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  2. 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.

  3. Golden rhombus - Wikipedia

    en.wikipedia.org/wiki/Golden_rhombus

    By using the area formula of the general rhombus in terms of its diagonal lengths and : The area of the golden rhombus in terms of its diagonal length d {\displaystyle d} is: [ 6 ] A = ( φ d ) ⋅ d 2 = φ 2 d 2 = 1 + 5 4 d 2 ≈ 0.80902 d 2 . {\displaystyle A={{(\varphi d)\cdot d} \over 2}={{\varphi } \over 2}~d^{2}={{1+{\sqrt {5}}} \over 4 ...

  4. Rhombus - Wikipedia

    en.wikipedia.org/wiki/Rhombus

    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).

  5. 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 ...

  6. Bellman's lost-in-a-forest problem - Wikipedia

    en.wikipedia.org/wiki/Bellman's_lost-in-a-forest...

    In particular, all shapes which can enclose a 60° rhombus with longer diagonal equal to the diameter have a solution of a straight line. The equilateral triangle is the only regular polygon which does not have this property, and has a solution consisting of a zig-zag line with three segments of equal length.

  7. Varignon's theorem - Wikipedia

    en.wikipedia.org/wiki/Varignon's_theorem

    An arbitrary quadrilateral and its diagonals. Bases of similar triangles are parallel to the blue diagonal. Ditto for the red diagonal. The base pairs form a parallelogram with half the area of the quadrilateral, A q, as the sum of the areas of the four large triangles, A l is 2 A q (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, A s is a quarter of A ...

  8. Dynamic rectangle - Wikipedia

    en.wikipedia.org/wiki/Dynamic_rectangle

    The root-3 rectangle is also called sixton, [6] and its short and longer sides are proportionally equivalent to the side and diameter of a hexagon. [7] Since 2 is the square root of 4, the root-4 rectangle has a proportion 1:2, which means that it is equivalent to two squares side-by-side. [7] The root-5 rectangle is related to the golden ratio ...

  9. Rhombic dodecahedron - Wikipedia

    en.wikipedia.org/wiki/Rhombic_dodecahedron

    The rhombic dodecahedron is a polyhedron with twelve rhombuses, each of which long face-diagonal length is exactly times the short face-diagonal length [1] and the acute angle measurement is ⁡ (/). Its dihedral angle between two rhombi is 120°. [2]