Elongated octahedron

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Elongated octahedron
Elongated octahedron.png
Elongated octahedron
TetOct2 solid2.png
Deltahedral hexadecahedron
Faces 4 {3}
4 traps
16 {3}
Edges 14 24
Vertices 8 10
Vertex configuration 4 (32.42)
4 (3.42)
4 (34)
4 (35)
2 (36)
Symmetry D2h, [2,2], (*222), order 8
Properties Convex Deltahedron
Elongated octahedron trapezoidal net.pngElongated octahedron net.png
Nets

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[edit]

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 Polyiamond-1-1.svg and 4 triamonds Polyiamond-3-1.svg. This construction has been called a triamond stretched octahedron.[1]

As a folded hexahedron[edit]

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[edit]

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[edit]

This polyhedron has a highest symmetry as D2h 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.

Tet-oct-wedge.png

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.

Rhombic prism triangles.png

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.png

See also[edit]

References[edit]

  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  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]
  • Johnson, Norman W. (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. ISSN 0008-414X. Zbl 0132.14603. doi:10.4153/cjm-1966-021-8.  Contains the original enumeration of the 92 solids and the conjecture that there are no others.
  • Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. Zbl 0177.24802. No ISBN.  The first proof that there are only 92 Johnson solids: see also Zalgaller, Victor A. (1967). "Convex Polyhedra with Regular Faces". Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova (in Russian). 2: 1–221. ISSN 0373-2703. Zbl 0165.56302. 

External links[edit]