THE 600-CELL AND THE 120-CELL  

AUGUST 31ST 2015

     


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 
THE 600-CELL: OLD AND NEW
best view is full screen - press F11


S.L.van Oss: Das regelmässige Sechshundertzell und seine Selbstdeckenden Bewegungen (1899)  -  PDF File (32p)  

All articles by S.L. van Oss in KNAW Digital Library  


P.H. Schoute: Regelmässige Schnitte und Projectionen des Hundertzwanzigzelles und Sechshundertzelles im vierdimensionalen Raume (1894)  -  PDF File (39p)  

All articles by P.H. Schoute in KNAW Digital Library  


The binary icosahedral group, Latest  the icosian ring, the 600-cell and the 120-cell at Wikipedia

The binary icosahedral group, the 600-cell and the 120-cell at InfoRapid Knowledge Portal



The Nottrot tables below show the numbers of vertices, edges, faces and cells and their incidence numbers for the six regular four-dimensional polytopes. 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.
More information about J.C.G. Notrott's book through Cyclopaedia.Net > Nottrot; on Notrott himself through Google > "Johan Christiaan Gerrit Nottrot".

REMARK: presumably the only existing reference to this book was as of August 2013 in a reader published by the Freudenthal Institute (=EN= and =NL= websites), named "de utopie van de vierde dimensie" (AD 2000; used in the NWD Lecture on 2000-02-04 and in the Studium Generale Utrecht on 2000-05-16). (local copy)

Locations previously mentioned (2013-08-08) on this page: 403.png - forbidden "de utopie van de vierde dimensie" - print version and display version: 403.png - forbidden "de utopie van de vierde dimensie"- display version, both at Utrecht University, are no longer accessible. Why! At least the print version was still accessible on August 8th 2013.


  Vertex Edge Face Cell
Vertex 120 m 12 m 30 m 20
Edge c 2 720 m 5 m 5
Face c 3 c 3 1200 m 2
Cell 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
Vertex 600 m 4 m 6 m 4
Edge c 2 1200 m 3 m 3
Face c 5 c 5 720 m 2
Cell c 20 c 30 c 12 120

THE 120-CELL
120 dodecahedral cells, 720 pentagonal faces
Schläfli symbol {5, 3, 3}
Wikipedia Lemma
 


600-CELL -
404
 Vertex table - 404  Edge table - 404  Face table - 404  Cell table (written by the late Dik T. Winter) 
120-CELL -
404
 Vertex table - 404  Edge table - 404  Face table - 404  Cell table (written by the late Dik T. Winter) 
(From 404  http://homepages.cwi.nl/~dik/english/mathematics/poly/index.html - Dik T. Winter's Polytope Database) (404  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)
 


2; 1/2;  GOLDEN-RATIO NUMBERS AND ROTATION ANGLES RELATED TO THE 600-CELL AND THE 120-CELL

0.1909830057  =  (3 - √5)/4  =  1/2τ2  = 

tan (10°.81231696)

 = 

cot (79°.18768304)

0.2360679775  =  √5 - 2  =  1/τ3  = 

tan (13°.28252559)

 = 

cot (76°.71747441)

0.2763932023  =  (10 - 2√5)/20  =  1/(τ2 + 1)  = 

tan (15°.45043709)

 = 

cot (74°.54956291)

0.3819660113  =  (3 - √5)/2  =  1/τ2  = 

tan (20°.90515745)

 = 

cot (69°.09484255)

0.5000000000  =  1/2  =  1/2  = 

tan (26°.56505118)

 = 

cot (63°.43494882)

0.6180339887  =  (√5 - 1)/2  =  1/τ  = 

tan (31°.71747441)

 = 

cot (58°.28252559)

0.7639320225  =  3 - √5  =  2/τ2  = 

tan (37°.37736814)

 = 

cot (52°.62263186)

1.3090169944  =  (3 + √5)/4  =  τ2/2  = 

tan (52°.62263186)

 = 

cot (37°.37736814)

1.6180339887  =  (√5 + 1)/2  =  τ  = 

tan (58°.28252559)

 = 

cot (31°.71747441)

2.0000000000  =  2  =  2  = 

tan (63°.43494882)

 = 

cot (26°.56505118)

2.6180339887  =  (3 + √5)/2  =  τ2  = 

tan (69°.09484255)

 = 

cot (20°.90515745)

3.6180339887  =  (10 + 2√5)/4  =  τ2 + 1  = 

tan (74°.54956291)

 = 

cot (15°.45043709)

4.2360679775  =  √5 + 2  =  τ3  = 

tan (76°.71747441)

 = 

cot (13°.28252559)

5.2360679775  =  3 + √5  =  2  = 

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