Compound of three octahedra

# Compound of three octahedra

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Compound of three octahedra
polyhedra 3 regular octahedra
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

faces 24 equilateral triangles
edges 36
vertices 18
Symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

(Single color)
Oh
Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

, order 48

In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound
Polyhedral compound
A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram....

formed from three regular octahedra
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...

, who used it in the central image of his 1948 woodcut Stars
Stars (M. C. Escher)
Stars is a wood engraving print by the Dutch artist M. C. Escher which was first printed in October 1948, depicting two chameleons in a polyhedral cage floating through space.-Description:...

.

## Construction

A regular octahedron can be circumscribed around a cube in such a way that the eight edges of two opposite squares of the cube lie on the eight faces of the octahedron. The three octahedra formed in this way from the three pairs of opposite cube squares form the compound of three octahedra. The eight cube vertices are the same as the eight points in the compound where three edges cross each other. Each of the octahedron edges that participates in these triple crossings is divided by the crossing point in the ratio 1:√2
Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...

. The remaining octahedron edges cross each other in pairs, within the interior of the compound; their crossings are at their midpoints and form right angles.

The compound of three octahedra can also be formed from three copies of a single octahedron by rotating each copy by an angle of π/4 around one of the three symmetry axes that pass through two opposite vertices of the starting octahedron. A third construction for the same compound of three octahedra is as the dual polyhedron
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

of the compound of three cubes
Compound of three cubes
This uniform polyhedron compound is a symmetric arrangement of 3 cubes, considered as square prisms. It can be constructed by superimposing three identical cubes, and then rotating each by 45 degrees about a separate axis .This compound famously appears in the lithograph print Waterfall by M.C....

, one of the uniform polyhedron compound
Uniform polyhedron compound
A uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.The uniform polyhedron compounds were first enumerated by John Skilling...

s.

The six vertices of one of the three octahedra may be given by the coordinates and . The other two octahedra have coordinates that may be obtained from these coordinates by exchanging the z coordinate for the x or y coordinate.

## Symmetries

The compound of three octahedra has the same symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

as a single octahedron.
It is an isohedral deltahedron
Deltahedron
A deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...

, meaning that its faces are equilateral triangles and that it has a symmetry taking every face to every other face. There is one known infinite family of isohedral deltahedra, and 36 more that do not fall into this family; the compound of three octahedra is one of the 36 sporadic examples. However, its symmetry group does not take every vertex to every other vertex, so it is not itself a uniform polyhedron compound.

The intersection of the three octahedra is a convex polyhedron with 14 vertices and 24 faces, a tetrakis hexahedron
Tetrakis hexahedron
In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube....

, formed by attaching a low square pyramid
Square pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...

to each face of the central cube. Thus, the compound can be seen as a stellation
Stellation
Stellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...

of the tetrakis hexahedron. A different form of the tetrakis hexahedron, formed by using taller pyramids on each face of the cube, is non-convex but has equilateral triangle faces that again lie on the same planes as the faces of the three octahedra; it is another of the known isohedral deltahedra. A third isohedral deltahedron sharing the same face planes, the compound of six tetrahedra
Compound of six tetrahedra
This uniform polyhedron compound is a symmetric arrangement of 6 tetrahedra. It can be constructed by inscribing a stella octangula within each cube in the compound of three cubes, or by stellating each octahedron in the compound of three octahedra....

, may be formed by stellating
Stellation
Stellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...

each face of the compound of three octahedra to form three stellae octangulae
Stella octangula
The stellated octahedron, or stella octangula, is the only stellation of the octahedron. It was named by Johannes Kepler in 1609, though it was known to earlier geometers...

. A fourth isohedral deltahedron with the same face planes, also a stellation of the compound of three octahedra, has the same combinatorial structure as the tetrakis hexahedron but with the cube faces dented inwards into intersecting pyramids rather than attaching the pyramids to the exterior of the cube.

The cube around which the three octahedra can be circumscribed has nine planes of reflection symmetry
Reflection symmetry
Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...

. Three of these reflection panes pass parallel to the sides of the cube, halfway between two opposite sides; the other six pass diagonally across the cube, through four of its vertices. These nine planes coincide with the nine equatorial planes of the three octahedra.

## History

In the 15th century manuscript Libellus De Quinque Corporibus Regularibus by Piero della Francesca
Piero della Francesca
Piero della Francesca was a painter of the Early Renaissance. As testified by Giorgio Vasari in his Lives of the Artists, to contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca is chiefly appreciated for his art. His painting was characterized by its...

, della Francesca already includes a drawing of an octahedron circumscribed around a cube, with eight of the cube edges lying in the octahedron's eight faces. Three octahedra circumscribed in this way around a single cube would form the compound of three octahedra, but della Francesca does not depict the compound.

The next appearance of the compound of three octahedra in the mathematical literature appears to be a 1900 work by Max Brückner, which mentions it and includes a photograph of a model of it.

Dutch artist M. C. Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...

, in his 1948 woodcut Stars
Stars (M. C. Escher)
Stars is a wood engraving print by the Dutch artist M. C. Escher which was first printed in October 1948, depicting two chameleons in a polyhedral cage floating through space.-Description:...

, used as the central figure of the woodcut a cage in this shape, containing two chameleons and floating through space. Escher would not have been familiar with Brückner's work and H. S. M. Coxeter
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, was a British-born Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....

writes that "It is remarkable that Escher, without any knowledge of algebra or analytic geometry, was able to rediscover this highly symmetrical figure." Earlier in 1948, Escher had made a preliminary woodcut with a similar theme, Study for Stars, but instead of using the compound of three regular octahedra in the study he used a related shape, a stellated rhombic dodecahedron
Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...

, which can be formed as a compound of three flattened octahedra. This form as a polyhedron is topologically identical to the disdyakis dodecahedron
Disdyakis dodecahedron
In geometry, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons...

, which can be seen as rhombic dodecahedron with shorter pyramids on the rhombic faces. The dual figure of the octahedral compound, the compound of three cubes, is also shown in a later Escher woodcut, Waterfall
Waterfall (M. C. Escher)
Waterfall is a lithography print by the Dutch artist M. C. Escher which was first printed in October, 1961. It shows an apparent paradox where water from the base of a waterfall appears to run downhill before reaching the top of the waterfall....

, next to the same stellated rhombic dodecahedron.

The compound of three octahedra re-entered the mathematical literature more properly with the work of , who observed its existence and provided coordinates for its vertices. It was studied in more detail by and .

• Compound of four octahedra
Compound of four octahedra
This uniform polyhedron compound is a symmetric arrangement of 4 octahedra, considered as triangular antiprisms. It can be constructed by superimposing four identical octahedra, and then rotating each by 60 degrees about a separate axis .- Cartesian coordinates :Cartesian coordinates for the...

• Compound of five octahedra
Compound of five octahedra
This polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876.- As a stellation :It is the second stellation of the icosahedron, and given as Wenninger model index 23....

• Compound of ten octahedra
Compound of ten octahedra
These uniform polyhedron compounds are symmetric arrangements of 10 octahedra, considered as triangular antiprisms, aligned with the axes of three-fold rotational symmetry of an icosahedron...

• Compound of twenty octahedra
Compound of twenty octahedra
This uniform polyhedron compound is a symmetric arrangement of 20 octahedra . It is a special case of the compound of 20 octahedra with rotational freedom, in which pairs of octahedral vertices coincide.- Related polyhedra :...