Journal Planetary Triangles: Hypothetical Designs and Uses

[StoatCatcher]

Disclaimer: Everything you're about to see here in this series of posts is more of a theoretical proposal than a tried and true method of harnessing planetary influences. I am posting this here to get some feedback on the ideas presented and to know if my research is onto something.

---<o>---​

---

Part 1: Building the Triangles​

1.1 The "3x3" Triangle​

In order to build the magic planetary triangles properly let us first consider the design of the magic planetary squares themselves, using the square of Saturn as a stand-in for the basic theory. The square of Saturn, as many of you already know, is built as a 3 by 3 square containing nine smaller squares inside of it with the numbers 1 through 9 inside of each smaller square. The numbers of the square are arranged in the following manner:

What is interesting is that for every single X by X square that contains X² smaller squares it's possible tile together X² equilateral triangles in a similar grid to form a larger triangle that can fit every number of the original square inside of it.

But where do we put all the numbers? The particular arrangement of numbers in the original square has the interesting mathematical property of producing rows, columns and diagonals with three numbers each whose sum is always equal to 15.

• Row A: 4 + 9 + 2 = 15​
• Row B: 3 + 5 + 7 = 15​
• Row C: 8 + 1 + 6 = 15​
• Column A: 4 + 3 + 8 = 15​
• Column B: 9 + 5 + 1 = 15​
• Column C: 2 + 7 + 6 = 16​
• Diagonal A: 4 + 5 + 6 = 15​
• Diagonal B: 8 + 5 + 2 = 15​

I hold that, in order for our planetary triangle of Saturn to be valid, this mathematical grouping of numbers must be respected and emulated in the design. Specifically, we must group the numbers in trios that correspond to those of the magic square and still sum up to 15. I naturally started looking around to see if someone had done this before and found a paper that seemingly described what I was looking for on arxiv:

Please, Log in or Register to view URLs content!

. The solution they propose in that paper is that a magic triangle can be defined as a triangle that contains three central lines (horizontal line, upwards slope and downwards slope) that all intersect each other and sum up to 15.

Unfortunately, as you can see in the illustration, this leaves out three numbers at the edges of the triangle that not only won't add up to 15 but are also not one of the valid trios form the original triangle. It's easier to see the unused trio when you transpose their solution back into the magical square like this:

In my opinion this mathematical solution to the problem is not valid for this particular design of magic triangle where we are trying to fit all of the magic trios together. However, keep this idea of building a triangle with three lines of numbers in mind. It's actually going to help us build another type of magic triangle later, and it's going to be mathematically valid one.

But back to the problem of fitting all the nine digits inside this triangle grid we have. I decided to keep looking. Eventually I found what I will politely describe as a "messy" blog by a man named Steven H. Cullinane who seems to be a retired mathematician that posts about various ideas involving geometry, somewhat at random and with little context. But regardless, here's his take on the subject:

Since he seemingly just posts images with no context or explanation for them, at random, I have no idea how he got these solutions in the first place. But regardless, I like his "3x3 Magic Triangle" arrangement. It uses two rows of the square to create two overlapping triangles in the middle that are inside of a larger triangle. And all the numbers that are on opposite sides of the square are also on opposite sides of the triangle. I don't know if there's some deeper math at work here but it does seem to have some intuitive sense to it.

​

You might be wondering "What about tracing a sigil by tracing a line from 1 to 9 just like in the square?" and the answer to that question is that the resulting sigil from trying to follow the path of the numbers is...a little underwhelming, actually. I do, however, see a downwards movement that starts at the top and spreads throughout the base of the triangle. This is significant because when we remove the lines and numbers from this triangle we end up with a structure that should be familiar to those who study Pythagorean ideas:

Please, Log in or Register to view URLs content!

.

There's a lot of symbolism with this figure but I just want to focus on the basic idea of energy coming from a singular point (the Monad, God, the One, pure potential) and reaching downwards through the levels of creation until it reaches the multiple points of the manifest world. This makes our planetary triangle act as some sort of circuit that is constantly pulling down the energy of the planet into our material world. Using this idea, we can actually reverse the orders of the numbers to make a triangle that sends energy out rather than receive it.

I'm going to call this particular magic triangle the "3x3" triangle from now on. Everything that I just detailed makes it feel just right to me and it gives interesting ideas for it's use that I'll elaborate on later. I especially like this concept of a planetary symbol that can be used to both receive energy and send energy away. But I'm getting ahead of myself.

I'm going to pause here as this post is already quite long. Do leave your thoughts and feedback below.

 

[StoatCatcher]

1.2 The "Perimeter" Triangle​

While I was looking for a solution for building a magic triangle that resembled it's corresponding magic square I easily came across another type of magic triangle that I initially dismissed: the perimeter magic triangle. To quote Wikipedia:

A magic triangle or perimeter magic triangle

Please, Log in or Register to view URLs content!

is an arrangement of the

Please, Log in or Register to view URLs content!

from 1 to n on the sides of a triangle with the same number of integers on each side, called the order of the triangle, so that the sum of integers on each side is a constant, the

Please, Log in or Register to view URLs content!

of the triangle.

If you recall that Magic Triangles article earlier their solution to finding magic triangles was to find three intersecting lines of numbers whose magic sum was equal to the sum of the original magic square. We could plot the numbers that they found for order 3 magic triangles into a perimeter triangle like this to get a triangle whose magic sum is the one for the magic square of Saturn... but that still leaves out the three unused numbers from the square. So what about using all the digits form the magic square and plotting them like this? The short answer is that there is no traditional perimeter triangle like that that will yield our desired sum. Not to mention we lose the groupings of three numbers that call back to the rows and columns of the square. But, if you click on that first citation in the quote from Wikipedia, we'll finally a valid solution in the form of balanced perimeter triangles!

This pattern was suggested by Charles Trigg [5].

The digits 1 to 9 may be partitions into 3 groups, each summing to 15. We can place these groups onto the 3 sides of a triangle, but with no numbers at the vertices. Thus each side will sum to 15.

The eight partitions for 3 sets of 3 are elegantly represented by the 3 x 3 magic square. The main diagonals cannot be used because the share the common number 5. The three rows may be placed on the 3 sides of a triangle for one arrangement. The three columns may be placed on the sides of a triangle for another solution.

Any position is permitted for the numbers on a side, so there are a lot of possible solutions.

Figure 1 is the set represented by the rows of the magic square. Figure 2 is represented by the columns. While this is the simple way to determine which side to place the numbers, there are two other methods (I know of) to do so. Place the numbers on the sides of the triangle in sequence, but at the end of each triad, skip one side (fig 1), or at the end of each triad, place the next number on the same side (fig 2).

For added interest, this problem may be generalized for polygons of any number of sides.

[5] Charles W. Trigg, Triangles With Balanced Perimeters, Journal of Recreational Mathematics, Vol. 3, No. 4, 1970, pp 255-256

Would you look at that, a triangle that is magic and contains all the magic square trios! You may recognize figure 1 as the triangle I used in the OP image for this thread, but for me figure 2 is where it's at magically speaking. Why? If we trace a line through the digits to figure out the energy flow of this triangle, as we did before with our 3x3 Triangle, we'll come to realize that the lines create a pattern where three more triangles emerge to keep the energy tightly bound inside of the perimeter of the triangle.

Whereas our previous magic triangle design is very good for moving energies from one point to another, this one is good for keeping the energy together in one place. Given that this is based on the square of Saturn you could make the argument that this is very fitting for channeling the quality of "restriction" or "arrested motion" that the planet is said to have. It could be very useful for containing volatile spirits within or as a protective barrier to put around yourself.

And...that's it! Since we have a mathematical source to draw from with very clear logic there's not much to speculate about this one. The next part will be focused on putting these triangular shapes in practice.

Addendum: Magic Perimeter Polygons in Planetary Magick​

You may have already noticed this little sentence from the quote about balanced perimeter polygons

For added interest, this problem may be generalized for polygons of any number of sides.

And yeah, this means we can build this type of "barrier" or shield-like sigil for each magic square of each planet. This would mean drawing the perimeters of a square for Jupiter, a pentagon for Mars, a hexagon for the Sun, a heptagon for Venus, an octagon for Mercury, and finally a nonagon for the Moon. The only planets that can be coerced into a triangle shape would be Saturn, the Sun (36 / 3 = 12 numbers on each side), and the Moon (81 / 3 = 27 numbers on each side) but you probably should stick to their natural shapes.

I think the best use for these magic perimeter polygons is for containing energy. You could place a tool inside one and consecrate it with the planet's energies or let they remain in there when you're not using them to keep them "charged". Another use is to use it as a sort of mouse-trap for planetary aligned spirits during evocation rituals (I'll dive on this aspect later) since the shape contains energy that would theoretically be favorable and attractive for the spirit but keeps it trapped within it's confines. You could also summon a spirit of the opposite planetary energy of the symbol to weaken it while it's inside, if it's a particularly troublesome intelligence. Or maybe you could just sit inside of the shape and do a planetary invocation to get a concentrated dose of it's influence. Lots of possibilities, let me know what you'd use them for.