E. L. Elte - AbsoluteAstronomy.com

Emanuel Lodewijk Elte was a Dutch

Dutch people

The Dutch people are an ethnic group native to the Netherlands. They share a common culture and speak the Dutch language. Dutch people and their descendants are found in migrant communities worldwide, notably in Suriname, Chile, Brazil, Canada, Australia, South Africa, New Zealand, and the United...

mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

. He is noted for discovering and classifying semiregular 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...

s in dimensions four and higher.

His work rediscovered the finite semiregular polytopes of Thorold Gosset

Thorold Gosset

Thorold Gosset 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.According to H. S. M...

, and further allowing not only regular facets, but recursively also allowing one or two semiregular ones. These were enumerated in his 1912 book, The Semiregular Polytopes of the Hyperspaces. He called them semiregular polytopes of the first kind, limiting his search to one or two types of regular or semiregular k-faces. These polytopes and more were rediscovered again by Coxeter, and renamed as a part of a larger class of uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

Elte's semiregular polytopes of the first kind

Summary of the semiregular polytopes of the first kind
n	Elte notation	Vertices	Edges	Faces	Cells	Facets	Coxeter notation
Polyhedra Semiregular polyhedron The term semiregular polyhedron is used variously by different authors.In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today as a uniform polyhedron... (Archimedean solid Archimedean solid In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices... s)
3	tT	12	18	4p₃+4p₆			t_0,1{3,3} Truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...
	tC	24	36	6p₈+8p₃			t_0,1{4,3} Truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....
	tO	24	36	6p₄+8p₆			t_0,1{3,4} Truncated octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
	tD	60	90	20p₃+12p₁₀			t_0,1{5,3} Truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :...
	tI	60	90	20p₆+12p₅			t_0,1{3,5} Truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids whose faces are two or more types of regular polygons.It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges....
	TT = O	6	12	(4+4)p₃			t₁{3,3}
	CO	12	24	6p₄+8p₃			t₁{3,4} Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...
	ID	30	60	20p₃+12p₅			t₁{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...
	P_q	2q	4q	2p_q+qp₄			{}x{q} Prism (geometry) In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
	AP_q	2q	4q	2p_q+2qp₃			s{2,q} Antiprism In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
semiregular 4-polytopes
4	tC₅	10	30	(10+20)p₃	5O+5T		t₁{3,3,3} Rectified 5-cell In four dimensional geometry, the rectified 5-cell is a uniform polychoron 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 total it has 30 triangle faces, 30 edges, and 10...
	tC₈	32	96	64p₃+24p₄	8CO+16T		t₁{4,3,3}
	tC₁₆=C₂₄(*)	48	96	96p₃	(16+8)O		t₁{3,4,3}
	tC₂₄	96	288	96p₃+144p₄	24CO+24C		t₁{3,4,3} Rectified 24-cell In geometry, the rectified 24-cell is a uniform 4-dimensional polytope , which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by reducing the icositetrachoron's cells to cubes or cuboctahedra....
	tC₆₀₀	720	3600	(1200+2400)p₃	600O+120I		t₁{3,3,5} 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...
	tC₁₂₀	1200	3600	2400p₃+720p₅	120ID+600T		t₁{5,3,3} Rectified 120-cell In geometry, a rectified 120-cell is a uniform polychoron formed as the rectification of the regular 120-cell.There are four rectifications of the 120-cell, including the zeroth, the 120-cell itself...
	HM₄=C₁₆(*)	8	24	32p₃	(8+8)T		1₁₁
	-	30	60	20p₃+20p₆	(5+5)tT		t_1,2{3,3,3}
	-	288	576	192p₃+144p₈	(24+24)tC		t_1,2{3,4,3}
	-	20	60	40p₃+30p₄	10T+20P₃		t_0,3{3,3,3}
	-	144	576	384p₃+288p₄	48O+192P₃		t_0,3{3,4,3} Runcinated 24-cell In four-dimensional geometry, a runcinated 24-cell is a convex uniform polychoron, being a runcination of the regular 24-cell....
	-	q²	2q²	q²p₄+2qp_q	(q+q)P_q		{q}x{q} Duoprism In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...
semiregular 5-polytope 5-polytope In five-dimensional geometry, a 5-polytope is a 5-dimensional polytope, bounded by facets. Each polyhedral cell being shared by exactly two polychoron facets. A proposed name for 5-polytopes is polyteron.-Definition:... s
5	S₅¹	15	60	(20+60)p₃	30T+15O	6C₅+6tC₅	t₁{3,3,3,3} Rectified hexateron In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the rectified 5-simplex are located at the edge-centers of the...
	S₅²	20	90	120p₃	30T+30O	(6+6)C₅	t₂{3,3,3,3}
	HM₅	16	80	160p₃	(80+40)T	16C₅+10C₁₆	1₂₁ Demipenteract In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...
	Cr₅¹	40	240	(80+320)p₃	160T+80O	32tC₅+10C₁₆	t₁{3,3,3,4} Rectified pentacross In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the...
	Cr₅²	80	480	(320+320)p₃	80T+200O	32tC₅+10C₂₄	t₂{3,3,3,4}
semiregular 6-polytope 6-polytope In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera.... s
6	S₆¹ (*)						t₁{3,3,3,3,3} Rectified 6-simplex In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the...
	S₆² (*)						t₂{3,3,3,3,3}
	HM₆	32	240	640p₃	(160+480)T	32S₅+12HM₅	1₃₁ Demihexeract In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube with alternate vertices deleted. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes....
	V₂₇	27	216	720p₃	1080T	72S₅+27HM₅	2₂₁
	V₇₂	72	720	2160p₃	2160T	(27+27)HM₆	1₂₂
semiregular 7-polytope 7-polytope In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.... s
7	S₇¹ (*)						t₁{3,3,3,3,3,3} Rectified 7-simplex In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the...
	S₇² (*)						t₂{3,3,3,3,3,3}
	S₇³ (*)						t₃{3,3,3,3,3,3}
	HM₇(*)	64	672	2240p₃	(560+2240)T	64S₆+14HM₆	1₄₁ Demihepteract In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube with alternated vertices deleted...
	V₅₆	56	756	4032p₃	10080T	576S₆+126Cr₆	3a₂₁
	V₁₂₆	126	2016	10080p₃	20160T	576S₆+56V₂₇	2₃₁
	V₅₇₆	576	10080	40320p₃	(30240+20160)T	126HM₆+56V₇₂	1₃₂
semiregular 8-polytope 8-polytope In eight-dimensional geometry, a polyzetton is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.... s
8	S₈¹ (*)						t₁{3,3,3,3,3,3,3} Rectified 8-simplex In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.There are unique 3 degrees of rectifications. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified...
	S₈² (*)						t₂{3,3,3,3,3,3,3}
	S₈³ (*)						t₃{3,3,3,3,3,3,3}
	HM₈(*)	128	1792	7168p₃	(1792+8960)T	128S₇+16HM₇	1₅₁ Demiocteract In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices deleted...
	V₂₁₆₀	2160	69120	483840p₃	1209600T	17280S₇+240V₁₂₆	2₄₁
	V₂₄₀	240	6720	60480p₃	241920T	17280S₇+2160Cr₇	4₂₁

Added in this table as a sequence Elte recognized but did not enumerate explicitly

Regular dimensional families:

S_n=n-simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

: S₃, S₄, S₅, S₆, S₇, S₈, ...
M_n=n-cube
Hypercube
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...

= measure polytope: M₃, M₄, M₅, M₆, M₇, M₈, ...
HM_n=n-demicube= half-measure polytope: HM₃, HM₄, M₅, M₆, HM₇, HM₈, ...
Cr_n=n-orthoplex= cross polytope: Cr₃, Cr₄, Cr₅, Cr₆, Cr₇, Cr₈, ...

Semiregular polytopes of first order:

V_n = semiregular polytope with n vertices

Polygons

P_n = regular n-gon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

Polyhedra:

Regular: T
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...

, C
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. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

, O
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

, I
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

, D
Truncated: tT
Truncated tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...

, tC
Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....

, tO
Truncated octahedron
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....

, tI
Truncated icosahedron
In geometry, the truncated icosahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids whose faces are two or more types of regular polygons.It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges....

, tD
Truncated dodecahedron
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :...
Quasiregular: CO
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

, ID
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...
rhombi: RCO
Rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles...

, RID
Rhombicosidodecahedron
In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces....
truncated quasi: tCO
Truncated cuboctahedron
In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges...

, tID
Truncated icosidodecahedron
In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces....
Prismatic: P_n
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

, AP_n
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...

4-polytopes:

C_n = Regular 4-polytopes with n cells: C₅, C₈
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

, C₁₆
16-cell
In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century....

, C₂₄, C₁₂₀, C₆₀₀
Rectified: tC₅
Rectified 5-cell
In four dimensional geometry, the rectified 5-cell is a uniform polychoron 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 total it has 30 triangle faces, 30 edges, and 10...

, tC₈
Rectified tesseract
In geometry, the rectified tesseract, or rectified 8-cell is a uniform polychoron bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra....

, tC₁₆, tC₂₄
Rectified 24-cell
In geometry, the rectified 24-cell is a uniform 4-dimensional polytope , which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by reducing the icositetrachoron's cells to cubes or cuboctahedra....

, tC₁₂₀
Rectified 120-cell
In geometry, a rectified 120-cell is a uniform polychoron formed as the rectification of the regular 120-cell.There are four rectifications of the 120-cell, including the zeroth, the 120-cell itself...

, tC₆₀₀
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...

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