geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a golden rhombus is a

rhombus In plane Euclidean geometry, a rhombus (plural 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 ...

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 sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

: :

= \varphi =  \approx 1.618~034

Equivalently, it is the

Varignon parallelogram Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof wa ...

formed from the edge midpoints of a

golden rectangle In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618. Golden rectangles exhibit a special form of self-similarity ...

. 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 A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any finite distance, without ...

, which are both related in other ways to the golden ratio but have different shapes than the golden rhombus.

Angles

(See the

characterizations Characterization or characterisation is the representation of persons (or other beings or creatures) in narrative and dramatic works. The term character development is sometimes used as a synonym. This representation may include direct methods ...

and the basic properties of the general

for angle properties.) The internal supplementary angles of the golden rhombus are:. See in particular table 1, p. 188. *Acute angle:

\alpha=2\arctan

; :by using the arctangent addition formula (see

inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...

): :

\alpha=\arctan=\arctan=\arctan2\approx63.43495^\circ.

: *Obtuse angle:

\beta=2\arctan\varphi=\pi-\arctan2\approx116.56505^\circ,

:which is also the

dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...

of the

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

. :Note: an "anecdotal" equality:

\pi-
\arctan2=\arctan1+
\arctan3~.

Edge and diagonals

By using the

parallelogram law In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...

(see the basic properties of the general

): The edge length of the golden rhombus in terms of the diagonal length

d

is: *

a=\sqrt=\sqrt~d=~d=\sqrt~d\approx0.95106~d~.~

Hence: : The diagonal lengths of the golden rhombus in terms of the edge length

a

are: *

d2\sqrt~a=\sqrt~a\approx1.05146~a~.

: *

D2\sqrt~a=\sqrt~a\approx1.70130~a~.

Area

*By using the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

formula of the general

in terms of its diagonal lengths

D

and

d

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

d

is: :

A =  = ~d^2 = ~d^2 \approx 0.80902~d^2~.

*By using the

formula of the general

in terms of its edge length

a

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

a

is: :

A = (\sin(\arctan2))~a^2 = ~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 In geometry, the Bilinski dodecahedron is a Convex set, convex polyhedron with twelve Congruence (geometry), congruent golden rhombus faces. It has the same topology but a different geometry than the face-transitive rhombic dodecahedron. It is a ...

(with 12 faces), the

rhombic icosahedron The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi; 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of its 2 poles; these 2 vertices lie on it ...

(with 20 faces), the

rhombic triacontahedron 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 Ca ...

(with 30 faces), and the nonconvex

rhombic hexecontahedron 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. It is topologically ident ...

(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.. File:Acute_golden_rhombohedron.png, Acute golden rhombohedron File:Flat_golden_rhombohedron.png, Obtuse golden rhombohedron File:Bilinski dodecahedron.png,

File:Rhombic icosahedron.png,

File:Rhombictriacontahedron.svg,

File:Rhombic hexecontahedron.png,

References

{{Metallic ratios Types of quadrilaterals Golden ratio

Angles

Edge and diagonals

Area

As the faces of polyhedra

See also

References