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Ancient geometry and the number 5:

The motion of matter through time is what proved to ancient peoples that something was alive. Motion was believed to have been given to nature by the "breath of God". This "divine breath" was called "aethyr" and was the fifth of the five elements. It was also called "quintessence", meaning "fifth essence"
The human body has five appendages (2 legs + 2 arms + head). We have 5 fingers and 5 toes. To ancient geometers, our 5-fold symmetry meant that the universe, too, had a 5-fold nature, since man was believed to be a microcosm of the universe.
The dodecahedron, comprised of 12 five-sided pentagons was the symbol for the whole of the cosmos.

Pentagons and Pentagrams exhibit self-symmetry and infinite fractal expansion. Fractal geometry is evident whenever a part of something resembles the whole. (ie. twigs look like branches, which look like trees, which look like large twigs, which look like branches etc....). Fractal geometry is found throughout plantlife and minerals. A seed of a plant contains enough data to construct an entirely new plant. And, in fact, a single strand of an animal's DNA contains enough data to construct the entire animal (this is why cloning is possible).
The fractal symmetry of the pentagon and pentagram is illustrated below:

The Golden Line:

If a part of a line is fractally related to the whole line, it is called a "golden line", and the cross which cuts the line into two fractally related sections is called the "golden cut". The line is seperated into two sections in such a way that (the whole line divided by the larger part) is equivalent to (the larger part divided by the smaller part). This ratio cuts the line into what has been called "the golden ratio". this golden ratio has been referred as "the section" by Plato, "extreme and mean ratio" by Euclid,and "the Divine Section" by Kepler. This ratio is infinite and non-repeating (like Pi). The golden ratio is written today as

, and is called "phi" (pronounced "fee").

phi = whole section/large section = large section/small section

"Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may compare to a precious jewel" - Johannes Kepler

"The carpenter stretcheth out his rule; he marketh it out with a line; he fitteth it with planes, and he marketh it out with a compass, and marketh it after the figure of a man according to the beauty of a man" - Isaiah 44:13

The Golden Rectangle:

A "golden rectangle" has one dimension that is in a ratio of phi to it's other dimension.

The golden rectangle was considered the most beautiful of all rectangles, because of it's fractal nature, and has therefore been used in paintings and architecture throughout history.

"The golden proportion is a scale of proportions which makes the bad difficult to produce and the good easy." - Albert Einstein

Fibonacci Sequence:

Leonardo de Pisa, filius Bonaccio (Leonard of Pisa, son of the man of good cheer), also known as "Leonardo Fibonacci" devised a sequence in which each subsequent digit contains the two previous digits. Each "child" digit is a representation of it's "parent" digits. Each digit contains it's two previous digits. Since each digit contains the data from the digits before it, this is a representation of fractal expansion.

The Fibonacci sequence is:
1,1,2,3,5,8,13,21,34,55, etc...

Notice that 1 plus 1 is 2, which is the next number in the sequence after 1 and 1. 1 plus 2 is 3, which is the next number in the sequence etc... Each number is the sum of the previous two digits.

A "golden spiral" can be drawn within a golden rectangle that is composed of a progression of squares of unit lengths according to the Fibonacci sequence.

Any section of this spiral will be in ratio to any other portion. The "golden spiral" extends infinitely inward and outward. It is a geometric representation of the fractal and infinite nature of phi. The Golden spiral is found in animal horns, conch shells, plants, storms and galaxies. It is prevalent throughout nature and throughout the universe. Branches grow around tree trunks in golden spirals. Fish, tortoise, rabbit, human and other vertebrate embryos begin life as an unfurling golden spiral, and our earlobes are golden spirals as well.

Spirals found in plants: flower petals, cacti spikes, asparagus, stawberry seeds, raspberries, corn, artichokes, pineapple, pinecones etc... - the spirals manifest in groups of 3, 5, 8, 13.... (the numbers of the Fibonacci sequence), and 5-pointed pentagram stars appear in the centers of fruits such as apples, cucumbers etc... (see: phi in nature)

"Where plants have 5-fold patterns, a consideration of their souls is in place, for patterns of five appear in the regular solids, and so involve the ratio called the golden section, which results from a self-developing series that is an image of the faculty of propogation in plants. Thus the flower carries the authentic flag of this faculty, the pentagon." - Johannes Kepler

"If measure and symmetry are absent from any composition in any degree, ruin awaits both the ingredients and the composition....measure and symmetry are beauty and virtue the world over" - Socrates

The Golden Triangle:

A triangle with sides in golden ratio is a "golden triangle" It's angle will always be 36, 72, 72. A "golden spiral" can be drawn within a series of consective golden triangles.

Not surprisingly, the pentagram is made up of golden triangles.

A progression of the ratios of each two successive Fibonacci numbers when plotted as a graph shows that the ratios are merging closer and closer to (but never reaching) a particular number. That number is the infinite non-repeating decimal known to ancient peoples around the globe as "the golden ratio" (phi).

the Golden Ratio = 1.6180339......

This decimal 1.6180339...... (phi) can also be represented as an infinite fraction or an infinite radical.

"The miracle is that the universe created a part of itself to study the rest of it, and that this part in studying itself finds the rest of the universe in it's own natural inner realities" - John Lilly