Rectified 120-cell

Four rectifications
Orthogonal projections in H₃ Coxeter plane
120-cell	Rectified 120-cell	Rectified 600-cell Rectified 600-cell In geometry, a rectified 600-cell is a uniform polychoron formed as the rectification of the regular 600-cell.There are four rectifications of the 600-cell, including the zeroth, the 600-cell itself...	600-cell

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a rectified 120-cell is a uniform polychoron

Uniform polychoron

In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....

(4-dimensional uniform polytope

Polytope

In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

) formed as the rectification

Rectification (geometry)

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...

of the regular 120-cell.

There are four rectifications of the 120-cell, including the zeroth, the 120-cell itself. Tbe birectified 120-cell is more easily seen as a rectified 600-cell, and the trirectified 120-cell is the same as the dual 600-cell.


Schlegel diagram, centered on icosidodecahedon, tetrahedral cells visible
Type	Uniform polychoron Uniform polychoron In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
Uniform index	33
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
Cells	720 total: 120 (3.5.3.5) Icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon... 600 (3.3.3) Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
Faces	3120 total: 2400 {3} Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted .... , 720 {5} Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
Edges	3600
Vertices	1200
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...	triangular prism Triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....
Schläfli symbol	t₁{5,3,3}
Symmetry group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...	H₄ or [3,3,5]
Properties	convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn... , edge-transitive

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the rectified
Rectification (geometry)
In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...

120-cell is a convex uniform polychoron

Uniform polychoron

In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....

composed of 600 regular tetrahedra

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

and 120 icosidodecahedra

Icosidodecahedron

In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon...

cells. Its vertex figure is a triangular prism

Triangular prism

In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....

, with 3 icosidodecahedra and 2 tetrahedra meeting at each vertex.

Alternative names:

Rectified 120-cell (Norman Johnson)
Rectified hecatonicosichoron
Rectified polydodecahedron
Icosidodecahedral hexacosihecatonicosachoron
Rahi (Jonathan Bowers: for rectified hecatonicosachoron)
Ambohecatonicosachoron (Neil Sloane & John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...

)

Projections

3D parallel projection
	Parallel projection of the rectified 120-cell into 3D, centered on an icosidodecahedral cell. Nearest cell to 4D viewpoint shown in orange, and tetrahedral cells shown in yellow. Remaining cells culled so that the structure of the projection is visible.

Orthographic projection
Orthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...

s by Coxeter planes
H₄	-	F₄
[30]	[20]	[12]
H₃	A₂ / B₃ / D₄	A₃ / B₂
[10]	[6]	[4]

External links

rectified 120-cell Marco Möller's Archimedean polytopes in R⁴ (German)

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.