Cuboctahedron

The cuboctahedron (also cubooctahedra or cubo- octahedra ) is a polyhedron with 14 faces ( six squares and eight regular triangles), twelve identical vertices and 24 identical edges.

Due to its regularity the cuboctahedron is one of the 13 Archimedean solids. In addition to the icosahedron, it is the only convex quasiregular body. The cuboctahedron is the only polyhedron, wherein the corner radius always corresponds to the edge length.

Its dual body is the rhombic dodecahedron.

• 2.1 cube and octahedron
• 2.2 tetrahedron
• 2.3 icosahedron
• 2.4 Kuboktaederstumpf
• 2.5 rhombododecahedral
• 2.6 Stellar cuboctahedron
• 2.7 Non -convex polyhedra
• 2.8 Johnson body

• 4.1 Chemistry
• 4.2 jitterbug transformation
• 4.3 Art

Mathematical properties

Symmetry

With 12 corners, 14 faces and 24 edges of the Euler Polyedersatz is satisfied:

In terms of its symmetric properties, the cuboctahedron be categorized as quasi- regular convex polyhedron surface:

• All surfaces are regular. Since the cuboctahedron over squares and triangles has, however, the surfaces are not uniform, and therefore it has no inscribed ball. This condition is met only by the Platonic and the Catalan bodies.
• All edges are symmetry equivalent, as touching on each edge exactly a square and a triangle. Apart from icosidodecahedra no other Archimedean body satisfies this condition. The cuboctahedron has an edge ball.
• All corners are symmetry equivalent since two triangles and two squares meeting at each corner. Therefore, the cuboctahedron has a circumscribed sphere.

Network

The web of cuboctahedron is constructed as follows: Starting from a square in each case, an equilateral triangle is applied to each of its edges. For each triangle, a square and at its opposite side is another triangle. Finally, a single square is still applied to an arbitrary triangle.

Orthogonal projection

For the cuboctahedron are four special orthogonal projections: for both surface types, for the edge and the corner. Six edges of the cuboctahedron form the edges of a regular hexagon. There are four such hexagons which are not planes of symmetry of the cuboctahedron, but Fixe planes of rotating mirror symmetries.

Sphere packing

In two dimensional Kiss number six equal circles are applied to a circuit. On the three-dimensional level twelve equal-sized balls are applied to a ball.

The two-dimensional kissing number is found at the cuboctahedron again: At one of its hexagons there are six balls to a ball around the origin arrange so that their centers correspond to the vertices of the polyhedron. Assuming the three-dimensional plane can be above and below each create three more balls at the origin ball in addition, their center points fall back on the corners of the cuboctahedron.

The cuboctahedron thus has the densest packing of spheres of all body. However, this applies also for the non-regular anticuboctahedron, in which the scale up and down six balls are vertically stacked and not offset like the cuboctahedron.

Formulas

Geometric relationship

Cube and octahedron

The cuboctahedron can be view as a derivation of two Platonic solid: to penetrate a cube ( cube ) and an octahedron arises as the intersection (core) a cuboctahedron. Its name is derived as a portmanteau of these two bodies. Also, the old designation means crystal refers to its role as an intermediate form. The faces of a cube ( six squares) and an octahedron ( eight triangles ) are the total of 14 faces of the cuboctahedron.

By blunting of the corners of a cuboctahedron can each produce from two main bodies: Blunts are the corners of a cube to the center of its edges from, on the one hand reduce his six squares; the other hand, are formed at the corners recent eight triangles. By blunting of the corners of an octahedron to the edge center its eight triangles are greatly reduced in size and the previous corners to six squares.

In the generation of a cuboctahedron by blunting of cube or octahedron produces two intermediate forms: When both base body is not up to edge center, but only partially blunted the two Archimedean solids Hexaederstumpf or truncated octahedron can be created.

Tetrahedron

Also from another Platonic solids can be derived the cuboctahedron: If a tetrahedron along its six edges extended arise six squares. Form on the previous corners of the tetrahedron four triangles, in addition to the four originally existing. Performs If this process continues until the squares are square, one obtains a cuboctahedron. Alternatively, can be thought of as a dulling of the edges of a tetrahedron this operation.

Icosahedron

The cuboctahedron is itself an output form for the derivation of other polyhedra. All Platonic and Archimedean solids can be derived either from cuboctahedron, icosahedron or tetra- tetrahedral ( octahedral ) by twisting (torsion ). In these three polyhedra are the possible penetrating the Platonic solids.

By turning a cuboctahedron can be with the icosahedron, a Platonic solid manufacture: The triangles of the cuboctahedron remain unchanged. Through a distortion of the squares created six diamonds. This is divided by new edges so that a total of twelve regular triangles, in addition to the original eight of the cuboctahedron. The new body thus has 20 triangles and is an icosahedron.

Kuboktaederstumpf

The Kuboktaederstumpf is an Archimedean body, but can not be produced by blunting of a cuboctahedron, as the name suggests and what is mistakenly thought occasionally. That it is not so, can easily recognize the type of colliding at the corners of the cuboctahedron surfaces. On the triangles each form two outgoing edges of a corner at an angle of 60 °, but the squares are 90 °. By blunting every corner would be a rectangle instead of a square, because the hypotenuse in an isosceles triangle is at a 90 -degree angle against more than under a 60 ° reverse angle.

However, this truncated cuboctahedron is topologically equivalent to Kuboktaederstumpf because it has the same number of faces, edges and corners.

Rhombic dodecahedron

The dual to the cuboctahedron body is the rhombic dodecahedron. This has twelve surfaces and fourteen vertices, thus the reverse ratio as the cuboctahedron. As with all Archimedean solids of the dual body is a Catalanischer body. During the cuboctahedron forms the interface with the penetration of cube and octahedron, the rhombic dodecahedron to the envelope body.

Stellar cuboctahedron

There are four different star shapes the cuboctahedron. The first stellar body is identical to the penetration of cube and octahedron.

Non-convex polyhedrons

Two non - convex bodies share the position of the edges and corners of the cuboctahedron: When Kubohemioktaeder consist of only the squares, the Oktahemioktaeder only the triangles. The remaining land is occupied by the four hexagons within the cuboctahedron. The cuboctahedron is the convex hull of the other two bodies.

Johnson body

If a cuboctahedron anlang one of its hexagons intersected, creating two triangular domes, the Johnson - body J3. Alternatively you can also the cuboctahedron of six square pyramids ( J1 ) and eight tetrahedra composed imagine.

History

As only one of the Archimedean solids, the cuboctahedron to have been already known to Plato.

Reference to the physical world

Chemistry

The crystalline structure of synthetic diamond is based ideally on the cube or the octahedron - but mostly on the cuboctahedron. Often these bodies are not regular, but only approximate shapes. Natural diamonds usually have an octahedral crystalline structure. The cuboctahedron is the crystal of the mineral argentite ( Ag 2 S ).

Jitterbug transformation

In so-called Buckminster Fuller Jitterbug transform the cuboctahedron with 24 edges is the most extensive stage. By twisting creates an icosahedron, with six of its 30 edges exist only virtually. After further twisting the edges come together in pairs, resulting in an octahedron. By a repeated twisting creates a tetrahedron, with four edges are collapsed. This can eventually be folded into a flat triangle, in which coincide eight edges.

If a surface model of the octahedron of the Jitterbug Transformation subject so can this transform on the icosahedron into an even bigger cuboctahedron. On the research exhibition Heureka in Zurich in 1991 was shown at the walk- eureka polyhedra this transformation. During the change, the visitors inside on a hoist were in sync with moves up and down.

Art

Leonardo da Vinci made ​​for Luca Pacioli's De divina proportione ( 1509 ) on the subscriptions of several polyhedra, including the cuboctahedron.

In MC Escher's 1948 wood engraving star at the bottom left is a small cuboctahedron among many other polyhedra appear.