Archimedean solid

From Academic Kids

In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices, and from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.

Origin of name

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. During the Renaissance, artists and mathematicians valued pure forms and rediscovered all of these forms. This search was completed around 1619 by Johannes Kepler, who defined prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot solids.

Classification

There are 13 Archimedean solids (15 if the mirror images of two enantiomorphs, see below, 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 and picture Faces Edges Vertices Vertex configuration Symmetry group
cuboctahedron
Cuboctahedron
(Video)
 14  8 triangles
6 squares
24 12 3,4,3,4 Oh
icosidodecahedron
Missing image
Icosidodecahedron.jpg
Icosidodecahedron


(Video)
32 20 triangles
12 pentagons
60 30 3,5,3,5 Ih
truncated tetrahedron
Missing image
Truncatedtetrahedron.jpg
Truncated tetrahedron


(Video)
8 4 triangles
4 hexagons
18 12 3,6,6 Td
truncated cube
or truncated hexahedron
Missing image
Truncatedhexahedron.jpg
Truncated hexahedron


(Video)
14 8 triangles
6 octagons
36 24 3,8,8 Oh
truncated octahedron
Truncated octahedron
(Video)
14 6 squares
8 hexagons
36 24 4,6,6 Oh
truncated dodecahedron
Missing image
Truncateddodecahedron.jpg
Truncated dodecahedron


(Video)
32 20 triangles
12 decagons
90 60 3,10,10 Ih
truncated icosahedron
or commonly football (soccer ball)
Missing image
Truncatedicosahedron.jpg
Truncated icosahedron


(Video)
32 12 pentagons
20 hexagons
90 60 5,6,6 Ih
rhombicuboctahedron
or small rhombicuboctahedron
Rhombicuboctahedron
(Video)
26 8 triangles
18 squares
48 24 3,4,4,4 Oh
truncated cuboctahedron
or great rhombicuboctahedron
Truncated cuboctahedron
(Video)
26 12 squares
8 hexagons
6 octagons
72 48 4,6,8 Oh
rhombicosidodecahedron
or small rhombicosidodecahedron
Rhombicosidodecahedron
(Video)
62 20 triangles
30 squares
12 pentagons
120 60 3,4,5,4 Ih
truncated icosidodecahedron
or great rhombicosidodecahedron
Missing image
Truncatedicosidodecahedron.jpg
Truncated icosidodecahedron


(Video)
62 30 squares
20 hexagons
12 decagons
180 120 4,6,10 Ih
snub cube
or snub cuboctahedron (2 chiral forms)
Snub hexahedron (Ccw)
(Video)
Snub hexahedron (Cw)
(Video)
38 32 triangles
6 squares
60 24 3,3,3,3,4 O
snub dodecahedron
or snub icosidodecahedron (2 chiral forms)
Snub dodecahedron (Ccw)
(Video)
Missing image
Snubdodecahedroncw.jpg
Snub dodecahedron (Cw)


(Video)
92 80 triangles
12 pentagons
150 60 3,3,3,3,5 I

The first two solids (cuboctahedron and icosidodecahedron) are edge-uniform and are called quasi-regular.

The last two (snub cube and snub dodecahedron) are known as chiral, as they come in a left-handed (Latin: levomorph or laevomorph) form and right-handed (Latin: dextromorph) form. 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 chemical compounds).

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.

External links

it:Solido archimedeo nl:Halfregelmatig veelvlak pl:Wielościan półforemny ru:Полуправильный многогранник zh:半正多面體

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