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Sunday, August 2, 2015

Golden Ratio ($\varphi$) Fractally embedded in Nature - Platonic Solids


From wikipedia page on Platonic solid:

Cartesian coordinates[edit]

For Platonic solids centered at the origin, simple Cartesian coordinates are given below. The Greek letter φ is used to represent the golden ratio \tfrac{1+\sqrt{5}}{2}.













Cartesian coordinates
FigureTetrahedronOctahedronCubeIcosahedronDodecahedron
Vertices46 (2×3)812 (4×3)20 (8+4×3)
Orientation
set
121212
Coordinates(1,1,1)
(1,−1,−1)
(−1,1,−1)
(−1,−1,1)
(-1,-1,-1)
(-1,1,1)
(1,-1,1)
(1,1,-1)

(±1, 0, 0)
(0, ±1, 0)
(0, 0, ±1)
(±1, ±1, ±1)
(0, ±1, ±φ)
(±1, ±φ, 0)
(±φ, 0, ±1)

(0, ±φ, ±1)
(±φ, ±1, 0)
(±1, 0, ±φ)
(±1, ±1, ±1)
(0, ±1/φ, ±φ)
(±1/φ, ±φ, 0)
φ, 0, ±1/φ)
(±1, ±1, ±1)
(0, ±φ, ±1/φ)
φ, ±1/φ, 0)
(±1/φ, 0, ±φ)
ImageCubeAndStel.svgDual Cube-Octahedron.svgIcosahedron-golden-rectangles.svgCube in dodecahedron.png
Note significantly $\varphi$,  the golden ratio \tfrac{1+\sqrt{5}}{2}.,  is all over the Platonic solids.

Considering that $\varphi$, the Golden ratio has been shown by many to be fractally embedded in all of Nature, Dan Winter's work is significanlty taking off where the ancients left off and giving meaning to:

...and more.
https://en.wikipedia.org/wiki/Toroidal_polyhedron <-- interesting as well!
https://en.wikipedia.org/wiki/Spherical_polyhedron

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