Small Rhombicuboctahedron

The small rhombicuboctahedron is an Archimedean solid, one of only 13. As such all the edges are the same length and it has two types of nonintersecting convex polygons arranged in the same formation about each vertex. There are 28 faces in total, 8 are triangular and 18 are square.

The small rhombicuboctahedron can be formed in (at least) 2 ways. One either starts with a cube with the desired edge length and extrude the 6 faces, joining up the adjacent edges to form 12 new square faces and 8 new triangular faces. The distance of the extruded faces is such as to result in a polyhedron where all the edges are of equal length, the length of the cube edges, denoted as "a" in the rest of this document. If the edges of the cube are +/- 1 along each axis, then the new position of the cube vertices is +/- (1+√2)

Alternatively, start with an octahedron and extrude each triangular face, joining up the adjacent edges to form 18 new square faces. The edges are the same length as above, the height of the octahedron is a/√2.

The following outlines a numbering scheme for a computer based model of a small rhombicuboctahedron. The geometry for vertices, edges and faces is provided.

The coordinates for a central cube of edge length 2 are given below.

// Vertices
// Coordinates with valiue +/- 3 should be remapped to +/- (1+sqrt(2))
#define NV 32
XYZ v[NV] = {
   {1,-1,3}, {1,1,3}, {-1,-1,3}, {-1,1,3},         // 0 to 3, upper z face
   {1,-3,1}, {1,-1,1}, {1,1,1}, {1,3,1},           // 4 to 7
   {-1,-3,1}, {-1,-1,1}, {-1,1,1}, {-1,3,1},       // 8 to 11
   {1,-3,-1}, {1,-1,-1}, {1,1,-1}, {1,3,-1},       // 12 to 15
   {-1,-3,-1}, {-1,-1,-1}, {-1,1,-1}, {-1,3,-1},   // 16 to 19
   {1,-1,-3}, {1,1,-3}, {-1,-1,-3}, {-1,1,-3},     // 20 to 23, lower z face
   {3,-1,1}, {3,1,1}, {3,-1,-1}, {3,1,-1},         // 24 to 27, right x face
   {-3,-1,1}, {-3,1,1}, {-3,-1,-1}, {-3,1,-1}      // 28 to 31, left x face
};

The edges as pairs of vertices, indexed into the above vertex list are given below.

// Edges
#define NE 48
EDGE e[NE] = {
   {0,1}, {1,3}, {3,2}, {2,0}, 
   {20,21}, {21,23}, {23,22}, {22,20}, 
   {24,25}, {25,27}, {27,26}, {26,24}, 
   {28,29}, {29,31}, {31,30}, {30,28}, 
   {4,8}, {8,16}, {16,12}, {12,4}, 
   {7,11}, {11,19}, {19,15}, {15,7},      
   {0,4}, {4,24}, {24,0}, {25,1}, {1,7}, {7,25}, 
   {11,3}, {3,29}, {29,11}, {28,2}, {2,8}, {8,28},
   {12,20}, {20,26}, {26,12}, {21,27}, {27,15}, {15,21}, 
   {19,23}, {23,31}, {31,19}, {16,30}, {30,22}, {22,16}
};

The faces, arranges clockwise looking from the outside, indexed into the above vertex list are given below. 18 of the faces have 4 vertices, 8 have only 3 vertices (triangles) and are indicated with a negative value for the 4th vertex.

// Faces
#define NF 26
FACE f[NF] = {
   {0,2,3,1}, {20,21,23,22},                                   // Extreme z faces
   {24,25,27,26}, {28,30,31,29},                               // Extreme x faces
   {8,4,12,16}, {7,11,19,15},                                  // Extreme y faces
   {0,1,25,24}, {1,3,11,7}, {3,2,28,29}, {2,0,4,8},            // Top wedge faces
   {20,26,27,21}, {15,19,23,21}, {31,30,22,23}, {16,12,20,22}, // Bottom wedge faces
   {4,24,26,12}, {25,7,15,27}, {11,29,31,19}, {28,8,16,30},    // Central wedge faces
   {4,0,24,-1}, {1,7,25,-1}, {11,3,29,-1}, {28,2,8,-1},        // Top pyramids
   {12,26,20,-1}, {27,15,21,-1}, {19,31,23,-1}, {30,16,22,-1}  // Lower pyramids
};

Expression for the surface area of each face and the volume for each component are presented in the following.

In some software packages the polyhedron is referred to as an "octotoad".

Equirectangular projection.

Historical usage

The rhombicuboctahedron features as a half fill glass container in the top left of the painting "Portrait of Luca Pacioli", circa 1500.

The rhombicuboctahedron is the basis for a sundial by Ludwig von Hohenfeld, with 17 faces for each region between Tubingen and Stuttgart. Circa 1596.