){{radic|2}}

|cantitruncated 16-cell (truncated 24-cell)

|{{CDD|node|4|node_1|3|node_1|3|node_1}}

|192

244!/2!
BGCOLOR="#f0e0e0"

!11

|(1,1,1,1) + (0,0,1,2){{radic|2}}

|Runcitruncated 16-cell

|{{CDD|node_1|4|node|3|node_1|3|node_1}}

|192

244!/2!
BGCOLOR="#e0e0f0"

!10

|(1,1,1,1) + (0,1,1,2){{radic|2}}

|Runcitruncated tesseract

|{{CDD|node_1|4|node_1|3|node|3|node_1}}

|192

244!/2!
BGCOLOR="#e0e0f0"

!9

|(1,1,1,1) + (0,1,2,2){{radic|2}}

|Cantitruncated tesseract

|{{CDD|node_1|4|node_1|3|node_1|3|node}}

|192

244!/2!
BGCOLOR="#e0f0e0"

!12

|(1,1,1,1) + (0,1,2,3){{radic|2}}

|Omnitruncated 16-cell

|{{CDD|node_1|4|node_1|3|node_1|3|node_1}}

|384

244!

References

{{reflist}}

  • J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26)
  • H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter]
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
  • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

External links

  • {{KlitzingPolytopes|polychora.htm|4D|uniform 4-polytopes}}
  • [http://www.polytope.de Uniform, convex polytopes in four dimensions:], Marco Möller {{in lang|de}}
  • {{Cite thesis |url=https://ediss.sub.uni-hamburg.de/volltexte/2004/2196/pdf/Dissertation.pdf |title=Vierdimensionale Archimedische Polytope |last=Möller |first=Marco |date=2004 |publisher=University of Hamburg |type=Doctoral dissertation |language=de }}
  • {{PolyCell | urlname = uniform.html| title = Uniform Polytopes in Four Dimensions}}
  • {{PolyCell | urlname = section2.html| title = Convex uniform polychora based on the tesserract/16-cell}}

{{Polytopes}}

Category:Uniform 4-polytopes