geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the snub disphenoid is a

convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

with 12

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

s as its

faces The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the ...

. It is an example of

deltahedron A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron was named by Martyn Cundy, after the Greek capital letter delta resembling a triangular shape Δ. Deltahedra can be categorized by the property of convexi ...

and

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

. It can be constructed in different approaches. This shape is also called Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron. The snub disphenoid can be visualized as an

atom cluster Nanoclusters are atomically precise, crystalline materials most often existing on the 0-2 nanometer scale. They are often considered kinetically stable intermediates that form during the synthesis of comparatively larger materials such as semic ...

surrounding a central atom, that is the

dodecahedral molecular geometry In chemistry, the dodecahedral molecular geometry describes the shape of compounds where eight atoms or groups of atoms or ligands are arranged around a central atom defining the vertices of a snub disphenoid (also known as a trigonal dodecahedron) ...

. Its vertices may be placed in a sphere and can also be used as a minimum possible

Lennard-Jones potential In computational chemistry, molecular physics, and physical chemistry, the Lennard-Jones potential (also termed the LJ potential or 12-6 potential; named for John Lennard-Jones) is an intermolecular pair potential. Out of all the intermolecul ...

among all eight-sphere clusters. The dual polyhedron of the snub disphenoid is the

elongated gyrobifastigium In geometry, the elongated gyrobifastigium or gabled rhombohedron is a space-filling octahedron with 4 rectangles and 4 right-angled pentagonal faces. Name The first name is from the regular-faced gyrobifastigium but Elongation (geometry), elon ...

Construction

The snub disphenoid can be constructed in different ways. As suggested by the name, the snub disphenoid is constructed from a

tetragonal disphenoid In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same s ...

by cutting all the edges from its faces, and adding

s (the light blue colors in the following image) that are twisted in a certain angle between them. This process construction is known as snubification. The snub disphenoid may also be constructed from a triangular bipyramid, by cutting its two edges along the apices. These apices can be pushed toward each other resulting in the new two vertices pushed away. Alternatively, the snub disphenoid can be constructed from

pentagonal bipyramid The pentagonal bipyramid (or pentagonal dipyramid) is a polyhedron with ten triangular faces. It is constructed by attaching two pentagonal pyramids to each of their bases. If the triangular faces are equilateral, the pentagonal bipyramid is an ...

by cutting the two edges along that connecting the base of the bipyramid and then inserting two equilateral triangles between them. Another way to construct the snub disphenoid is started from the

square antiprism In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even number, even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an ''anticube''. If all its faces are regular ...

, by replacing the two square faces with pairs of equilateral triangles. Another construction of the ''snub disphenoid'' is as a digonal gyrobianticupola. It has the same topology and symmetry but without equilateral triangles. It has 4 vertices in a

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

on a center plane as two

anticupola In geometry, a cupola is a solid formed by joining two polygons, one (the base) with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are square ...

e attached with rotational symmetry. A physical model of the snub disphenoid can be formed by folding a

net NET may refer to: Broadcast media United States * National Educational Television, the predecessor of the Public Broadcasting Service (PBS) in the United States * National Empowerment Television, a politically conservative cable TV network ...

formed by 12 equilateral triangles (a 12-iamond), shown. An alternative net suggested by

John Montroll John Montroll is an American origami artist, author, teacher, and mathematician. He has written many books on origami, promoting the single-square, no-cut, no glue approach. Montroll taught mathematics at St. Anselm's Abbey School in Washington, ...

has fewer concave vertices on its boundary, making it more convenient for

origami ) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a ...

construction. The eight vertices of the snub disphenoid may then be given

Cartesian coordinate In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

\begin
 (\pm t, r, 0), &\qquad (0, -r, \pm t), \\
 (\pm 1, -s, 0), &\qquad (0, s, \pm 1).
\end

Here,

q \approx 0.16902

is the positive real solution of the cubic polynomial

2x^3 +11x^2+4x-1.

. The three variables

r

s

, and

t

is the expression of:

r = \sqrt \approx 0.41112, \qquad s = \sqrt \approx 1.56786, \qquad t = 2rs = \sqrt \approx 1.28917.

Because this construction involves the solution to a cubic equation, the snub disphenoid cannot be constructed with a compass and straightedge, unlike the other seven deltahedra.

Properties

As a consequence of such constructions, the snub disphenoid has 12 equilateral triangles. A

is a polyhedron in which all faces are equilateral triangles. There are eight convex deltahedra, one of which is the snub disphenoid. More generally, the

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

polyhedron in which all faces are

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

are the

s, and every convex deltahedron is Johnson solid. The snub disphenoid is among them, enumerated as the 84th Johnson solid

J_

. The dual polyhedron of the snub disphenoid is the

Measurement

A snub disphenoid with edge length

a

has a surface area:

A = 3 \sqrta^2 \approx 5.19615a^2,

the area of 12 equilateral triangles. Its volume can be calculated as the formula:

V \approx 0.85949 a^3.

Symmetry and geodesic

The snub disphenoid has the same symmetries as a

, the

antiprismatic symmetry In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih''n'' (for ''n'' ≥ 2). Types Ther ...

D_

of order 8: it has an axis of 180° rotational symmetry through the midpoints of its two opposite edges, two perpendicular planes of

reflection symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...

through this axis, and four additional symmetry operations given by a reflection perpendicular to the axis followed by a quarter-turn and possibly another reflection parallel to the axis.. Up to symmetries and parallel translation, the snub disphenoid has five types of simple (non-self-crossing)

closed geodesic In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic f ...

s. These are paths on the surface of the polyhedron that avoid the vertices and locally look like the shortest path: they follow straight line segments across each face of the polyhedron that they intersect, and when they cross an edge of the polyhedron they make complementary angles on the two incident faces to the edge. Intuitively, one could stretch a rubber band around the polyhedron along this path and it would stay in place: there is no way to locally change the path and make it shorter. For example, one type of geodesic crosses the two opposite edges of the snub disphenoid at their midpoints (where the symmetry axis exits the polytope) at an angle of

\pi/3

. A second type of geodesic passes near the intersection of the snub disphenoid with the plane that perpendicularly bisects the symmetry axis (the

equator The equator is the circle of latitude that divides Earth into the Northern Hemisphere, Northern and Southern Hemisphere, Southern Hemispheres of Earth, hemispheres. It is an imaginary line located at 0 degrees latitude, about in circumferen ...

of the polyhedron), crossing the edges of eight triangles at angles that alternate between

\pi/2

and

\pi/6

. Shifting a geodesic on the surface of the polyhedron by a small amount (small enough that the shift does not cause it to cross any vertices) preserves the property of being a geodesic and preserves its length, so both of these examples have shifted versions of the same type that are less symmetrically placed. The lengths of the five simple closed geodesics on a snub disphenoid with unit-length edges are :

2\sqrt\approx 3.464

(for the equatorial geodesic),

\sqrt\approx 3.606

4

(for the geodesic through the midpoints of opposite edges),

2\sqrt\approx 5.292

, and

\sqrt\approx 4.359

. Except for the tetrahedron, which has infinitely many types of simple closed geodesics, the snub disphenoid has the most types of geodesics of any deltahedron.

Representation by the graph

The snub disphenoid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected

simplicial In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

well-covered polyhedra, meaning that all of the

maximal independent set In graph theory, a maximal independent set (MIS) or maximal stable set is an Independent set (graph theory), independent set that is not a subset of any other independent set. In other words, there is no Vertex (graph theory), vertex outside th ...

s of its vertices have the same size. The other three polyhedra with this property are the

regular octahedron In geometry, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyh ...

, the

, and an irregular polyhedron with 12 vertices and 20 triangular faces.

Applications

In the study of

computational chemistry Computational chemistry is a branch of chemistry that uses computer simulations to assist in solving chemical problems. It uses methods of theoretical chemistry incorporated into computer programs to calculate the structures and properties of mol ...

and

molecular physics Molecular physics is the study of the physical properties of molecules and molecular dynamics. The field overlaps significantly with physical chemistry, chemical physics, and quantum chemistry. It is often considered as a sub-field of atomic, mo ...

, spheres centered at the vertices of the snub disphenoid form a cluster that, according to numerical experiments, has the minimum possible

among all eight-sphere clusters. In the geometry of

chemical compound A chemical compound is a chemical substance composed of many identical molecules (or molecular entities) containing atoms from more than one chemical element held together by chemical bonds. A molecule consisting of atoms of only one element ...

s, a polyhedron may be visualized as the

surrounding a central atom. The

describes the cluster for which it is a snub disphenoid.

History and naming

The ''snub disphenoid'' name comes from classification of the

. However, this solid was first studied by . It was studied again in the paper by , which first described the set of eight convex

deltahedra A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron was named by Martyn Cundy, after the Greek capital letter Delta (letter), delta resembling a triangular shape Δ. Deltahedra can be categorized by the prope ...

, and named it the ''Siamese dodecahedron''. The ''dodecadeltahedron'' name was given to the same shape by , referring to the fact that it is a 12-sided deltahedron. There are other simplicial dodecahedra, such as the

hexagonal bipyramid A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles. Although it is face-transiti ...

, but this is the only one that can be realized with equilateral faces. Bernal was interested in the shapes of holes left in irregular close-packed arrangements of spheres, so he used a restrictive definition of deltahedra, in which a deltahedron is a convex polyhedron with triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres. This restrictive definition disallows the

triangular bipyramid A triangular bipyramid is a hexahedron with six triangular faces constructed by attaching two tetrahedra face-to-face. The same shape is also known as a triangular dipyramid or trigonal bipyramid. If these tetrahedra are regular, all faces of a t ...

(as forming two tetrahedral holes rather than a single hole),

(because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and

regular icosahedron The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting ...

(because it has interior room for another sphere). Bernal writes that the snub disphenoid is "a very common

coordination Coordination may refer to: * Coordination (linguistics), a compound grammatical construction * Coordination complex, consisting of a central atom or ion and a surrounding array of bound molecules or ions ** A chemical reaction to form a coordinati ...

for the

calcium ion Calcium is a chemical element; it has symbol Ca and atomic number 20. As an alkaline earth metal, calcium is a reactive metal that forms a dark oxide-nitride layer when exposed to air. Its physical and chemical properties are most similar to it ...

crystallography Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...

". In coordination geometry, it is usually known as the trigonal dodecahedron or simply as the dodecahedron.

References

External links

* {{Johnson solids navigator Johnson solids Deltahedra