The Measure of Life - Introduction to Sacred Geometry

Geometry is defined as "measure of earth." Since ancient times, civilizations have observed geometric relationships in the world around them. At the time, they were simple observations, but as time passed and technology developed, these observations were quantified and documented. Eventually, commonly reoccurring proportions for noticed. These were not dismissed as mere coincidence, but rather embraced as the workings of the gods. Thus sacred geometry was born, a philosophy in which the world and all of its components are made up of specific geometric patterns and proportions. Throughout history, civilizations have observed geometric patterns and proportions repeatedly present in life itself; such a development would catalyze a method of looking at the world unlike any other in history.

Sacred geometry is found in many forms, spanning across a vast range of subjects. Nature itself exhibits geometric novelties, such as the Chambered Nautilus. The shell of this sea creature forms a perfect logarithmic spiral. This enables the creature to grow without having to obtain a new shell. The hexagonal structure of a bee hive is another common example of geometry at work. Hexagons provide a steadfast method of construction, allowing the hive to fit together in a tightly compacted fashion ("Sacred Geometry").

Cosmology implements geometry as well. Through the findings of Johannes Kepler, a German mathematician and astronomer ("Johannes Kepler"), and other similar scientists, the orbital paths of the planets in the solar system were analyzed. According to Kepler, planets sustain an orbit about the sun that are proportionate to the other planets. He attempted to identify shapes and patterns based on Platonic solids, the five basic three-dimensional polyhedrons, due to the fact that the planets do not move in a purely two dimensional fashion. Kepler's theories, although heavily scrutinized, are prominently studied by scientists even today ("Sacred Geometry").

Even music has roots in geometry. Musical intervals are created by fractionally dividing strings in various proportions. According to Pythagoras, an Ionian Greek philosopher ("Pythagoras"), a string pressed at its midpoint creates an octave, while 3/2 ratio produces a fifth interval and a ratio of 4/3 produces a fourth interval. Sound waves themselves are geometric figures, as they are commonly modeled in modern times with sine and cosine waves ("Sacred Geometry").

Architecture illustrate perhaps the most prominent use of geometry in ancient and modern-day societies. Cultures long before the first century have constructed buildings with certain proportions, such as the Phi ratio, the Golden Rectangle, and the logarithmic spiral (all of which will be elaborated on later). These proportions and patterns were commonly found in Egyptian, Greek, Indian, and Roman architecture, as well as countless others. The Pyramids of Giza are a prominent example of such construction, as is Stonehenge ("Sacred Geometry").

However, sacred geometry is not limited to the past. There are many contemporary usages of the philosophy. Many of these uses are based upon religion and superstition, much as earlier accounts of sacred geometry were. The vesica piscis, which has since been claimed by Edward Venn as the Venn Diagram, is an immensely popular form of geometry in modern society ("Sacred Geometry"). Various company logos, such as MasterCard and Trans World Airlines, implement the vesica piscis (Schneider 32). The music scene has been influenced by sacred geometry as well. The progressive rock band Tool depicts many forms of geometry when playing live: the drum set and stage are embellished with geometric designs such as the unicursal hexagram, and a large portion of their lyrics are heavily laced with facets of sacred geometry such as the merkaba, logarithmic spirals, the Golden Rectangle, and the flower of life (Gomes par. 2). Sacred geometry continues to be a widely studied philosophy, even amongst those outside of scientific professions.

Sacred geometry begins with life itself. Most with secondary education have been exposed to the three-dimensional coordinate grid (Fig. 1-1). Each axis represents a dimension of the world: the x-axis and y-axis make up the two-dimensional plane that most students graph on paper at some point in their education. The z-axis is then introduced as the third dimension, giving flat objects depth. Knowing this qualifies the fact that people live in a three-dimensional world. However, sacred geometry expands on this core idea. Two-dimensional circles are placed on the three-dimensional coordinate plane to form the "fruit of life," the basis of all life on earth, as shown in figure 1-2.

Note that the number of circles superimposed on the three-dimensional coordinate plane is thirteen. There are a number of superstitions surround the number. Hotels avoid the thirteenth floor, skipping from twelve to fourteen. Friday the thirteenth is an unlucky day. These superstitions, ironically, all carry a negative connotation. However, contrary to popular belief, the number thirteen is a fundamental number of life, as the thirteen circles are the thirteen systems of information. These systems are everything from taste, smell, and sight to the actual atomic structure (Frissell 85).

Rounded lines are female, whereas straight lines are male, thus the fruit of life is composed of female energies. However, life is composed of both male and female energies, thus more needs to be added to the three-dimensional coordinate plane. By connecting each and every circle at their centers, male and female energies are infused. This process forms Metatron's Cube (Fig. 1-3). Metatron's Cube is the source for all geometric figures known to

humans. At first glance, it appears to be a smaller cube inside a larger cube. While this is true, there is also much more; each of the five Platonic Solids-the tetrahedron, hexahedron (or cube), octahedron, dodecahedron, and icosahedrons-can be found in two-dimensional forms. Metatron's Cube was commonly found in religious art, often times in the background of paintings. This is an immensely significant fact, for Metatron was an angel in Judaism and Metatron claimed that he formed the cube from his soul. Therefore, religion has scientific and mathematic roots. Another noteworthy symbolic reference is the presence of the Star of David. Each vertex of the star is located at the center of the six outer circles. This furthers the evidence that Metatron's Cube is a religious figure based on mathematics and supports the fact that it has Judaist meaning. The Star of David itself is a fundamental symbol as well. It is composed of two congruent triangles: one facing with the point up and the other with the point down. The upward pointing triangle represents the male energy, while the downward pointing triangle is its counterpart, the female energy. As is the common theme with sacred geometry, the two energies combine to form something even more significant (Frissell 85-86).

In terms of architecture, the Golden Rectangle has had a great deal of influence over the course of human history (Fig. 2-1). This figure can be identified by certain proportions between the side lengths. It begins with a square (ABCD), with a vertical line (EH) extending

perpendicular from the center of line AB to the center of line DC. A diagonal line (BH) is then drawn and swung downward with H as its rotation point until it lines up with line CD. This forms line DG. Line AB is then extended out until it is the same length as line DG (thus forming line AF). Points F and G are then connected to complete the rectangle. An interesting note to point out is that the newly created rectangle BFGC is

also a Golden Rectangle. Because of this, the entire process can be repeated, thus creating numerous Golden Rectangles nested within one another. This plays a critical role in the formation of the logarithmic spiral, which will be elaborated on later (Lawlor 68-69).

The Golden Rectangle is not simply a concocted shape with a majestic name, however, for it has been widely used since ancient times. Perhaps the most significant is the Greek Parthenon. As figure 2-2 depicts, the Parthenon was constructed based on the Golden Rectangle. Notice that the largest Golden Rectangle has within it several smaller Golden Rectangles. Although they are not the same size, the dimensions of each and every Golden Rectangle are proportionate.

The proportion of the sides of the Golden Rectangle is a critical one known as the Phi Ratio. Phi is an irrational number, as it does not come to a definite conclusion, but rather continues on for infinity (1.6180339...). However, its most common abbreviation is 1.618. Using diagram 2-1 as

a reference, line CD divided by line HD equals Phi. Likewise, line GD divided by line HD equals Phi. This number has been coined a transcendental number, as it as some unique properties that no other number is known to display. For example: consider the following algebra:

Phi - 1 = 1 / Phi = .618...

Phi + 1 = Phi * Phi = 2.618...

According to those examples, adding and multiplying are the same operation in the case of Phi, as are subtracting and dividing (Schneider 141; Frissell 92-93).

The Phi Ratio is present in various aspects of life. In fact, the human body itself is built around the transcendental number. This is depicted in Leonardo da Vinci's drawing, Vitruvian Man. The length of the first section of the human finger (the closest to the hand) divided by the length of the middle segment, Da Vinci observed, equals Phi. Similarly, the length of the middle portion of the finger divided by the length of the last section also equates to Phi. The same is true with toes. The length of the upper leg (the thighs) divided by the length of the lower leg (the shins) is once again the Phi Ratio. Of course, this varies from person to person, as everyone develops slightly differently, but most people are within a reasonable distance from this number; the prototype human being has these exact proportions. Da Vinci made other notable observations as well. First, the naval is in the exact center of the circle in which the Vitruvian Man resides. The arms, spread out parallel to the ground, are identical in length as the height of the human being. In this same position, the man fits perfectly into a cube that can be drawn around him. Likewise, if the second position is observed-with arms and legs spread apart-the man becomes encompassed in a sphere (Frissell 94-95).

From the Golden Rectangle and the Phi Ratio comes the logarithmic spiral. Using figure 2-1 as a reference, one can see that multiple Golden Rectangles are created. AFGD is the first and BFGC is the second, smaller rectangle. From here, BFGC can be divided into a square and an even smaller rectangle, which as expected is another Golden Rectangle with the constant Phi Ratio. This can keep reoccurring infinitely, as more and more Golden Rectangles can be produced. Figure 3-1 demonstrates the infinite Golden Rectangles. As one can see, the spiral will continue to grow at an exponential rate infinitely (Frissell 89).

Similar to the logarithmic spiral is the Fibonacci Spiral. Leonardo Fibonacci, an Italian mathematician, derived a series of numbers that proceed in a limitless pattern. This pattern models the logarithmic spiral in a varied way, thus creating his own version of the spiral. Figure 3-2 depicts the Fibonacci Spiral superimposed on the logarithmic spiral.

The Fibonacci Spiral shows the lengths of a diagonal of each segment of the logarithmic spiral. The Fibonacci Sequence, his most famous work, is as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.... This is no random sequence of numbers, as it has a direct relationship with the Phi Ratio:

1 divided by 1 = 1, 2 divided by 1 = 2, 3 divided by 2 = 1.5, 5 divided by 3 = 1.66, 8 divided by 5 = 1.60, 13 divided by 8 = 1.625, 21 divided by 13 = 1.615

34 divided by 21 = 1.619, 55 divided by 34 = 1.617, 89 divided by 55 = 1.6181...

As one divides the next term in the sequence by the previous term, the result is a number that gets infinitely closer to Phi. However, Phi is never precisely reached, but the result is more accurate with every division that takes place. Fibonacci implemented this pattern with a hypothetical rabbit problem. Assuming that rabbits can breed after one month of living, each pair of rabbits consists of a male and female, and that the rabbits will mate at the beginning of each month, he created a hypothesis. At the end of the first month, the first pair mates, so the total pairs of rabbits is one. At the end of the second month, a pair of rabbits is born; there are now two pairs of rabbits. At the end of the third month, another pair of rabbits is born; there are now three pairs of rabbits. At the end of the fourth month, yet another pair of rabbits are born, and in addition, the first pair of birthed rabbits produced a pair of rabbits; there are now five pairs of rabbits total. As one can see, as more rabbits are introduced into the population, there will be in infinitely increasing amount of rabbits. Of course, this is purely hypothetical, for rabbits will die from old age and predators which, in turn, keeps the population relatively stable. However, for the sake of implementing his sequence, Fibonacci's hypothesis proved to be reasonable (Frissell 98-101). Ever since the time of the first civilizations, people began to notice reoccurring patterns in the world around them. These geometric occurrences were found, and continue to be observed, in a variety of places, from nature itself to cosmology, to music, and to architecture. These observations were heavily studied and by ancient cultures. Modern day enthusiasts continue the pursuit of knowledge on these topics. The fruit of life has been considered the basis of all life on Earth. Every known Platonic solid can be derived in a two-dimensional form from Metradon's Cube. Male and female energies combine to produce something even more spectacular. The Golden Rectangle, Phi Ratio, and the logarithmic spirals provided an immense influence on a variety of things, most notably architecture and theories on the construction of the solar system and human body. Then Fibonacci contributed his sequence of numbers relating to the logarithmic spiral and transcendental Phi Ratio. Countless other patterns, ratios, seemingly coincidental occurrences continue to puzzle the world's most brilliant minds even today. Every new discovery seems to unlock another secret of the universe, but in the process reveals three more new uncertainties. Sacred geometry aids the human understanding the Earth and the construction of life in ways people never imagined.

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