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Phi and Geometry
Phi is one of the two great treasures of geometry
Phi or
 , which is 1.618 0339 887 ..., was described by Johannes Kepler as one of the "two great
treasures of geometry." (The other is the Theorem of Pythagoras.)
Phi appears in many basic geometric constructions:
| Shape |
Geometry |
Construction |
| 3 lines |
 |
Take 3 equal lines. Lay the 2nd line against
the midpoint of the 1st.
Lay the 3rd line against
the midpoint of the 2nd.
The ratio of AG to AB is Phi.
Contributed by
Jo Niemeyer
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3 sides:
Triangle |
 |
Insert an equilateral triangle
inside a circle.Find the midpoints
of the two sides at A and B.
Extend the line to the circle.
The ratio of AG to AB is Phi. |
4 sides:
Square |
 |
Insert a square
inside a semi-circle.The ratio of AG to AB is Phi. |
5 sides:
Pentagon |
 |
Insert a pentagon
inside a circle.Connect three
of the five points
to cut one line
into three sections.
The ratio of AG to AB is Phi. |
When the basic phi relationships are used to create a right triangle, it forms the dimensions of the great pyramids of Egypt,
with the geometry shown below creating an angle of 51.83 degrees, the
cosine of which is phi, or 0.618. A ruler and
compass can be used to construct the "golden rectangle," as shown
by the animations below, which was used by the Greeks in the
Parthenon. (See also the
Orthogons page.) Phi also defines other dimensions of a pentagon.
There are also a number of geometric constructions using a circle which
produce phi relationships, as shown on the Geometric
Construction of Phi in Circles page.
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:
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This
is the basis for two geometric solids
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| 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
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Icosahedron
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| 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.
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Investors:
Apply
Phi and
Fibonacci
principles
to the
stock market |
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