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In geometry, the Bilinski dodecahedron is a
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 with twelve
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
golden rhombus In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio: : = \varphi = \approx 1.618~034 Equivalently, it is the Varignon parallelogram formed from the edge midpoints of a golden rectangle. Rhombi with this shape f ...
faces. It has the same topology as the
face-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its Face (geometry), faces are the same. More specifically, all faces must be not ...
rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...
, but a different geometry. It is a
parallelohedron In geometry, a parallelohedron or Fedorov polyhedron is a convex polyhedron that can be Translation (geometry), translated without rotations to fill Euclidean space, producing a Honeycomb (geometry), honeycomb in which all copies of the polyhed ...
, a polyhedron that can tile space with translated copies of itself.


History

This shape appears in a 1752 book by John Lodge Cowley, labeled as the dodecarhombus. It is named after
Stanko Bilinski Stanko Bilinski (22 April 1909 in Našice – 6 April 1998 in Zagreb) was a Croatian mathematician and academician. He was a professor at the University of Zagreb and a fellow of the Croatian Academy of Sciences and Arts. In 1960, he discovered ...
, who rediscovered it in 1960. Bilinski himself called it the rhombic dodecahedron of the second kind.. Bilinski's discovery corrected a 75-year-old omission in Evgraf Fedorov's classification of convex polyhedra with congruent rhombic faces.


Definition and properties


Definition

The Bilinski dodecahedron is formed by gluing together twelve
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
golden rhombi. These are
rhombi 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 ...
whose diagonals are in the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
: :\varphi = \approx 1.618~034 . The graph of the resulting polyhedron is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the graph of the
rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...
, but the faces are oriented differently: one pair of opposite rhombi has their long and short diagonals reversed, relatively to the orientation of the corresponding rhombi in the rhombic dodecahedron.


Symmetry

Because of its reversal, the Bilinski dodecahedron has a lower order of symmetry; its symmetry group is that of a
rectangular cuboid A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped. Many writers just call these ...
: of order 8. This is a subgroup of
octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
; its elements are three 2-fold symmetry axes, three symmetry planes (which are also the axial planes of this solid), and a center of inversion symmetry. The rotation group of the Bilinski dodecahedron is of order 4.


Vertices

Like the rhombic dodecahedron, the Bilinski dodecahedron has eight vertices of degree 3 and six of degree 4. It has two apices on the vertical axis, and four vertices on each axial plane. But due to the reversal, its non-apical vertices form two squares (red and green) and one rectangle (blue), and its fourteen vertices in all are of four different kinds: *two degree-4 apices surrounded by four acute face angles (vertical-axis vertices, black in first figure); *four degree-4 vertices surrounded by three acute and one obtuse face angles (horizontal-axial-plane vertices, blue in first figure); *four degree-3 vertices surrounded by three obtuse face angles (one vertical-axial-plane vertices, red in first figure); *four degree-3 vertices surrounded by two obtuse and one acute face angles (other vertical-axial-plane vertices, green in first figure).


Faces

The supplementary internal angles of a golden rhombus are:. See in particular table 1, p. 188. *acute angle: ::\alpha = \arctan 2 \approx 63.434~949 ^ \circ ; *obtuse angle: ::\beta = \pi - \arctan 2 \approx 116.565~051 ^ \circ . The faces of the Bilinski dodecahedron are twelve congruent golden rhombi; but due to the reversal, they are of three different kinds: *eight apical faces with all four kinds of vertices, *two side faces with alternate blue and red vertices (front and back in first figure), *two side faces with alternating blue and green vertices (left and right in first figure). (See also the figure with edges and front faces colored.)


Edges

The 24 edges of the Bilinski dodecahedron have the same length; but due to the reversal, they are of four different kinds: *four apical edges with black and red vertices (in first figure), *four apical edges with black and green vertices (in first figure), *eight side edges with blue and red vertices (in first figure), *eight side edges with blue and green vertices (in first figure). (See also the figure with edges and front faces colored.)


Cartesian coordinates

The vertices of a Bilinski dodecahedron with thickness 2 has the following
Cartesian coordinates 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 ...
, where is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
:


In families of polyhedra

The Bilinski dodecahedron is a
parallelohedron In geometry, a parallelohedron or Fedorov polyhedron is a convex polyhedron that can be Translation (geometry), translated without rotations to fill Euclidean space, producing a Honeycomb (geometry), honeycomb in which all copies of the polyhed ...
; thus it is also 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 ...
, and a
zonohedron In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...
.


Relation to rhombic dodecahedron

In a 1962 paper, H. S. M. Coxeter claimed that the Bilinski dodecahedron could be obtained by an
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...
from the rhombic dodecahedron, but this is false. In the rhombic dodecahedron: every long body diagonal (i.e. lying on opposite degree-4 vertices) is parallel to the short diagonals of four faces. In the Bilinski dodecahedron: the longest body diagonal (i.e. lying on opposite black degree-4 vertices) is parallel to the short diagonals of two faces, and to the long diagonals of two other faces; the shorter body diagonals (i.e. lying on opposite blue degree-4 vertices) are not parallel to the diagonal of any face. In any affine transformation of the rhombic dodecahedron: every long body diagonal (i.e. lying on opposite degree-4 vertices) remains parallel to four face diagonals, and these remain of the same (new) length.


Zonohedra with golden rhombic faces

The Bilinski dodecahedron can be formed from the
rhombic triacontahedron The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombus, rhombic face (geometry), faces. It has 60 edge (geometry), edges and 32 vertex ...
(another zonohedron, with thirty
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
golden rhombic faces) by removing or collapsing two zones or belts of ten and eight golden rhombic faces with parallel edges. Removing only one zone of ten faces produces the
rhombic icosahedron The rhombic icosahedron is a polyhedron shaped like an Oblate spheroid, oblate sphere. Its 20 faces are Congruence (geometry), congruent golden rhombi; 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of ...
. Removing three zones of ten, eight, and six faces produces a golden rhombohedron.. Thus removing a zone of six faces from the Bilinski dodecahedron produces a golden rhombohedron. The Bilinski dodecahedron can be dissected into four golden rhombohedra, two of each type. The vertices of the zonohedra with golden rhombic faces can be computed by linear combinations of two to six generating edge vectors with coefficients 0 or 1.Let ''V'' denote the number of vertices and ''ek'' denote the ''k''-th generating edge vector, where 1 ≤ ''k'' ≤ ''n'';
for 2 ≤ ''n'' ≤ 3, ''V'' = card(𝒫 ) = 2''n'';
for 4 ≤ ''n'' ≤ 6, ''V'' < 2''n'', because some of the linear combinations of four to six generating edge vectors with coefficients 0 or 1 end strictly inside the golden rhombic zonohedron.
A belt means a belt representing directional vectors, and containing coparallel edges with same length. The Bilinski dodecahedron has four belts of six coparallel edges. These zonohedra are projection envelopes of the
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
s, with -dimensional projection basis, with
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
(). For the specific basis is: : : : For the basis is the same with the sixth column removed. For the fifth and sixth columns are removed.


References


External links

* VRML model, George W. Hart: * animation and coordinates, David I. McCooey: {{URL, http://dmccooey.com/polyhedra/BilinskiDodecahedron.html
''A new Rhombic Dodecahedron from Croatia!''
YouTube video by
Matt Parker Matthew Thomas Parker (born 22 December 1980) is an Australian recreational mathematics, recreational mathematician, author, comedian, YouTube personality and Science communication, science communicator based in the United Kingdom. His book ''H ...
Zonohedra Golden ratio