In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the snub
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
or snub disicositetrachoron is a convex
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 polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
composed of 120 regular
tetrahedral
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
and 24
icosahedral
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.
Topologically, under its highest symmetry,
+,4,3">+,4,3 as an alternation of a
truncated 24-cell
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.
There are two degrees of truncations, including a bitruncation.
Truncated 24-cell
The truncated 24- ...
, it contains 24 pyritohedra (an icosahedron with T
h symmetry), 24 regular tetrahedra, and 96 triangular pyramids.
Semiregular polytope
It is one of three
semiregular 4-polytopes
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polytop ...
made of two or more cells which are
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s, discovered by
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and ...
in his 1900 paper. He called it a ''tetricosahedric'' for being made of
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
and
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
cells. (The other two are the
rectified 5-cell
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In t ...
and
rectified 600-cell
In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two ico ...
.)
Alternative names
* Snub icositetrachoron
* Snub demitesseract
* Semi-snub polyoctahedron (
John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
)
* Sadi (Jonathan Bowers) for snub disicositetrachoron
* Tetricosahedric (
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and ...
)
Geometry
Coordinates
The vertices of a snub 24-cell centered at the origin of 4-space, with edges of length 2, are obtained by taking
even permutation
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total or ...
s of
:(0, ±1, ±φ, ±φ
2)
(where φ = (1+)/2 ≈ 1.618 is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
).
These 96 vertices can be found by partitioning each of the 96 edges of a
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
in the golden ratio in a consistent manner, in much the same way that the 12 vertices of an
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
or "snub octahedron" can be produced by partitioning the 12 edges of an octahedron in the golden ratio. This can be done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. This is equivalent to the snub truncation of the 24-cell described below.
The 96 vertices of the snub 24-cell, together with the 24 vertices of a 24-cell, form the 120 vertices of 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 " ...
.
Constructions
The snub 24-cell is derived from the 24-cell by a special form of
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 ...
.
Truncations remove vertices by cutting through the edges incident to the vertex; forms of truncation differ by where on the edge the cut is made. The common truncations of the 24-cell include the
recitified 24-cell (which cuts each edge at its midpoint, producing a polytope bounded by 24 cubes and 24
cuboctahedra
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a t ...
), and the
truncated 24-cell
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.
There are two degrees of truncations, including a bitruncation.
Truncated 24-cell
The truncated 24- ...
(which cuts each edge one-third of its length from the vertex, producing a polytope bounded by 24 cubes and 24
truncated octahedra
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
). In these truncations a cube is produced in place of the removed vertex, because the vertex figure of the 24-cell is a cube and the cuts are equidistant from the vertex.
The snub truncation of the 24-cell cuts each edge into two golden sections (such that the larger section is in the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
~1.618 to the smaller section, and the original edge is in the golden ratio to the larger section). The cut must be made in alternate directions on alternate edges incident to each vertex, in order to have a coherent result. The edges incident to a vertex in the 24-cell are the 8 radii of its cubical vertex figure. The only way to choose alternate radii of a cube is to choose the four radii of a tetrahedron (inscribed in the cube) to be cut at the smaller section of their length from the vertex, and the opposite four radii (of the other tetrahedron that can be inscribed in the cube) to be cut at the larger section of their length from the vertex. There are of course two ways to do this; both produce a cluster of five regular tetrahedra in place of the removed vertex, rather than a cube.
This construction has an analogy in 3 dimensions: the construction of the icosahedron (the "
snub octahedron") from the octahedron, by the same method. That is how the snub-24 cell's icosahedra are produced from the 24-cell's octahedra during truncation.
The snub 24-cell is related to the
truncated 24-cell
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.
There are two degrees of truncations, including a bitruncation.
Truncated 24-cell
The truncated 24- ...
by an
alternation operation. Half the vertices are deleted, the 24
truncated octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
cells become 24
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
cells, the 24
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
s become 24
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
cells, and the 96 deleted vertex voids create 96 new tetrahedron cells.
The snub 24-cell may also be constructed by a particular ''diminishing'' of 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 removing 24 vertices from the 600-cell corresponding to those of an inscribed
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
, and then taking the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of the remaining vertices. This is equivalent to removing 24
icosahedral pyramids from the 600-cell.
Structure
The icosahedral cells fit together face-to-face leaving voids between them filled by clusters of five tetrahedral cells.
Each icosahedral cell is joined to 8 other icosahedral cells at 8 triangular faces in the positions corresponding to an inscribing octahedron. The remaining triangular faces are joined to tetrahedral cells, which occur in pairs that share an edge on the icosahedral cell.
The tetrahedral cells may be divided into two groups, of 96 yellow cells and 24 red cells respectively (as colored in the net illustration). Each yellow tetrahedral cell is joined via its triangular faces to 3 blue icosahedral cells and one red tetrahedral cell, while each red tetrahedral cell is joined to 4 yellow tetrahedra. Thus, the tetrahedral cells occur in clusters of five (four yellow cells face-bonded around a red central one, each red/yellow pair lying in a different hyperplane). The red central tetrahedron of the five shares each of its six edges with a different icosahedral cell, and with the pair of yellow tetrahedral cells which shares that edge on the icosahedral cell.
Symmetry
The snub 24-cell has three
vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
colorings based on a
Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.
Construction process
...
on a
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 reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...
from which it is
alternated: F
4 defines 24 interchangeable icosahedra, while the B
4 group defines two groups of icosahedra in an 8:16 counts, and finally the D
4 group has 3 groups of icosahedra with 8:8:8 counts.
Conversely, the 600-cell may be constructed from the snub 24-cell by augmenting it with 24 icosahedral pyramids.
Projections
Orthographic projections
Perspective projections
Dual
The
Dual snub 24-cell
In geometry, the dual snub 24-cell is a 144 vertex convex 4-polytope composed of 96 irregular cell (mathematics), cells. Each cell has faces of two kinds: 3 Kite (geometry), kites and 6 isosceles triangles. The polytope has a total of 432 faces (14 ...
has 144 identical irregular cells. Each cell has faces of two kinds: 3
kites
A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...
and 6 isosceles triangles. The polytope has a total of 432 faces (144 kites and 288 isosceles triangles) and 480 edges.
Related polytopes
The ''snub 24-cell'' can be obtained as a diminishing of 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 " ...
at 24 of its vertices, in fact those of a vertex inscribed
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
. There is also a further such ''bi-''diminishing, when the vertices of a second vertex inscribed 24-cell would be diminished as well. Accordingly, this one is known as the
bi-24-diminished 600-cell.
The ''snub 24-cell'' is also called a semi-snub 24-cell because it is not a true
snub
A snub, cut or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub.
In Awards and Lists
For awards, the term "snub" ...
(alternation of an omnitruncated 24-cell). The
full snub 24-cell can also be constructed although it is not uniform, being composed of irregular tetrahedra on the alternated vertices.
The snub 24-cell is the largest facet of the 4-dimensional honeycomb, the
snub 24-cell honeycomb
In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 ...
.
The snub 24-cell is a part of the F
4 symmetry family of uniform 4-polytopes.
See also
*
Snub 24-cell honeycomb
In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 ...
*
Dual snub 24-cell
In geometry, the dual snub 24-cell is a 144 vertex convex 4-polytope composed of 96 irregular cell (mathematics), cells. Each cell has faces of two kinds: 3 Kite (geometry), kites and 6 isosceles triangles. The polytope has a total of 432 faces (14 ...
Citations
References
*
*
*
*
*
*
External links
*
*
Snub icositetrachoron - Data and images
{{DEFAULTSORT:Snub 24-Cell
4-polytopes