Archimedean solid

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Polyhedron truncated 4a max.png
Polyhedron 6-8 max.png
Polyhedron great rhombi 12-20 max.png
Truncated tetrahedron, cuboctahedron and truncated icosidodecahedron. The first and the last one can be described as the smallest and the largest Archimedean solid, respectively.

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed of only one type of polygon), excluding the prisms and antiprisms, and excluding the pseudorhombicuboctahedron. [1] They are a subset of the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.

Contents

"Identical vertices" means that each two vertices are symmetric to each other: A global isometry of the entire solid takes one vertex to the other while laying the solid directly on its initial position. BrankoGrünbaum  ( 2009 ) observed that a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a local isometry is required. Grünbaum pointed out a frequent error in which authors define Archimedean solids using this local definition but omit the 14th polyhedron. If only 13 polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods.

Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, even though their faces are regular polygons and their symmetry groups act transitively on their vertices. Excluding these two infinite families, there are 13 Archimedean solids. All the Archimedean solids (but not the elongated square gyrobicupola) can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.

Origin of name

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra. [2] During the Renaissance, artists and mathematicians valued pure forms with high symmetry, and by around 1620 Johannes Kepler had completed the rediscovery of the 13 polyhedra, [3] as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot polyhedra. (See Schreiber, Fischer & Sternath 2008 for more information about the rediscovery of the Archimedean solids during the renaissance.)

Kepler may have also found the elongated square gyrobicupola (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, and the first clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by Duncan Sommerville. [2]

Classification

There are 13 Archimedean solids (not counting the elongated square gyrobicupola; 15 if the mirror images of two enantiomorphs, the snub cube and snub dodecahedron, are counted separately).

Here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a vertex configuration of 4.6.8 means that a square, hexagon, and octagon meet at a vertex (with the order taken to be clockwise around the vertex).

Name/
(alternative name)
Schläfli
Coxeter
TransparentSolid Net Vertex
conf./fig.
FacesEdgesVert.Volume
(unit edges)
Point
group
Sphericity
Truncated tetrahedron t{3,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncatedtetrahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron truncated 4a max.png Polyhedron truncated 4a net.svg 3.6.6
Polyhedron truncated 4a vertfig.png
84 triangles
4 hexagons
18122.710576Td0.7754132
Cuboctahedron
(rhombitetratetrahedron, triangular gyrobicupola)
r{4,3} or rr{3,3}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png or CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cuboctahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron 6-8 max.png Polyhedron 6-8 net.svg 3.4.3.4
Polyhedron 6-8 vertfig.png
148 triangles
6 squares
24122.357023Oh0.9049972
Truncated cube t{4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncatedhexahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron truncated 6 max.png Polyhedron truncated 6 net.svg 3.8.8
Polyhedron truncated 6 vertfig.png
148 triangles
6 octagons
362413.599663Oh0.8494937
Truncated octahedron
(truncated tetratetrahedron)
t{3,4} or tr{3,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png or CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Truncatedoctahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron truncated 8 max.png Polyhedron truncated 8 net.svg 4.6.6
Polyhedron truncated 8 vertfig.png
146 squares
8 hexagons
362411.313709Oh0.9099178
Rhombicuboctahedron
(small rhombicuboctahedron, elongated square orthobicupola)
rr{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Rhombicuboctahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron small rhombi 6-8 max.png Polyhedron small rhombi 6-8 net.svg 3.4.4.4
Polyhedron small rhombi 6-8 vertfig.png
268 triangles
18 squares
48248.714045Oh0.9540796
Truncated cuboctahedron
(great rhombicuboctahedron)
tr{4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Truncatedcuboctahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron great rhombi 6-8 max.png Polyhedron great rhombi 6-8 net.svg 4.6.8
Polyhedron great rhombi 6-8 vertfig light.png
2612 squares
8 hexagons
6 octagons
724841.798990Oh0.9431657
Snub cube
(snub cuboctahedron)
sr{4,3}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Snubhexahedronccw.jpg   Cog-scripted-svg-blue.svg Polyhedron snub 6-8 left max.png Polyhedron snub 6-8 left net.svg 3.3.3.3.4
Polyhedron snub 6-8 left vertfig.png
3832 triangles
6 squares
60247.889295O0.9651814
Icosidodecahedron
(pentagonal gyrobirotunda)
r{5,3}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Icosidodecahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron 12-20 max.png Polyhedron 12-20 net.svg 3.5.3.5
Polyhedron 12-20 vertfig.png
3220 triangles
12 pentagons
603013.835526Ih0.9510243
Truncated dodecahedron t{5,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncateddodecahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron truncated 12 max.png Polyhedron truncated 12 net.svg 3.10.10
Polyhedron truncated 12 vertfig.png
3220 triangles
12 decagons
906085.039665Ih0.9260125
Truncated icosahedron t{3,5}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Truncatedicosahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron truncated 20 max.png Polyhedron truncated 20 net compact.svg 5.6.6
Polyhedron truncated 20 vertfig.png
3212 pentagons
20 hexagons
906055.287731Ih0.9666219
Rhombicosidodecahedron
(small rhombicosidodecahedron)
rr{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Rhombicosidodecahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron small rhombi 12-20 max.png Polyhedron small rhombi 12-20 net.svg 3.4.5.4
Polyhedron small rhombi 12-20 vertfig.png
6220 triangles
30 squares
12 pentagons
1206041.615324Ih0.9792370
Truncated icosidodecahedron
(great rhombicosidodecahedron)
tr{5,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Truncatedicosidodecahedron.jpg   Cog-scripted-svg-blue.svg Polyhedron great rhombi 12-20 max.png Polyhedron great rhombi 12-20 net.svg 4.6.10
Polyhedron great rhombi 12-20 vertfig light.png
6230 squares
20 hexagons
12 decagons
180120206.803399Ih0.9703127
Snub dodecahedron
(snub icosidodecahedron)
sr{5,3}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Snubdodecahedroncw.jpg   Cog-scripted-svg-blue.svg Polyhedron snub 12-20 left max.png Polyhedron snub 12-20 left net.svg 3.3.3.3.5
Polyhedron snub 12-20 left vertfig.png
9280 triangles
12 pentagons
1506037.616650I0.9820114

Some definitions of Semiregular polyhedron include one more figure, the Elongated square gyrobicupola or "pseudo-rhombicuboctahedron". [4]

Properties

The number of vertices is 720° divided by the vertex angle defect.

The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular.

The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices.

Chirality

The snub cube and snub dodecahedron are known as chiral , as they come in a left-handed form (Latin: levomorph or laevomorph) and right-handed form (Latin: dextromorph). When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of certain chemical compounds.)

Construction of Archimedean solids

The Archimedean solids can be constructed as generator positions in a kaleidoscope. Polyhedron truncation example3.png
The Archimedean solids can be constructed as generator positions in a kaleidoscope.

The different Archimedean and Platonic solids can be related to each other using a handful of general constructions. Starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated (see table below), different Platonic and Archimedean (and other) solids can be created. If the truncation is exactly deep enough such that each pair of faces from adjacent vertices shares exactly one point, it is known as a rectification. An expansion, or cantellation, involves moving each face away from the center (by the same distance so as to preserve the symmetry of the Platonic solid) and taking the convex hull. Expansion with twisting also involves rotating the faces, thus splitting each rectangle corresponding to an edge into two triangles by one of the diagonals of the rectangle. The last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed the rectification of the rectification. Likewise, the cantitruncation can be viewed as the truncation of the rectification.

Construction of Archimedean Solids
Symmetry Tetrahedral
Tetrahedral reflection domains.png
Octahedral
Octahedral reflection domains.png
Icosahedral
Icosahedral reflection domains.png
Starting solid
Operation
Symbol
{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
Tetrahedron
{3,3}
Uniform polyhedron-33-t0.png
Cube
{4,3}
Uniform polyhedron-43-t0.svg
Octahedron
{3,4}
Uniform polyhedron-43-t2.svg
Dodecahedron
{5,3}
Uniform polyhedron-53-t0.svg
Icosahedron
{3,5}
Uniform polyhedron-53-t2.svg
Truncation (t)t{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
truncated tetrahedron
Uniform polyhedron-33-t01.png
truncated cube
Uniform polyhedron-43-t01.svg
truncated octahedron
Uniform polyhedron-43-t12.svg
truncated dodecahedron
Uniform polyhedron-53-t01.svg
truncated icosahedron
Uniform polyhedron-53-t12.svg
Rectification (r)
Ambo (a)
r{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
tetratetrahedron
(octahedron)
Uniform polyhedron-33-t1.png
cuboctahedron
Uniform polyhedron-43-t1.svg
icosidodecahedron
Uniform polyhedron-53-t1.svg
Bitruncation (2t)
Dual kis (dk)
2t{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
truncated tetrahedron
Uniform polyhedron-33-t12.png
truncated octahedron
Uniform polyhedron-43-t12.png
truncated cube
Uniform polyhedron-43-t01.svg
truncated icosahedron
Uniform polyhedron-53-t12.svg
truncated dodecahedron
Uniform polyhedron-53-t01.svg
Birectification (2r)
Dual (d)
2r{p,q}
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
tetrahedron
Uniform polyhedron-33-t2.png
octahedron
Uniform polyhedron-43-t2.svg
cube
Uniform polyhedron-43-t0.svg
icosahedron
Uniform polyhedron-53-t2.svg
dodecahedron
Uniform polyhedron-53-t0.svg
cantellation (rr)
Expansion (e)
rr{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
rhombitetratetrahedron
(cuboctahedron)
Uniform polyhedron-33-t02.png
rhombicuboctahedron
Uniform polyhedron-43-t02.png
rhombicosidodecahedron
Uniform polyhedron-53-t02.png
Snub rectified (sr)
Snub (s)
sr{p,q}
CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
snub tetratetrahedron
(icosahedron)
Uniform polyhedron-33-s012.svg
snub cuboctahedron
Uniform polyhedron-43-s012.png
snub icosidodecahedron
Uniform polyhedron-53-s012.png
Cantitruncation (tr)
Bevel (b)
tr{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
truncated tetratetrahedron
(truncated octahedron)
Uniform polyhedron-33-t012.png
truncated cuboctahedron
Uniform polyhedron-43-t012.png
truncated icosidodecahedron
Uniform polyhedron-53-t012.png

Note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron. Also, partially because the tetrahedron is self-dual, only one Archimedean solid that has at most tetrahedral symmetry. (All Platonic solids have at least tetrahedral symmetry, as tetrahedral symmetry is a symmetry operation of (i.e. is included in) octahedral and isohedral symmetries, which is demonstrated by the fact that an octahedron can be viewed as a rectified tetrahedron, and an icosahedron can be used as a snub tetrahedron.)

Stereographic projection

truncated tetrahedron truncated cube truncated octahedron truncated dodecahedron truncated icosahedron
Truncated tetrahedron stereographic projection triangle.png
triangle-centered
Truncated tetrahedron stereographic projection hexagon.png
hexagon-centered
Truncated cube stereographic projection octagon.png
octagon-centered
Truncated cube stereographic projection triangle.png
triangle-centered
Truncated octahedron stereographic projection square.png
square-centered
Truncated octahedron stereographic projection hexagon.png
hexagon-centered
Truncated dodecahedron stereographic projection decagon.png
Decagon-centered
Truncated dodecahedron stereographic projection triangle.png
Triangle-centered
Truncated icosahedron stereographic projection pentagon.png
pentagon-centered
Truncated icosahedron stereographic projection hexagon.png
hexagon-centered
cuboctahedron icosidodecahedron rhombicuboctahedron rhombicosidodecahedron
Cuboctahedron stereographic projection square.png
square-centered
Cuboctahedron stereographic projection triangle.png
triangle-centered
Cuboctahedron stereographic projection vertex.png
vertex-centered
Icosidodecahedron stereographic projection pentagon.png
pentagon-centered
Icosidodecahedron stereographic projection triangle.png
triangle-centered
Rhombicuboctahedron stereographic projection square.png
square-centered
Rhombicuboctahedron stereographic projection square2.png
square-centered
Rhombicuboctahedron stereographic projection triangle.png
triangle-centered
Rhombicosidodecahedron stereographic projection pentagon'.png
Pentagon-centered
Rhombicosidodecahedron stereographic projection triangle.png
Triangle-centered
Rhombicosidodecahedron stereographic projection square.png
Square-centered
truncated cuboctahedron truncated icosidodecahedron snub cube
Truncated cuboctahedron stereographic projection square.png
square-centered
Truncated cuboctahedron stereographic projection hexagon.png
hexagon-centered
Truncated cuboctahedron stereographic projection octagon.png
octagon-centered
Truncated icosidodecahedron stereographic projection decagon.png Truncated icosidodecahedron stereographic projection hexagon.png Truncated icosidodecahedron stereographic projection square.png Snub cube stereographic projection.png
square-centered

See also

Citations

  1. Steckles, Katie. "The Unwanted Shape". YouTube . Retrieved 20 January 2022.
  2. 1 2 Grünbaum (2009).
  3. Field J., Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler, Archive for History of Exact Sciences, 50, 1997, 227
  4. Malkevitch (1988), p. 85

General references

Related Research Articles

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

Regular icosahedron One of the five Platonic solids

In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

Johnson solid Non-uniform convex polyhedron, with each face a regular polygon

In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (J1); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform before they refer to it as a “Johnson solid”.

Polyhedron Three-dimensional shape with flat polygonal faces, straight edges and sharp corners

In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron.

A Platonic solid is a convex regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

Rhombicuboctahedron Archimedean solid with eight triangular, six square, and twelve rectangular faces

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

Truncated cuboctahedron Archimedean solid in geometry

In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.

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.

Elongated square gyrobicupola

In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (J37). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pattern at each of its vertices, because unlike the 13 Archimedean solids, it lacks a set of global symmetries that take every vertex to every other vertex. It strongly resembles, but should not be mistaken for, the small rhombicuboctahedron, which is an Archimedean solid. It is also a canonical polyhedron.

Rectification (geometry) Operation in Euclidean geometry

In Euclidean geometry, rectification, also known as critical truncation or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

Uniform polyhedron Class of mathematical solids

A uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

Great icosahedron

In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra, with Schläfli symbol {3,52} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

Vertex configuration

In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron.

Truncation (geometry) Operation that cuts polytope vertices, creating a new facet in place of each vertex

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

Alternation (geometry) Operation on a polyhedron or tiling that removes alternate vertices

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

Conway polyhedron notation

In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

Uniform polytope

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

A pseudo-uniform polyhedron is a polyhedron which has regular polygons as faces and has the same vertex configuration at all vertices but is not vertex-transitive: it is not true that for any two vertices, there exists a symmetry of the polyhedron mapping the first isometrically onto the second. Thus, although all the vertices of a pseudo-uniform polyhedron appear the same, it is not isogonal. They are called pseudo-uniform polyhedra due to their resemblance to some true uniform polyhedra.