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  2. Rhombus - Wikipedia

    en.wikipedia.org/wiki/Rhombus

    Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties: Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral. Its diagonals bisect opposite ...

  3. Rhombic triacontahedron - Wikipedia

    en.wikipedia.org/wiki/Rhombic_triacontahedron

    The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 arctan(⁠ 1 / φ ⁠) = arctan(2), or approximately 63.43°. A rhombus so obtained is called a golden rhombus.

  4. Rhombic dodecahedron - Wikipedia

    en.wikipedia.org/wiki/Rhombic_dodecahedron

    In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb.

  5. Rhombicosidodecahedron - Wikipedia

    en.wikipedia.org/wiki/Rhombicosidodecahedron

    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. It has a total of 62 faces: 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, with 60 vertices , and 120 edges .

  6. Rhombohedron - Wikipedia

    en.wikipedia.org/wiki/Rhombohedron

    In geometry, a rhombohedron (also called a rhombic hexahedron [1] [2] or, inaccurately, a rhomboid [a]) is a special case of a parallelepiped in which all six faces are congruent rhombi. [3] It can be used to define the rhombohedral lattice system , a honeycomb with rhombohedral cells.

  7. Golden ratio - Wikipedia

    en.wikipedia.org/wiki/Golden_ratio

    The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of ⁠ ⁠ and ⁠ ⁠, and a thick rhombus with angles of ⁠ ⁠ and ⁠ ⁠. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals ⁠ 1 : φ {\displaystyle 1\mathbin {:} \varphi } ⁠ , as does the ...

  8. Bilinski dodecahedron - Wikipedia

    en.wikipedia.org/wiki/Bilinski_dodecahedron

    The Bilinski dodecahedron is formed by gluing together twelve congruent golden rhombi.These are rhombi whose diagonals are in the golden ratio: = + The graph of the resulting polyhedron is isomorphic to the graph of the rhombic dodecahedron, but the faces are oriented differently: one pair of opposite rhombi has their long and short diagonals reversed, relatively to the orientation of the ...

  9. Diagonal - Wikipedia

    en.wikipedia.org/wiki/Diagonal

    The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length , while AC (shown in red) is a face diagonal and has length .. In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge.