Elongated octahedron

In geometry, an elongated octahedron (also trapezoidal octahedron) is a polyhedron with 8 faces (4 triangular, 4 isosceles trapezoidal), 14 edges, and 8 vertices.

As a deltahedral hexadecahedron

It can also be constructed as a hexadecahedron, with 16 triangular faces, 24 edges, and 10 vertices. Starting with the regular octahedron, it is elongated along one axes, adding 8 new triangles. It has 2 sets of 3 coplanar equilateral triangles (each forming a half-hexagon), and thus is not a Johnson solid.

If the sets of coplanar triangles are considered a single isosceles trapezoidal face (a triamond), it has 8 vertices, 14 edges, and 8 faces - 4 triangles and 4 triamonds . This construction has been called a triamond stretched octahedron.

As a folded hexahedron

Another interpretation can represent this solid as a hexahedron, by considering pairs of trapezoids as a folded regular hexagon. It will have 6 faces (4 triangles, and 2 hexagons), 12 edges, and 8 vertices.

Cartesian coordinates

The Cartesian coordinates of the 8 vertices of an elongated octahedron, elongated in the x-axis, with edge length 2 are:

( ±1, 0, ±2 )

( ±2, ±1, 0 ).

The 2 extra vertices of the deltahedral variation are:

( 0, ±1, 0 ).

Related polyhedra and honeycombs

This polyhedron has a highest symmetry as D_2h symmetry, order 8, representing 3 orthogonal mirrors. Removing one mirror between the pairs of triangles divides the polyhedron into two identical wedges, giving the names octahedral wedge, or double wedge. The half-model has 8 triangles and 2 squares.

In the special case, where the trapezoid faces are squares or rectangles, the pairs of triangles becoming coplanar and the polyhedron's geometry is more specifically a right rhombic prism.

It can also be seen as the augmentation of 2 octahedrons, sharing a common edge, with 2 tetrahedrons filling in the gaps. This represents a section of a tetrahedral-octahedral honeycomb. The elongated octahedron can thus be used with the tetrahedron as a space-filling honeycomb.

See also

• Edge-contracted icosahedron
• Elongated dodecahedron

References

• p.172 tetrahedra-octahedral packing

• H. Martyn Cundy Deltahedra. Math. Gaz. 36, 263-266, Dec 1952. 1
• H. Martyn Cundy and A. Rollett Deltahedra. §3.11 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 142-144, 1989.
• Charles W. Trigg An Infinite Class of Deltahedra, Mathematics Magazine, Vol. 51, No. 1 (Jan., 1978), pp. 55-57 2

• Contains the original enumeration of the 92 solids and the conjecture that there are no others.

• The first proof that there are only 92 Johnson solids: see also

External links

• The Convex Deltahedra, And the Allowance of Coplanar Faces