The Pentagon & Pentagram: 5, Phi, Harmony


Locked within the pentagon—the 5-sided regular polygon—lies its 5-pointed partner, the star-shaped pentagram familiar to Wiccans and pagans of many related paths as their principal symbol. From their proportions arises a number that rivals pi in its importance in sacred geometry: the constant known as phi (about 1.6180339887), represented by the Greek letter:


–and often called the golden ratio or golden section.

Why is this number so special? Let’s say you split something into a small part and a big part; if the relative size of the small part to the big part is the same as the relative size of the big part to the whole you started with, you’ll find that size ratio is equal to phi. Always. [I]Phi is the only number with which this is possible.[/I]

The pentagram, riddled through and through with golden ratios, provides a great example. In the pentagram below, examine the relative sizes of the colored lines: The purple line (A; the small part) is to the blue line (B; the big part) as B is to A+B (or the green line, C; the whole).


Written mathematically:

A:B = B:(A+B) = phi

And the same proportion continues in the relationship of B (in this case, the small part), which is to C (the big part) as C is to B+C (or the red line, D; the whole).

Because phi divides the whole while maintaining the original proportions, it can represent the evolution of multiple forms from a single point of origin. And, you guessed it: phi is an irrational number.

Here’s another way to look at it. Make a series of numbers by adding the previous two numbers to get the next: 0, 1, 1, 2, 3, 5, 8, 13, 21,… This is called a Fibonacci series, and the ratio between any two consecutive numbers gets closer and closer to phi as the numbers get bigger, without ever actually reaching phi.

As well as emerging from a Fibonacci series, phi has been applied to rectangles and spirals (referred to as “golden” in such cases).

The fact that, with phi, the relative proportions are always the same—no matter how large or small the segments being compared—symbolizes how the patterns we perceive in nature repeat, whether on a large scale or a small scale.


Phi has undergone a bit of a renaissance lately, entering the popular consciousness through a flurry of books devoted to it. This popularity has come with a price: aggrandized claims for phi, claims that ultimately may be of dubious merit. For example, there is the oft repeated assertion that anything conforming to golden proportions is innately more pleasing to the human eye. Repetition, however, is not evidence, and to date no scientific study has ever demonstrated this alleged innate esthetic bias.

How often phi pops up in the universe could be the subject of endless debate. In living things that grow by the addition of new material to an existing core, phi may emerge if the new growth is going to maintain the existing proportions. Many find the golden proportion in the shape of a nautilus shell:


This Wikipedia and Wikimedia Commons image is from the user Chris 73 and is freely available at https://commons.wikimedia.org/wiki/File:NautilusCutawayLogarithmicSpiral.jpg under the creative commons cc-by-sa 3.0 license.

Unfortunately, it’s not there, as is easily seen by superimposing a golden spiral over it. The border of the nautilus curl quickly departs from the red line:


This nautilus is a logarithmic spiral, but not a golden spiral (whose proportions would be governed by phi). All golden spirals are logarithmic spirals, which is why the nautilus looks so similar; but not all logarithmic spirals are golden.

Some find Fibonacci numbers in the planets; in examples of art and architecture from all ages, some claim phi is incorporated in the underlying design (the Parthenon, the Great Pyramid of Giza). Even when such designs aren’t deliberate, some say phi is applied subconsciously, falling back on the claim that golden proportions are the most pleasing to the human eye. Whether these claims are true or not, the mathematical properties of phi make it of great symbolic value.

The 3-dimensional analogue of the pentagon is the dodecahedron (12 sides consisting of pentagons).


Next–Sacred Geometry in 3D
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  • geometry/pentagram.txt
  • Last modified: 2019/08/05 11:51
  • by RandallS