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

As a deltahedral hexadecahedron

A related construction is a hexadecahedron, 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 In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

. If the sets of coplanar triangles are considered a single isosceles trapezoidal face (a

triamond A polyiamond (also polyamond or simply iamond, or sometimes triangular polyomino) is a polyform whose base form is an equilateral triangle. The word ''polyiamond'' is a back-formation from ''diamond'', because this word is often used to describe ...

), it has 8 vertices, 14 edges, and 8 faces - 4 triangles Polyiamond-1-1

and 4 triamonds Polyiamond-3-1

. This construction has been called a triamond stretched octahedron.Convex Triamond Regular Polyhedra
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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. It could also be seen as a ''folded tetrahedron'' also seeing pairs of end triangles as a folded rhombus. It would have 8 vertices, 10 edges, and 4 faces.

Cartesian coordinates

The

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

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

In the special case, where the trapezoid faces are squares or

rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...

s, the pairs of triangles becoming coplanar and the polyhedron's geometry is more specifically a right rhombic prism. : Rhombic prism triangles

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 A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by conv ...

, giving the names octahedral wedge, or double wedge. The half-model has 8 triangles and 2 squares. : Tet-oct-wedge

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. : HC P1-P3

References

* p.172 tetrahedra-octahedral packing * H. Martyn Cundy ''Deltahedra.'' Math. Gaz. 36, 263-266, Dec 1952

* H. Martyn Cundy and A. Rollett. "Deltahedra". §3.11 in '' Mathematical Models (Cundy and Rollett), 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. 55–5

* 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