The rhombic hexecontahedron is a stellation of the rhombic triacontahedron. ![]() The great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron with 30 intersecting rhombic faces.The rhombic triacontahedron is a convex polyhedron with 30 golden rhombi (rhombi whose diagonals are in the golden ratio) as its faces.The rhombic dodecahedron is a convex polyhedron with 12 congruent rhombi as its faces.A rhombohedron (also called a rhombic hexahedron) is a three-dimensional figure like a cuboid (also called a rectangular parallelepiped), except that its 3 pairs of parallel faces are up to 3 types of rhombi instead of rectangles.Three-dimensional analogues of a rhombus include the bipyramid and the bicone as a surface of revolution.Ĭonvex polyhedra with rhombi include the infinite set of rhombic zonohedrons, which can be seen as projective envelopes of hypercubes. ![]() Identical rhombi can tile the 2D plane in three different ways, including, for the 60° rhombus, the rhombille tiling.One of the five 2D lattice types is the rhombic lattice, also called centered rectangular lattice.This is a special case of the superellipse, with exponent 1. Thus denoting the common side as a and the diagonals as p and q, in every rhombusĤ a 2 = p 2 + q 2. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel adjacent angles are supplementary the two diagonals bisect one another any line through the midpoint bisects the area and the sum of the squares of the sides equals the sum of the squares of the diagonals (the parallelogram law). The first property implies that every rhombus is a parallelogram. The two diagonals of a rhombus are perpendicular that is, a rhombus is an orthodiagonal quadrilateral.Opposite angles of a rhombus have equal measure.It follows that any rhombus has the following properties: Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point Įvery rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides.a quadrilateral ABCD possessing a point P in its plane such that the four triangles ABP, BCP, CDP, and DAP are all congruent.a quadrilateral in which each diagonal bisects two opposite interior angles.a quadrilateral in which the diagonals are perpendicular and bisect each other.a quadrilateral with four sides of equal length (by definition).a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram).a parallelogram in which at least two consecutive sides are equal in length.a parallelogram in which a diagonal bisects an interior angle.The surface we refer to as rhombus today is a cross section of the bicone on a plane through the apexes of the two cones.Ī simple (non- self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following: The word "rhombus" comes from Ancient Greek: ῥόμβος, romanized: rhombos, meaning something that spins, which derives from the verb ῥέμβω, romanized: rhémbō, meaning "to turn round and round." The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for a bicone, two right circular cones sharing a common base. 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, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after the French sweet – also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.Įvery rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. ![]() Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. The rhombus has a square as a special case, and is a special case of a kite and parallelogram.
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