POLYHEDRONS AND THE FIVE PLATONIC SOLIDS
The Platonic solids, also called the regular solids or regular polyhedra, are composed of congruent convex regular polyhedrons.
They were proved by Euclid in the last proposition of the Elements, to be: the Cube, Dodecahedron, Icosahedron, Octahedron and Tetrahedron. (shown in this order in the Bryce picture).
The Platonic solids were known to the ancient Greeks, and were described by Plato in his Timaeus ca. 350 BC. In this work, Plato corresponded the tetrahedron with the "element" fire, the cube with earth, the icosahedron with water, the octahedron with air, and the dodecahedron with the ether.
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The Golden Triangle
The golden triangle can be characterized as an isosceles triangle ABC with the property that bisecting the angle C produces a new triangle CXB which is a similar triangle to the original.
If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°-36°-108°.
Suppose XB has length 1, and we call BC length φ. Because of the isosceles triangles BC=XC and XC=XA, so these are also length φ. Length AC = AB, therefore equals φ+1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, and so AC also equals φ2. Thus φ2 = φ+1, confirming that φ is indeed the golden ratio.
Pentagram
The golden ratio plays an important role in regular pentagons and pentagrams. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the 2 intersecting edges (a side of the pentagon in the pentagram's center) is φ, as the four-color illustration shows.
The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomon.
Ptolemy's Theorem
The golden ratio can also be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one vertex from a regular pentagon. If the quadrilateral's long edge and diagonals are b, and short edges are a, then Ptolemy's theorem gives b2 = a2 + ab which yields
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Multiplication sign |
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degree |
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pi symbol (3.14159...) |
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