What is the difference between platonic solids and archimedean solids

The following table gives all possible regular and semiregular polyhedra and tessellations. In the table, 'P' denotes Platonic solid , 'M' denotes a prism or antiprism , 'A' denotes an Archimedean solid, and 'T' a plane tessellation. As shown in the above table, there are exactly 13 Archimedean solids Walsh , Ball and Coxeter They are called the cuboctahedron , great rhombicosidodecahedron , great rhombicuboctahedron , icosidodecahedron , small rhombicosidodecahedron , small rhombicuboctahedron , snub cube , snub dodecahedron , truncated cube , truncated dodecahedron , truncated icosahedron soccer ball , truncated octahedron , and truncated tetrahedron.

Let be the inradius of the dual polyhedron corresponding to the insphere , which touches the faces of the dual solid , be the midradius of both the polyhedron and its dual corresponding to the midsphere , which touches the edges of both the polyhedron and its duals , the circumradius corresponding to the circumsphere of the solid which touches the vertices of the solid of the Archimedean solid, and the edge length of the solid Since the circumsphere and insphere are dual to each other, they obey the relationship.

The following tables give the analytic and numerical values of , , and for the Archimedean solids with polyhedron edges of unit length Coxeter et al. Hume gives approximate expressions for the dihedral angles of the Archimedean solid and exact expressions for their duals. The Archimedean solids and their duals are all canonical polyhedra. Since the Archimedean solids are convex, the convex hull of each Archimedean solid is the solid itself.

Ball, W. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. Behnke, H. Fundamentals of Mathematics, Vol. Catalan, E. Coxeter, H. Cambridge Phil. London Ser. A , , Critchlow, K. New York: Viking Press, Cromwell, P. New York: Cambridge University Press, pp. Cundy, H. Stradbroke, England: Tarquin Pub. Oxford, England: Pergamon Press, Geometry Technologies. Holden, A.

Shapes, Space, and Symmetry. Hovinga, S. Hume, A. Kepler, J. Frankfurt, pp. Kraitchik, M. Mathematical Recreations. New York: W. Norton, pp. The following figure ilustrates this process for the case of the Snub Cube, which starts from the Truncated Cuboctahedron also called Great Rhombicuboctahedron :. You can find a summarizing table on Archimedean Solids on the article related to Catalan Solids , which are their duals. Skip to main content. Sacred Geometry.

You are here Home » Sacred Solids. Archimedean Solids. Hence their "double name": Cuboctahedron Icosidodecahedron The next two solids, the Truncated Cuboctahedron also called Great Rhombicuboctahedron and the Truncated Icosidodecahedron also called Great Rhombicosidodecahedron apparently seem to be derived from truncating the two preceding ones.

This process is illustrated in the following figure: Rhombicuboctahedron Expansion process from Wikipedia Rhombicosidodecahedron Expansion process from Wikipedia The name of the Truncated Cuboctahedron also called Great Rhombicuboctahedron and of the Truncated Icosidodecahedron also called Great Rhombicosidodecahedron again seem to indicate that they can be derived from truncating the Cuboctahedron and the Icosidodecahedron.

Truncated Cuboctahedron Truncated Icosidodecahedron Finally, there are two special solids which have two chiral specular symmetric variations: the Snub Cube and the Snub Dodecahedron. Here we show only one chiral form of each: Snub Cube Snub Dodecahedron These solids can be constructed as an alternation of another Archimedean solid. The following figure ilustrates this process for the case of the Snub Cube, which starts from the Truncated Cuboctahedron also called Great Rhombicuboctahedron : You can find a summarizing table on Archimedean Solids on the article related to Catalan Solids , which are their duals.

Powered by Drupal. Expansion process from Wikipedia. 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. Sometimes it is instead only required that the faces that meet at one vertex are related isometrically to the faces that meet at the other. Platonic solid, any of the five geometric solids whose faces are all identical, regular polygons meeting at the same three-dimensional angles.

Also known as the five regular polyhedra, they consist of the tetrahedron or pyramid , cube, octahedron, dodecahedron, and icosahedron. These solids, now called the Platonic solids, are the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.

Some of the Archimedean solids can be thought of as variations on the Platonic solids. Non-polyhedrons are cones, spheres, and cylinders because they have sides that are not polygons.