Take a long look at this image. In this image is contained the beauty of the number twelve and the reason the number twelve had throughout history been considered a mystical number.
Notice first the three squares about the outside of the figure, each rotated so their points are evenly spaced. Just within those squares are four triangles, each rotated so their points are evenly spaced, aligned with the directions of the points of the squares. Nothing is left out, and every point is met with another between the two rings. But more. Notice, for example, the yellow square. Each of its four points is met with a point of each of the four triangles.
It’s beautiful, right? But you can do it with any composite number. Why twelve?
To start, the prime factorization of twelve is [2, 2, 3]. Alternatively, you could say twelve is two squared times three, which is 4 x 3. Contained within twelve are the numbers 1, 2, 3, 4, 6, and 12. What I am most interested in, though, are the numbers right in the middle there: 3 and 4.
The fewest points necessary to make a two-dimensional figure is three: the triangle. The number three has been widely recognized as significant throughout history and throughout the world. Look into any culture and you’re likely to find three things with a significant role mythologically, cosmogonically, cosmologically, theologically, ontologically, what-have-you. The number three is universally given a place of importance. In Christianity we find the Holy Trinity. The Taoists have the Three Pure Ones. The Hindus have the three primary gunas. The Greeks mentioned the three Fates. Matter on Earth naturally comes in the forms solid, liquid, and gas. Atoms are comprised of protons, electrons, and neutrons, each proton and neutron composed of three quarks, which come in three generations. There are indigenous languages which don’t get past three; essentially, they count, “One, two, many.” And let us not forget the significance of beginning, middle, and end. The list goes on and on.
We also have this. Perhaps you recognize it.
One more than three is the square: Four. Also, two. Four is two twice, and the significance of the number two is unmistakable. If anything is to exist at all, the number two is implied. This is the ancient notion of yin and yang.
If a thing is to be, there must be both that thing and not-that-thing. If there existed only a zero-dimensional object — a point — the number two would be implied as the point would not exist as a point if it had no space to surround it; if there were nothing that was not the point there would be no point at all. Two is fundamental to existence, and very clearly so.
Notice also that the fewest points necessary for a three-dimensional object to exist is four: a tetrahedron. We might get to that later. We probably won’t. The reason we probably won’t get to that later is that we are not dealing with three dimensions, not directly, at least. While we live in a three dimensional world, our direct visual experience of it is always two dimensional. More specifically, when you look up at the sky, what you see is the apparent inside of a sphere, which is two-dimensional, its apparent motions projections of the motion of the Earth, whose surface is the two-dimensional surface of a sphere. Astrology is not concerned about the planets of themselves, not really. It is, rather, concerned about our relationship to the planets. After all these years, this is why astrology is still geocentric. Astrology is, in principle, a matter of learning about the sky by looking at it as it presents itself to us. The most ancient astrologers, who did not apply complex mathematical calculations to these observations, were very likely most in tune with this — but do not interpret this as though I am saying that the mathematical calculations were a mistake! They most certainly were not. Nor am I saying that we as astrologers should not be concerned about the planets of themselves at all. I am, rather, saying that astrology as an art is a study of particular types of relationships.
As we know, the Earth is tilted about 23.5°. As the Earth orbits the Sun, or perhaps as the Sun transits the sky, it is sometimes — during the summer — further north and sometimes — during the winter — further south in such a way that it forms a ring at an angle of 23.5° to the angle of the apparent rotation of the stars about the northern star, Polaris. This ring we call the ecliptic. It is defined by the Sun, and the Moon and the rest of the planets follow along the path with minor deviations. There’s a lot to say about this, but for now I’m still talking about number.
I’ll point out here that we’re dealing with one, two, and three dimensional objects at once. I hope you can see how and I hope it helps you to visualize. Given the rotation of the Earth, all by itself, any observer has four significant points: North, East, South, and West — the cardinal directions. Given the angle of the ecliptic, the tilt of the Earth, the cardinal directions also imply Summer, Autumn, Winter, and Spring. These are the cardinal points not of the Earth but of the Ecliptic — the Zodiac.
The number four is inherent to the tilt of a ring relative to another ring.
At first glance we have the solstice points and the equinox points: Cancer and Capricorn, Aries and Libra. Now, there are a number of ways to make this next step. Possibly the least elegant method, but the simplest, is simply to consider each quadrant with a beginning, a middle, and an end. Thus we have twelve divisions of the ecliptic. I’m not going to stop here. This explanation clearly isn’t satisfactory.
The ancient Phoenicians used a base twelve number system. Actually, they wound up with a base sixty number system when they combined it with a base five system. This is still apparent in the way we measure time. Phoenicia was a major center of trade in its time, and some part of its influence probably stretched far and wide with high regard. This is likely the real reason that the Zodiac consists of twelve signs, but not just because of the influence of a base-twelve system necessary for trade. The base twelve system itself came about simply out of the realization that twelve is a small highly composite number. Fifteen might have been chosen instead, but at the cost of the number two, and twenty might have been chosen, but with the cost of losing the number three and taking an unwieldy size.
One step further. We’ll get there.
If there is one of something, you can see so immediately. You wouldn’t mistake a single thing for several of them and wouldn’t even need to count it. It is clearly and unmistakably one. Now, if there are two things, you likely have exactly the same experience. You don’t even need to count or do any sort of math to see that there are two things when presented with two things, and the same is true for the number three. When we get to the number four, we might have to look a bit longer, but the result is the same. We can see immediately the quantity four.
At about five, everything starts to change. Immediately, you probably start to view five as 2 + 3. Try it now. Imagine a pentagon in your mind. Just five points. There’s a good chance that you’re imagining three things, and then two more, or two things, two things, and one more, unless you’ve spent a lot of time trying to visualize pentagons. At six, we can still glance at a thing and know it’s six, but we do so with more difficulty and most likely by seeing two sets of three. Seven brings us to about our limit: three and four. We likely have to see three and four or possibly three, three, and one; possibly three, two, and two.
Notice what happens at eight. At eight you’re forced to see four and four, probably, and possibly two, two, two, and two. You might be less likely to view eight as five and three because you’re probably already seeing five as three and two, which would force the image to be two, three, and three. This doesn’t sound more difficult than recognizing seven when explained this way, but try it yourself as I just did. Pull out a handful of coins or whatever you have on hand and try to figure out different quantities without counting.
This is called subitizing and it is essentially an arational appreciation of mathematics built into our perceptual capacities. We can subitize up to three very easily, and nearly as easily we can subitize the number four. Five through seven are more difficult, and the difficulty then increases at an increasing rate from there. Though I have pointed out that we might tend to recognize the number five as two and three, it is entirely possible to view it simply as five. A psychologist might run an experiment in which a subject would be required to state the number of dots that had flashed on a screen. There would be no time to mentally split the dots into groups; you just have to see the quantity for what it is. Children and very small babies, even as young as six months old, have the ability to subitize. As I’ve been showing, subitization comes in at least two forms: the subitizing one can do up to the number three, which infants can do, and more complex sorts which involve essentially subitizing twice, which is probably what a person does beyond the number four and almost certainly what the person does as we venture very far from that.
I have two purposes for explaining this. The first is to point out that every factor of the number twelve is within the subitizing range, even to include six, and, as such, the number twelve is readily subitized in a slightly more complex fashion. This is less true for a number such as fifteen simply because the number five is not as readily subitized, and, again, we lose our two for a five — conceptually, perceptually, not a fair trade.
The second reason for doing this is to get you to see, to feel, the qualities of the numbers. Just focus on the numbers up to four for now. Focus on each in turn and feel what the arational apprehension of each number is like. The number three has a distinct personality, I find, as do the numbers two and four and five and six and seven. And especially the number one.
There is a qualitative aspect to quantity. This is what I would like you to recognize, and regarding the understanding of the essence of astrology, the fact is crucial: there is a qualitative aspect to number.
My next post will be a more in-depth look into the twelve divisions of the ecliptic and more insight into why all of this matters.