Archimedean solid

one of the 13 solids (semi-regular convex polyhedrons composed of regular polygons meeting in identical vertices, excluding the 5 Platonic solids (which are composed of only one type of polygon) and excluding the prisms and antiprisms)
(Redirected from Cuboctahedron)

In geometry, an Archimedean solid is a convex shape which is composed of polygons. It is a polyhedron, with the following properties:

  • Each face is made of a regular polygon
  • All the corners of the shape look the same
  • The shape is neither a platonic solid, nor a prism, nor an antiprism.
A truncated icosahedron looks like a soccer ball. It is made of 12 equilateral pentagons and 20 regular hexagons. It has 60 vertices and 90 edges. It is an Archimedean solid

Depending on the way there are counted, there are thirteen or fifteen such shapes. Of two of these shapes, there are two versions, which cannot be made congruent using rotation. The Archimedean solids are named after the Ancient Greek mathematician Archimedes, who probably discovered them in the 3rd century BC. The writings of Archimedes have been lost, but Pappus of Alexandria summarized them in the 4th century.[1] During the Renaissance, artists and mathematicians valued pure forms and rediscovered all of these forms. Johannes Kepler probably completed this search around 1620.[2]

Constructing an Archimedean solid takes at least two different polygons.

Properties change

  • Archimedean solids are made of regular polygons, therefore all edges have the same length.
  • All Archimedean solids can be produced from Platonic solids, by "cutting the edges" of the platonic solid.
  • The type of polygons meting at a corner ("vertex") characterizes both the archimedean and platonic solid

Relationship with Platonic solids change

The Archimedeans solids can be constructed as generator positions in a kaleidoscope

Platonic solids can be turned into Archimedean solids by following a series of rules for their construction.

Listing of Archimedean solids change

The following is a listing of all Archimedean solids

Image Name Faces Type Edges Vertices
  Truncated tetrahedron 8 4 triangles

4 hexagons

18 12
  Cuboctahedron 14 8 triangles

6 squares

24 12
  Truncated cube 14 8 triangles

6 octagons

36 24
  Truncated octahedron 14 6 squares

8 hexagons

36 24
  Rhombicuboctahedron 26 8 triangles

18 squares

48 24
  Truncated cuboctahedron 26 12 squares

8 hexagons

6 octagons

72 48
Snub cube (2 mirrored versions) 38 32 triangles

6 squares

60 24
  Icosidodecahedron 32 20 triangles

12 pentagons

60 30
  Truncated dodecahedron 32 20 triangles

12 decagons

90 60
  Truncated icosahedron 32 12 pentagons

20 hexagons

90 60
  Rhombicosidodecahedron 62 20 triangles
30 squares
12 pentagons
120 60
  Truncated icosidodecahedron 62 30 squares

20 hexagons

12 decagons

180 120
Snub dodecahedron (2 mirrored versions) 92 80 triangles

12 pentagons

150 60

References change

  1. Grünbaum, Branko 2009. An enduring error. Elemente der Mathematik 64 (3): 89–101. Reprinted in Pitici, Mircea, ed. 2011. The best writing on mathematics, 2010. Princeton University Press, pp. 18–31.
  2. Field J. 1997. 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, 227.