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# 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.

### 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 Quasi-crystals pages.

Phi - The Golden Number
A source to some of Net's "phi-nest" information on the
Golden Section / Mean / Proportion / Ratio / Number,
Divine Proportion, Fibonacci Series and Phi (1.6180339887...)