Thursday, June 8, 2017

More Math in Magic Squares

This entry is a continuation of my last blog entry on planetary sigils and kamea. It is the entry with all the math and odd speculation that makes a blog entry boring. So if reading about investigation and the process of decoding puzzles isn’t your thing, feel free to skip over this one. But if you are the least bit curious about the Solomonic sigils, you might want to take a peek.

First, let’s remember that there are sigils for both the planetary intelligences and the planetary spirits. We saw in the last blog entry how those two groups of sigils are created using numerology and names.

But there are also sigils simply called “planetary seals.” Here are images from around the web of how each of the planetary sigils are traditionally drawn. I will note that the seal of Saturn is my own drawing because the web is currently circulating all sorts of non-traditional garbage for what is probably the easiest drawn seal in all of ceremonial magic. (Witches, please check your work!)

Sigil of the Moon (9)

Sigil of Mercury (8)

Sigil of Venus (7)

Sigil of the Sun (6)

Sigil of Mars (5)

Sigil of Jupiter (4)

Sigil of Saturn (3)

These sigils are not drawn using the same method we examined previously, but I did get curious how they were created. I also began to notice some odd characteristics of the seals and the kameas.

I first noticed that the sigil of Saturn is drawn over that planet’s kamea using a very obvious technique. The numerals are divided into three sets of three. The numerals of each of the sets are connected with lines, separate from each other. A circle starts and ends each connecting line. I’ll illustrate that here by removing the numbers and using colors so it’s easy to highlight some of the characteristics of this seal. We will observe those characteristics later.

Saturn Drawing

Notice how the seal has bilateral symmetry with the axes of symmetry falling on the whole kamea's diagonals. The middle set of numerals also creates a line of symmetry (in red). The sets before and after the middle set make designs that are reflections of each other. Also, there is a convergence of lines on both sides of the center block.

I don’t know why this one planetary sigil was drawn using this technique, while the others were not. But I began to wonder what would happen if I applied these drawing methods to the other kameas. Allow me to walk you through my process.

So let’s draw on the other kameas, but only those that are based on odd numbers for now. Using the same methods we observed in the sigil of Saturn, we first write down all the numerals of each square and divide them into sets appropriate for the planetary number, then we draw out each numeral set using circles to start and end each set. Some interesting designs appear. We can see that there is bilateral symmetry in them. Again, the numeral set in the middle creates one of the lines of symmetry (in red). Each of the pairs of numeral sets on both sides of the middle set create designs that are reflections of each other. Also, all the beginnings and endings of the numeral sets straddle the diagonal (with the exception of the middle set). I’ve illustrated all of this with colors again to make it all easier to see.

Mars Drawing

Venus Drawing

Moon Drawing

What this shows is that the odd numbered kameas represent the symmetry inherent in mathematics. In order for all the rows and columns of blocks to sum the same, the very largest number must be placed opposite to the very smallest, then the next largest and smallest, and so on until the middle of the available numbers is reached. This isn’t magic, it’s just math. We can also see from the patterns that all of the odd numbered kameas are based on a similar numeral arrangement, but just increase their complexity as the kameas get larger. This makes the convergence that we saw on both sides of the middle block no longer the only convergence.

Let’s apply the same drawing methods to the even numbered kameas and see what happens. Jupiter and Mercury also show a bilateral symmetry. Since there is no middle numeral set, the sets simply pair up evenly without it. However, now the axes of symmetry have been rotated 45 degrees from the diagonals to the vertical and horizontal. Also, the start and end of each set has moved to the outer blocks. A convergence of lines becomes prevalent, only now all of the numeral sets is contributing. Here are more colors to illustrate all of this, though Mercury started to get visually busy, so I separated the first four numeral sets from their mates. This also helps to show the symmetry that is happening, as the first four sets create a mirror image of the last four sets. Notice also that a pentagram is starting to appear in Mercury's crossing lines.

Jupiter Drawing

Mercury Drawing

The odd man out in all of this is the kamea of the Sun. When we start to draw the lines from the sets, we quickly notice that there is no symmetry! Drawing it to the end shows that there is only one point of convergence to the right of center and there are several sets that force a drawn line to be back-tracked, which never happens in any of the other kameas. However, the sets do begin and end only on the outer blocks. I've separated the mates of the pairs, as I did in the Mercury drawing, to show the lack of symmetry.

Sun Drawing

Since I know that there are many different arrangements of numbers that can create a magic square still fulfilling the row and column summation rule, my first thought was that tradition had brought down to us the wrong 36-block kamea for this planet – one without symmetry. So, I tried to build a kamea that fit the rules displayed by all the other kamea. This proved impossible. Here’s why it’s impossible.

For an even numbered kamea to have bilateral symmetry, the base number must be divisible by 4, because bilateral symmetry creates a reflection (two images) on two planes, which creates quadrants of images (2 images x 2 planes = 4 images). Imagine a kaleidoscope that only produced four images inside it. Odd numbered kameas don’t have this problem because one of their numeral sets creates one of the needed lines of symmetry; a midline will always be built into those drawings. If we had kameas with a base of 10, 14, 18, 22, 26, etc., they would all show the same problem with bilateral symmetry, because they are all even numbers that are not divisible by 4. Again, this isn’t magic, it’s just math, though I will agree that math, particularly as it applies to shapes (geometry) can be very sacred.

By now you have likely wondered exactly what technique was used to create all those sigils of the planets that have come down to us - the ones at the very beginning of this entry. Well, unfortunately, no one knows. The best hypothesis I’ve seen to date was done by a clever Rosicrucian, who speculated that a counting technique was applied to the kameas. That information can be found here. For now, we simply accept that it is knowledge lost to the past.

If any of you have done work with kameas and planetary sigils, please comment your insights below or link the rest of us to your pages.

Thanks for indulging my mathematical deviation.

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