Phi and Geometry

Phi is one of the two great treasures of geometry

Phi or Φ, 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. (See also the Orthogons page.) Phi also defines the dimensions of a pentagon.

Pyramid based on phi

Forming a golden rectangle

Construction of 1/Φ (phi)

The pentagon is based on phi

Phi can be constructed from concentric circles whose sizes are in a ratio of 1 : 2 : 4. Draw a tangent from the small circle through the other two and the ratio of AC to AB is Φ, as AB = 2 * 3^½ and AC = 15^½ + 3^½, which by factoring out the 3^½can be reduced to a ratio of 2 to (5^½+1), or Φ.

Note: The construction of phi with concentric circles was developed by Sam Kutler and submitted by Steve Lautizar.

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:

Dodecahedron / Icosahedron from 3 golden rectangles

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

dodecahedron

Icosahedron

icosahedron

Solid	Dodecahedron	Icosahedron
Face shape	Pentagon	Triangle
Faces	12	20
Points	20	12
Edges	30	30

Learn more about phi and geometry on the Penrose Tiling and Quasi-crystals pages.