Phi and Geometry
Phi is one of the two great treasures of geometry
Phi or Ψ, was described by Johannes Kepler as one of the "two great
treasures of geometry." (The other is the Theorem of Pythagoras.)
In a triangle it forms the dimensions of the great pyramids of Egypt. A ruler and
compass can be used to form the "golden rectangle" used by the Greeks in the
Parthenon. It also defines the dimensions of a pentagon.
Phi can be related to Pi through trigonometric functions
Note: Above formulas expressed in radians, not
degrees
Phi appears in 3D geometric solids
Take three golden rectangles and assemble them at 90 degree
angles to get a 3D shape with 12 corners: 
Click on the shape below and the print the page to do it yourself:



This
is the basis for two geometric solids

The 12 corners become the 12 centers of
each of the 12 pentagons that form the faces of a dodecahedron. 
The 12 corners can also become the 12
points of each of the 20 triangles that form the faces of a icosahedron. 
Dodecahedron

Icosahedron

Solid 
Dodecahedron 
Icosahedron 
Face shape 
Pentagon 
Triangle 
Faces 
12 
20 
Points 
20 
12 
Edges 
20 
30 
Learn more about phi and geometry on the Penrose
Tiling and Quasicrystals pages.

