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 ten-of-diamonds decahedron is a

space-filling polyhedron In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where ''filling'' means that; taken together, all the instances of the polyhedron c ...

with 10 faces, 2 opposite rhombi with orthogonal major axes, connected by 8 identical

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

faces. Although it is convex, it is not a

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 ...

because its faces are not composed entirely of regular polygons. Michael Goldberg named it after a

playing card A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a f ...

, as a 10-faced polyhedron with two opposite rhombic (diamond-shaped) faces. He catalogued it in a 1982 paper as 10-II, the second in a list of 26 known space-filling decahedra.

Coordinates

If the space-filling polyhedron is placed in a 3-D coordinate grid, the coordinates for the 8 vertices can be given as: (0, ±2, −1), (±2, 0, 1), (±1, 0, −1), (0, ±1, 1). : Ten-of-diamonds_decahedron_in_cube

Symmetry

The ''ten-of-diamonds'' has D_2d symmetry, which projects as order-4 dihedral (square) symmetry in two dimensions. It can be seen as a

triakis tetrahedron In geometry, a triakis tetrahedron (or tristetrahedron, or kistetrahedron) is a solid constructed by attaching four triangular pyramids onto the triangular faces of a regular tetrahedron, a Kleetope of a tetrahedron. This replaces the equilateral ...

, with two pairs of coplanar triangles merged into rhombic faces. The dual is similar to a

truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncation (geometry), truncating all 4 vertices of ...

, except two edges from the original tetrahedron are reduced to zero length making pentagonal faces. The dual polyhedra can be called a skew-truncated tetragonal disphenoid, where 2 edges along the symmetry axis completely truncated down to the edge midpoint.

Honeycomb

The ''ten-of-diamonds'' is used in the honeycomb with

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

, being the dual of an alternated bitruncated cubic honeycomb, . Since the ''alternated bitruncated cubic honeycomb'' fills space by pyritohedral icosahedra, , and

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 ...

al tetrahedra,

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

s of this honeycomb are their duals – pyritohedra, and

s. Cells can be seen as the cells of the

tetragonal disphenoid honeycomb The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Horton ...

, , with alternate cells removed and augmented into neighboring cells by a center vertex. The rhombic faces in the honeycomb are aligned along 3 orthogonal planes.

Related space-filling polyhedra

The ''ten-of-diamonds'' can be dissected in an

octagon In geometry, an octagon () is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a ...

al cross-section between the two rhombic faces. It is a decahedron with 12 vertices, 20 edges, and 10 faces (4

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

s, 4

trapezoid In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are ...

s, 1

rhombus In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...

, and 1 isotoxal octagon). Michael Goldberg labels this polyhedron 10-XXV, the 25th in a list of space-filling decahedra. The ''ten-of-diamonds'' can be dissected as a half-model on a symmetry plane into a space-filling heptahedron with 6 vertices, 11 edges, and 7 faces (6 triangles and 1 trapezoid). Michael Goldberg identifies this polyhedron as a ''triply truncated quadrilateral prism'', type 7-XXIV, the 24th in a list of space-fillering heptahedra. It can be further dissected as a quarter-model by another symmetry plane into a space-filling

hexahedron A hexahedron (: hexahedra or hexahedrons) or sexahedron (: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex. There are seven ...

with 6 vertices, 10 edges, and 6 faces (4 triangles, 2 right trapezoids). Michael Goldberg identifies this polyhedron as an ''ungulated quadrilateral pyramid'', type 6-X, the 10th in a list of space-filling hexahedron.

Rhombic bowtie

Pairs of ''ten-of-diamonds'' can be attached as a nonconvex ''bow-tie'' space-filler, called a rhombic bowtie for its cross-sectional appearance. The two right-most symmetric projections below show the rhombi edge-on on the top, bottom and a middle ''neck'' where the two halves are connected. The 2D projections can look convex or concave. It has 12 vertices, 28 edges, and 18 faces (16 triangles and 2 rhombi) within D_2h symmetry. These paired-cells stack more easily as inter-locking elements. Long sequences of these can be stacked together in 3 axes to fill space.Robert Reid, Anthony Stee
Bowties: A Novel Class of Space Filling Polyhedron
2003 The 12 vertex coordinates in a 2-

unit cube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.. See in particulap. 671. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.. Unit hypercube The term '' ...

. (further augmentations on the rhombi can be done with 2 unit translation in ''z''.) :(0, ±1, −1), (±1, 0, 0), (0, ±1, 1), :(±1/2, 0, −1), (0, ±1/2, 0), (±1/2, 0, 1)

References

{{reflist * Koch 1972 Koch, Elke, Wirkungsbereichspolyeder und Wirkungsbereichsteilunger zukubischen Gitterkomplexen mit weniger als drei Freiheitsgraden (Efficiency Polyhedra, and Efficiency Dividers, cubic lattice complexes with less than three degrees of freedom) Dissertation, University Marburg/Lahn 1972 - Model 10/8–1, 28–404. Space-filling polyhedra