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The golden ratio φ and its negative reciprocal −φ −1 are the two roots of the quadratic polynomial x 2 − x − 1. The golden ratio's negative −φ and reciprocal φ −1 are the two roots of the quadratic polynomial x 2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer.
Let φ be the golden ratio.The 12 points given by (0, ±1, ±φ) and cyclic permutations of these coordinates are the vertices of a regular icosahedron.Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points (±1, ±1, ±1) together with the 12 points (0, ±φ, ± 1 / φ ) and cyclic permutations of these coordinates.
The ratio between the lengths of the long and short diagonal of the rhombs equals the golden ratio . Convex, medial and great rhombic triacontahedron on the right (shown with pyritohedral symmetry ) and the corresponding dual compounds of regular solids on the left
Each face has two angles of (+) and two angles of (+).The diagonals of each antiparallelogram intersect at an angle of ().The dihedral angle equals (+).The ratio between the lengths of the long edges and the short ones equals +, which is the golden ratio.
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 ...
Denoting the golden ratio by and putting = + +, the stars have five equal angles of and one of (). Each face has four long and two short edges. Each face has four long and two short edges. The ratio between the edge lengths is
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Denoting the golden ratio by and putting = +, the hexagons have five equal angles of and one of (). Each face has four long and two short edges. Each face has four long and two short edges. The ratio between the edge lengths is