Rhombus

Rhombus
(OFF file)
Rank	2
Type	Isotopic
Space	Spherical
Notation
Bowers style acronym	Rhomb
Coxeter diagram	m2m
Elements
Edges	4
Vertices	2+2
Vertex figure	Dyad
Measures (edge length 1, angle α)
Inradius	$\frac{\sin\alpha}{2}$
Area	$\sin\alpha$
Angles	Acute: $\alpha$
	Obtuse: $\pi-\alpha$
Height	$\sin\alpha$
Central density	1
Related polytopes
Army	Rhomb
Regiment	Rhomb
Dual	Rectangle
Conjugate	None
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	K2, order 4
Convex	Yes
Nature	Tame

The rhombus, or rhomb, is a quadrilateral with all four edges of the same length. It has two different angles, and its diagonals are always at right angles. It is a special case of a parallelogram.

The two angles of a rhombus add up to 180°, and one is always acute, the other is obtuse. Rhombi occur as faces in two of the Catalan solids, namely the rhombic dodecahedron and rhombic triacontahedron.

A rhombus can be considered to be the tegum product of two dyads of different lengths. These two dyads then form the two diagonals of the rhombus.

Golden rhombus

The golden rhombus is a rhombus whose diagonals have the golden ratio. It appears as a face of the golden isozonohedra as well as other polyhedra such as the rhombic hexecontahedron.

Vertex coordinates

The coordinates of a golden rhombus centered at the origin with side lengths equal to 1:

• $\left(\pm\frac{1}{2}\sqrt{\frac{8}{5+\sqrt{5}}},0\right)$,
• $\left(0,\pm\frac{1}{2}\sqrt{2+\frac{2}{\sqrt{5}}}\right)$.

External links

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".