Golden rhombus

Last updated May 31, 2022

In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio:^[1]

Equivalently, it is the Varignon parallelogram formed from the edge midpoints of a golden rectangle.^[1] Rhombi with this shape form the faces of several notable polyhedra. The golden rhombus should be distinguished from the two rhombi of the Penrose tiling, which are both related in other ways to the golden ratio but have different shapes than the golden rhombus.^[2]

Angles

(See the characterizations and the basic properties of the general rhombus for angle properties.)

The internal supplementary angles of the golden rhombus are:^[3]

Acute angle: $\alpha =2\arctan {1 \over \varphi }$ ;

by using the arctangent addition formula (see inverse trigonometric functions):

\alpha =\arctan {{2 \over \varphi } \over {1-({1 \over \varphi })^{2}}}=\arctan {{2 \over \varphi } \over {1 \over \varphi }}=\arctan 2\approx 63.43495^{\circ }.

Obtuse angle: $\beta =2\arctan \varphi =\pi -\arctan 2\approx 116.56505^{\circ },$

which is also the dihedral angle of the dodecahedron.^[4]

Note: an "anecdotal" equality:

\pi -\arctan 2=\arctan 1+\arctan 3~.

Edge and diagonals

By using the parallelogram law (see the basic properties of the general rhombus):^[5]

The edge length of the golden rhombus in terms of the diagonal length $d$ is:

$a={1 \over 2}{\sqrt {d^{2}+(\varphi d)^{2}}}={1 \over 2}{\sqrt {1+\varphi ^{2}}}~d={{\sqrt {2+\varphi }} \over 2}~d={1 \over 4}{\sqrt {10+2{\sqrt {5}}}}~d\approx 0.95106~d~.~$ Hence:

The diagonal lengths of the golden rhombus in terms of the edge length $a$ are:^[3]

$d={2a \over {\sqrt {2+\varphi }}}=2{\sqrt {{3-\varphi } \over 5}}~a={\sqrt {2-{2 \over {\sqrt {5}}}}}~a\approx 1.05146~a~.$

$D={2\varphi a \over {\sqrt {2+\varphi }}}=2{\sqrt {{2+\varphi } \over 5}}~a={\sqrt {2+{2 \over {\sqrt {5}}}}}~a\approx 1.70130~a~.$

Area

By using the area formula of the general rhombus in terms of its diagonal lengths $D$ and $d$ :

The area of the golden rhombus in terms of its diagonal length

d

is:^[6]

A={{(\varphi d)\cdot d} \over 2}={{\varphi } \over 2}~d^{2}={{1+{\sqrt {5}}} \over 4}~d^{2}\approx 0.80902~d^{2}~.

By using the area formula of the general rhombus in terms of its edge length $a$ :

The area of the golden rhombus in terms of its edge length

a

is:^[3]^[6]

A=(\sin(\arctan 2))~a^{2}={2 \over {\sqrt {5}}}~a^{2}\approx 0.89443~a^{2}~.

Note: $\alpha +\beta =\pi$ , hence: $\sin \alpha =\sin \beta ~.$

As the faces of polyhedra

Several notable polyhedra have golden rhombi as their faces. They include the two golden rhombohedra (with six faces each), the Bilinski dodecahedron (with 12 faces), the rhombic icosahedron (with 20 faces), the rhombic triacontahedron (with 30 faces), and the nonconvex rhombic hexecontahedron (with 60 faces). The first five of these are the only convex polyhedra with golden rhomb faces, but there exist infinitely many nonconvex polyhedra having this shape for all of their faces.^[7]

Related Research Articles

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities $and with$

Regular icosahedron One of the five Platonic solids

In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

In geometry, an icosidodecahedron is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

Rhombus Quadrilateral in which all sides have the same length

In plane Euclidean geometry, a rhombus 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 rhombus is often called a 'diamond', after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge (shape), though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

Rhombicosidodecahedron Archimedean solid

In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865.

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

Disdyakis triacontahedron Catalan solid with 120 faces

In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It also has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.

A rhombic enneacontahedron is a polyhedron composed of 90 rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron.

Rhombohedron Polyhedron with six rhombi as faces

In geometry, a rhombohedron is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A cube is a special case of a rhombohedron with all sides square.

In geometry, a trigonal trapezohedron is a rhombohedron in which, additionally, all six faces are congruent. Alternative names for the same shape are the trigonal deltohedron or isohedral rhombohedron. Some sources just call them rhombohedra.

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.

In geometry, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.

In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices.

In geometry, the great rhombihexacron (or great dipteral disdodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U₂₁). It has 24 identical bow-tie-shaped faces, 18 vertices, and 48 edges.

In geometry, the truncated triangular trapezohedron is the first in an infinite series of truncated trapezohedron polyhedra. It has 6 pentagon and 2 triangle faces.

In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. It was described mathematically in 1940 by Helmut Unkelbach.

In geometry, the Bilinski dodecahedron is a convex polyhedron with twelve congruent golden rhombic faces. It has the same topology but different geometry from the face-transitive rhombic dodecahedron. It is a zonohedron.

References

1 2 Senechal, Marjorie (2006), "Donald and the golden rhombohedra", in Davis, Chandler; Ellers, Erich W. (eds.), The Coxeter Legacy, American Mathematical Society, Providence, RI, pp. 159–177, ISBN 0-8218-3722-2, MR 2209027
↑ For instance, an incorrect identification between the golden rhombus and one of the Penrose rhombi can be found in Livio, Mario (2002), The Golden Ratio: The Story of Phi, the World's Most Astonishing Number, New York: Broadway Books, p. 206
1 2 3 Ogawa, Tohru (January 1987), "Symmetry of three-dimensional quasicrystals", Materials Science Forum, 22–24: 187–200, doi:10.4028/www.scientific.net/msf.22-24.187 . See in particular table 1, p. 188.
↑ Gevay, G. (June 1993), "Non-metallic quasicrystals: Hypothesis or reality?", Phase Transitions, 44 (1–3): 47–50, doi:10.1080/01411599308210255
↑ Weisstein, Eric W. "Rhombus". MathWorld .
1 2 Weisstein, Eric W. "Golden Rhombus". MathWorld .
↑ Grünbaum, Branko (2010), "The Bilinski dodecahedron and assorted parallelohedra, zonohedra, monohedra, isozonohedra, and otherhedra" (PDF), The Mathematical Intelligencer , 32 (4): 5–15, doi:10.1007/s00283-010-9138-7, hdl: 1773/15593 , MR 2747698, archived from the original (PDF) on 2015-04-02.

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[senechal-1] 1 2 Senechal, Marjorie (2006), "Donald and the golden rhombohedra", in Davis, Chandler; Ellers, Erich W. (eds.), The Coxeter Legacy, American Mathematical Society, Providence, RI, pp. 159–177, ISBN 0-8218-3722-2, MR 2209027

[2] For instance, an incorrect identification between the golden rhombus and one of the Penrose rhombi can be found in Livio, Mario (2002), The Golden Ratio: The Story of Phi, the World's Most Astonishing Number, New York: Broadway Books, p. 206

[ogawa-3] 1 2 3 Ogawa, Tohru (January 1987), "Symmetry of three-dimensional quasicrystals", Materials Science Forum, 22–24: 187–200, doi:10.4028/www.scientific.net/msf.22-24.187 . See in particular table 1, p. 188.

[4] Gevay, G. (June 1993), "Non-metallic quasicrystals: Hypothesis or reality?", Phase Transitions, 44 (1–3): 47–50, doi:10.1080/01411599308210255

[5] Weisstein, Eric W. "Rhombus". MathWorld .

[Mathworld-6] 1 2 Weisstein, Eric W. "Golden Rhombus". MathWorld .

[7] Grünbaum, Branko (2010), "The Bilinski dodecahedron and assorted parallelohedra, zonohedra, monohedra, isozonohedra, and otherhedra" (PDF), The Mathematical Intelligencer , 32 (4): 5–15, doi:10.1007/s00283-010-9138-7, hdl: 1773/15593 , MR 2747698, archived from the original (PDF) on 2015-04-02.

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