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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 ...
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.
The golden rhombus. In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio: [1] = = + Equivalently, it is the Varignon parallelogram formed from the edge midpoints of a golden rectangle. [1]
All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals : , as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of the two rhombi are in the golden ratio to each other.
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.
For example, a circular disk of radius 1/2 can accommodate any plane curve of length 1 by placing the midpoint of the curve at the center of the disk. Another possible solution has the shape of a rhombus with vertex angles of 60° and 120° and with a long diagonal of unit length. [2]
From April 2012 to December 2012, if you bought shares in companies when William R. Loomis Jr. joined the board, and sold them when he left, you would have a 56.2 percent return on your investment, compared to a 2.8 percent return from the S&P 500.
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 ...