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The truncated 120-cell or truncated hecatonicosachoron is a
The bitruncated 120-cell or hexacosihecatonicosachoron is a
The truncated 600-cell or truncated hexacosichoron is a
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, ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45*
Four-dimensional Archimedean Polytopes
(German), Marco Möller, 2004 PhD dissertation
m58
* * o3o3x5x - thi, o3x3x5o - xhi, x3x3o5o - tex
Four-Dimensional Polytope Projection Barn Raisings
(A Zometool construction of the ''truncated 120-cell''),
tt2t
{{DEFAULTSORT:Truncated 120-Cell Uniform 4-polytopes
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a truncated 120-cell is a uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons.
There are 47 non-Prism (geometry), prism ...
formed as the truncation
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
Truncation and floor function
Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...
of the regular 120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hec ...
.
There are three truncations, including a bitruncation, and a tritruncation, which creates the ''truncated 600-cell''.
Truncated 120-cell
The truncated 120-cell or truncated hecatonicosachoron is a uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons.
There are 47 non-Prism (geometry), prism ...
, constructed by a uniform truncation
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
Truncation and floor function
Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...
of the regular 120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hec ...
4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: Vertex (geometry), vertices, Edge (geo ...
.
It is made of 120 truncated dodecahedral and 600 tetrahedral
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
cells. It has 3120 faces: 2400 being triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s and 720 being decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. Regular decagon
A '' regular decagon'' has a ...
s. There are 4800 edges of two types: 3600 shared by three truncated dodecahedra and 1200 are shared by two truncated dodecahedra and one tetrahedron. Each vertex has 3 truncated dodecahedra and one tetrahedron around it. Its vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is an equilateral triangular pyramid.
Alternate names
* Truncated 120-cell ( Norman W. Johnson) ** Tuncated hecatonicosachoron / Truncated dodecacontachoron / Truncated polydodecahedron * Truncated-icosahedral hexacosihecatonicosachoron (Acronym thi) (George Olshevsky, and Jonathan Bowers)Images
Bitruncated 120-cell
The bitruncated 120-cell or hexacosihecatonicosachoron is a uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons.
There are 47 non-Prism (geometry), prism ...
. It has 720 cells: 120 truncated icosahedra, and 600 truncated tetrahedra. Its vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is a digonal disphenoid
In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same ...
, with two truncated icosahedra and two truncated tetrahedra around it.
Alternate names
* Bitruncated 120-cell / Bitruncated 600-cell ( Norman W. Johnson) ** Bitruncated hecatonicosachoron / Bitruncated hexacosichoron / Bitruncated polydodecahedron / Bitruncated polytetrahedron * Truncated-icosahedral hexacosihecatonicosachoron (Acronym Xhi) (George Olshevsky, and Jonathan Bowers)Images
Truncated 600-cell
The truncated 600-cell or truncated hexacosichoron is a uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons.
There are 47 non-Prism (geometry), prism ...
. It is derived from the 600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol .
It is also known as the C600, hexacosichoron and hexacosihedroid.
It is also called a tetraplex (abbreviated from ...
by truncation
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
Truncation and floor function
Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...
. It has 720 cells: 120 icosahedra and 600 truncated tetrahedra. Its vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is a pentagonal pyramid
In geometry, a pentagonal pyramid is a Pyramid (geometry), pyramid with a pentagon base and five triangular faces, having a total of six faces. It is categorized as a Johnson solid if all of the edges are equal in length, forming Equilateral tria ...
, with one icosahedron on the base, and 5 truncated tetrahedra around the sides.
Alternate names
* Truncated 600-cell ( Norman W. Johnson) * Truncated hexacosichoron (Acronym tex) (George Olshevsky, and Jonathan Bowers)Klitizing, (x3x3o5o - tex) * Truncated tetraplex (Conway)Structure
The truncated 600-cell consists of 600 truncated tetrahedra and 120 icosahedra. The truncated tetrahedral cells are joined to each other via their hexagonal faces, and to the icosahedral cells via their triangular faces. Each icosahedron is surrounded by 20 truncated tetrahedra.Images
Related polytopes
Notes
References
*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, ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45*
J.H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many br ...
and M.J.T. Guy: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
* N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966Four-dimensional Archimedean Polytopes
(German), Marco Möller, 2004 PhD dissertation
* * o3o3x5x - thi, o3x3x5o - xhi, x3x3o5o - tex
Four-Dimensional Polytope Projection Barn Raisings
(A Zometool construction of the ''truncated 120-cell''),
George W. Hart
George William Hart (born 1955) is an American sculptor and geometer. Before retiring, he was an associate professor of Electrical Engineering at Columbia University in New York City and then an interdepartmental research professor at Stony Bro ...
External links
* H4 uniform polytopes with coordinatest
{{DEFAULTSORT:Truncated 120-Cell Uniform 4-polytopes