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QUASI-PETRIE ICOSAGONS IN THE 600-CELL - PICTURE GALLERY
WORKING ON THE 120-CELL - PHOTO GALLERY
THE 120-CELL WITH A PAIR OF OPPOSITE CELLS
best view is full screen - press F11
The binary icosahedral group, the 600-cell and the 120-cell at Wikipedia
The binary icosahedral group, the 600-cell and the 120-cell at InfoRapid Knowledge Portal
Remark: InfoRapid is too slow at times to be practical.
The Nottrot tables below show the numbers of vertices, edges, faces and cells and their incidence numbers for the 600-cell and the 120-cell. Explanation: "c" means "contains"; "m" means "is met by". For instance, in the 600-cell a face contains three vertices, and at a vertex 30 faces meet each other (a
vertex is met by 30 faces).
The format is adapted from Nottrot's book "Inzicht in de Vierde Dimensie", published in 1954 under the pen name Nothing All by P. Noordhoff N.V., Groningen - Djakarta.
Remark: presumably the only reference to this book is in a reader published by the Freudenthal Institute, named "de utopie van de vierde dimensie" (2000).
| |
Vertex |
Edge |
Face |
Cell |
| V |
120 |
m 12 |
m 30 |
m 20 |
| E |
c 2 |
720 |
m 5 |
m 5 |
| F |
c 3 |
c 3 |
1200 |
m 2 |
| C |
c 4 |
c 6 |
c 4 |
600 |
THE 600-CELL
600 tetrahedral cells, 1200 triangular faces
Schläfli symbol {3, 3, 5}
 Wikipedia Lemma
|
| |
Vertex |
Edge |
Face |
Cell |
| V |
600 |
m 4 |
m 6 |
m 4 |
| E |
c 2 |
1200 |
m 3 |
m 3 |
| F |
c 5 |
c 5 |
720 |
m 2 |
| C |
c 20 |
c 30 |
c 12 |
120 |
THE 120-CELL
120 dodecahedral cells, 720 pentagonal faces
Schläfli symbol {5, 3, 3}
Wikipedia Lemma
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600-CELL -
 Vertex table - Edge table - Face table - Cell table (written by the late Dik T. Winter)
120-CELL -
Vertex table - Edge table - Face table - Cell table (written by the late Dik T. Winter)
(From http://homepages.cwi.nl/~dik/english/mathematics/poly/index.html - Dik T. Winter's Polytope Database) ( Dik T. Winter - born March 21st 1945 Warffum - died December 28th 2009 Amsterdam)
In this database "tau" denotes the Golden Ratio: tau = sqrt(5/4) + 1/2 = 1.618033989 to 9 decimals.
Erratum & Correction in Dik T. Winter's 120-cell vertex table
Discovered January 31st 2010
Entry nr 129 in 120-cell vertex table is mistaken; must be the opposite of vertex nr 128.
Excerpt from original file:
V128: ( tau^3, -1, -tau^3, -tau^3)
V129: (-tau^3, -1, tau^3, tau^3)
With corrected vertex nr 129:
V128: ( tau^3, -1, -tau^3, -tau^3)
V129: (-tau^3, 1, tau^3, tau^3)
GOLDEN-RATIO NUMBERS AND ANGLES RELATED TO THE 600-CELL AND THE 120-CELL
| 0.3819660113 |
= |
3/2 - sqrt(5/4) |
= |
τ-2 |
= |
tan (20°.90515745)
|
= |
cot (69°.09484255)
|
| 0.6180339887 |
= |
sqrt(5/4) - 1/2 |
= |
τ-1 |
= |
tan (31°.71747441)
|
= |
cot (58°.28252559)
|
| 0.7639320225 |
= |
3 - sqrt(5) |
= |
2τ-2 |
= |
tan (37°.37736814)
|
= |
cot (52°.62263186)
|
| 1.309016994 |
= |
(3 + sqrt(5))/4 |
= |
τ2/2 |
= |
tan (52°.62263186)
|
= |
cot (37°.37736814)
|
| 1.618033989 |
= |
sqrt(5/4) + 1/2 |
= |
τ |
= |
tan (58°.28252559)
|
= |
cot (31°.71747441)
|
| 2.618033989 |
= |
3/2 + sqrt(5/4) |
= |
τ2 |
= |
tan (69°.09484255)
|
= |
cot (20°.90515745)
|
| 5.236067977 |
= |
3 + sqrt(5) |
= |
2τ2 |
= |
tan (79°.18768304)
|
= |
cot (10°.81231696)
|
Knowing the Greek names of polygons appears to pay off: a Google search for "Petrie triacontagon" yields in first and second positions
B.L.Chilton: On the projection of the regular polytope {5, 3, 3} into a regular triacontagon.
Canad. Math. Bull. Vol 7, No 3, July 1964 - open access - local copy - in Google:
Canadian Mathematical Bulletin - 1964 - Google Books Result
Vol. 7, No. 3 - 168 pages - Magazine
3} INTO A REGULAR TRIACONTAGON BL Chilton (received November 29, 1963) 1 ... This polygon arises as the plane projection of a Petrie polygon of the 120-cell ...
books.google.com/books?id=dxlND0gdXjsC... - More
book results »
H.S.M.Coxeter: Regular honeycombs in elliptic space.
Proc. London Math. Soc. (3) 4 (1954) - open access - local copy - in Google:
[PDF] REGULAR HONEYCOMBS IN ELLIPTIC SPACE
File Format: PDF/Adobe Acrobat - Quick View
by HSM COXETER - Cited
by 7 - Related articles
is a Petrie polygon of e, derived from the corresponding skew triacontagon in {3,3,5} by identifying
antipodes. It follows that the centres of the fifteen ...
citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.125.6188...
(Contains a definition of "Petrie polygon"; is this the first-ever that appeared in the literature?)
Websites and web pages on 4D rotations and 4D polytopes
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