The Basics - Part 4: Meaning in Shapes and Forms --
Numbers and the Pythagoreans
Pythagoras was born on the Greek island of Samos off the coast of Asia Minor but moved to Croton in southern Italy as a young man. It is believed his teacher was the mathematician Thales of Miletus. It is also thought that Pythagoras lived in Egypt for at least 20 years where he was taught religion, philosophy and mathematics by Egyptian priests. When Persia invaded Egypt, many of the priests were exiled to Babylon, and Pythagoras may have been taken with them.
While Pythagoras undoubtedly learned much about mathematics from his Egyptian and Babylonian teachers, he was the first to go beyond using numbers just for calculations. He began to look at numbers as symbols in themselves. In Croton, Pythagoras gathered large crowds to listen to his lectures on philosophy and mathematics. While we have no recorded writings of Pythagoras, his more enthusiastic followers formed a secret academic brotherhood known as the Pythagoreans, and many of them recorded his teachings. His school was open to both male and female students. Those who lived at the school were called Mathematikoi, while those who attended the school but commuted from elsewhere were called Akousmatikoi. (Notice the Greek word roots for mathematics and acoustics. It is significant that the Pythagorean school taught that music was based upon fundamental mathematical principles of harmony. They linked that harmony with the order of the cosmos, and we shall deal more with that subject in subsequent articles.)
The Pythagorean Theorem
While Pythagoras is given credit for inventing the words philosophy (love of wisdom) and mathematics (that which is learned), he is best remembered for the so-called Pythagorean Theorem that we all encountered in basic geometry class. You'll recall that it deals with the simple trigonometric function that lets us derive the length of any side of a triangle with one 90 degree (right) angle if we know the lengths of the other two sides. Stated simply, the square of the hypotenuse c (the side opposite the right angle) is equal to the sum of the squares of the other two sides, a and b. The familiar algebraic equation is: a2 + b2 = c2. It's graphically represented here:
What is significant for our purposes in dealing with the subject of Sacred Geometry is that we need to think of the Pythagorean Theorem in terms of 2-dimensional square surfaces representing each side, not just the algebraic equation relative to the simple diagram above. As in my case, you probably were not exposed to the slightly extended veiwpoint by your high school math teacher of showing the square associated with each side. So, graphically, here it is:
This makes visualizing the process a whole lot simpler. It shows the equation pictorially.
Integers (whole numbers) that solve the equation and form lengths of the three sides of a right triangle, when shown grouped together, are called Pythagorean Triples. Examples of such Triples are 3, 4, 5, and 7, 24, 25.
Here are two additional graphic methods that further illustrate the Pythagorean Theorem:
Put together in the manner above, these two diagrams show clearly that the square of the hypotenuse (the long side c) is equal in area to the sum of the squares of the two other shorter sides of the triangle (a and b). There are, in fact, more than 300 mathematical and geometric proofs of the Pythagorean Theorem.
Pythagorean Properties of Numbers
The Pythagoreans assigned gender to numbers. They said odd numbers had male qualities and even numbers possessed female qualities. They also associated specific properties to certain numbers. For instance, the number 1 was thought of as the prime source for all numbers, but not as a number in itself. To them, it represented unity and reason. In geometry, the number 1 was illustrated by a point from which all dimensions could then be produced and extended. It was seen as the active, positive and affirming force, so to speak.
The number 2, the first female number, was illustrated by a line between two points. 2 represented to the Pythagoreans the passive, negative and denying force. Of course, a line is the first dimension of measurement (1D). It also has two sides, and it can connect whatever may be at its two ends. It is synonymous with the concept of polarity. A line can also be used to divide one area into two areas. This may sound rudimentary and self-evident, but it has rich psychological and metaphysical meaning. Two represents the opposites in general: good and bad, up and down, right and left, light and dark, Yin and Yang, etc.
The number 3 was said to represent the number of harmony because it combined the number 1 (unity) and the number 2 (division). Three represented the neutralizing and reconciling force. Geometrically, three is shown as a triangle shape, which has 3 sides and 3 angles. The area of a triangle, as with the area of any flat-surfaced geometric figure, is expressed in two dimensions (2D). In fact, other than a circle (which is really just an exploded point), a triangle is the first regular geometric shape we can draw that truly possesses two dimensions. It is also significant that, in engineering, we know we can get more stability by "triangulating" the placement of structural members.
The number 4 was, according to the Pythagoreans, the number of order and of justice. It is the number of the four directions on the face of the earth by which we can determine our location. In three dimensions, 4 points and 4 triangular faces make up the solid figure called a tetrahedron:
We'll talk about the tetrahedron later in this series as the first of the so-called Platonic Solids. But, for the moment, let's just say it possesses perhaps the greatest strength from a structural standpoint of any 3D shape. The number 4 represented for the Pythagoreans the three dimensions of space and matter. Thus, it is the first number to denote the characteristic of volume.
We've already discussed at length the number 5 in previous articles, so we'll skip to the number 6. It is the product of the first female number 2 and the first male number 3. Six was considered the first perfect number. A perfect number was defined as one that is equal to the sum of all the numbers that can be divided into it. So, 1 + 2 + 3 = 6. Other examples of perfect numbers are 28 = 1 + 2 + 4 + 7 + 14, and 496 = 1 + 2 + 8 + 16 + 31 + 62 + 124 + 248.
We'll stop here with our account of the Pythagorean attributes of numbers and continue this discussion in the next article.
As Above, So Below Redux
To wind up this month's article, I want to mention two more examples that occurred to me to illustrate the Fibonacci Series and the number 5 that we discussed last month, the number 6 we briefly talked about this month, as well as the Hermetic principle As Above, So Below discussed in our first installment.
One example of the Fibonacci Spiral which I neglected to mention is the obvious overall pattern of giant, swirling storms such as hurricanes. And, recent discoveries by scientists also establish that the eye of such storms can display a pentagon shape. In fact, Fibonacci Spirals and pentagonal shapes are now found at both the macro and the micro scale by today's physicists.
So is there any proof that may corroborate the validity of this Hermetic principle in our universe?
The answer is yes there is; the amazing fact of the matter is that recent discoveries reveal that the Hermetic principle and the sacred geometry involved can be demonstrated in both the theoretical unobservable subatomic vortexes of the atom, and the very real observable largest vortexes of hurricanes! Richard C. Hoagland and David Wilcock discovered the pentagonal and hexagonal structures in the eyes of violent category 4 and 5 (Saffir-Simpson scale) hurricanes that have threatened the United States in the last few years. Satellite photos beautifully display the five spoked-wheel observed in the eye of some of these hurricanes. The satellite photo below shows the pentagon in the eye of hurricane Isabel of September 2003."
Five-sided pentagon in hurricane Isabel
The second example that occurred to me is the pentagonal arrangement of the lug bolts and nuts mounting for most automobile wheels (not small cars, big SUV's, trucks and larger vehicles). Since a 4-lug mounting as seen on some small autos would tend to be inherently unstable for heavy loads, rocking on both the vertical and horizontal axes, a 5-lug mounting would seem to employ the least number of fasteners to provide necessary stability. Three lugs would be much too flimsy and not provide the required factor of safety for a standard weight car, and six would seem to be overkill and an added expense. I'm certainly no auto mechanic, but that guess seems logical to me, and it reinforces for us the characteristics of strength and economy of design that can be attributed to the pentagon shape when used for such purposes. (I'm sure the spacing of lugs to suit wrench size and operation is also a major factor.)