This generated a surprising amount of interest, so part 2! People who have read Tysons Agrippa book are going to be bored.
“Whats in the margins of the Tyson book?”
Agrippas method of generating the squares is mechanical. While programmers today would recognize his method as a series of algorithmic shifts in a grid, this was hot esoteric stuff back in the day. Other known examples of this are the Knights Tour and the Uncrossed Knights Path as an open shape. People working over transformations in magic squares were shuffling the tiles. Was there one move, like a chess piece, that generated valid boards?
I’ve been talking to Aleph Tzaddie quite a bit and he gave me permission to use the ThAOI images. Which is nice, because he’s worked through them and provides alternatives to the published ones. He also suggested for people to read The Key to it All which I haven’t personally read but apparently has helped him put this stuff together. The ThAOI magic square page suffers from facebook formatting, so I’ve pulled in the good stuff. He is working from the back of the Three Books, so those versions of the squares. Additionally his squares have the Hebrew on them. I would suggest his squares with the Hebrew if you prefer to work that way since he’s got a good grasp of the topic. I wholly admit to being utterly lacking in that area beyond barely sounding out words. The ThAOI presentation is as follows:
That brings us to the next piece of the puzzle – are there non-Agrippa/Tyson/ThAOI squares? Yes, but they’re reflections or rotations of the existing squares. Which is to say that a hypothetical order 13 square (calculated):
Will retain the values in the corners to be a valid magic square. What does that mean for us magicians? It means that reflected magic squares are just as valid. People seem to get hung up on “upside down” and “left-to-right”, the point here is the math doesn’t change. A knight’s tour can be mirrored vertically, horizontally, and rotated 90, 180, and 270 degrees (or a combination of them) and the sequence remains the same.
Now, since we haven’t discussed this to hell yet – how do we generate the even-nominal squares? Well, there’s the curiously named LUX method. Which is a little more akin to the knights-tour method, and was the method I didn’t think was particularly well represented by Tyson. The quick hack is to generate four 3×3 (1-9,10-19,20-29,30-29) and tile the squares to make a 6×6 square.
Where does that leave poor Jupiter? There are no 2×2 magic squares, so we can’t tile up to 4×4, but since we know rotation is implied, and we know mirroring is a quick hack for putting these together, it turns out that is exactly how to generate the 4×4 square.
Mirrors? Rotation? That sounds like an X and a circle…
That probably shouldn’t have surprised anyone. So are these seals shorthand for the math? Well, the answer is maybe… that works for Saturn, Jupiter, and Mercury, but for Mars, Venus, and the Moon, the seals given do not reflect the rotations and mirroring very well.
This is, in my opinion, either someone mixing up the folios or a blind, but saying it’s a blind requires the allegation of malice towards some class of reader, and I don’t like to make that accusation. But what if this is a map?
Points E and C are something we could attribute to the Jupiter rotation scheme. D? Might indicate that the vertical mirroring and the horizontal mirroring are not supposed to be interpreted at the same time, which would differentiate A and B. If we want to explore possible meanings here, A might indicate the shift solution while B would indicate the solution for the Square of Mercury:
So which is it? Can we nail down one or the other? I was inclined to make the Venus seal look like the seal of Mercury if only because Mercury very nicely has this pattern already in the inner square. It would yield this:
And now we see the problem! That’s the luna square! So here’s the thing about the shift solution, it only applies to even-numbered squares. Venus is 7 along the axis, so this makes me fairly confident that Agrippa (or whoever made it) might have known something was amiss but couldn’t solve it. Luna is 9 along the axis, so Luna can’t be a shift solution either. The problem is, whatever the solution is if we’re talking derivation through shifts, it would apply to both Luna and Venus and generate the same map.
That’s the fundamental problem here in my opinion – if we only have four operations, and we want to call them Pandiagonal (Crescent), Diagonal (Circle), Mirror X, and Mirror Y, we quickly fall into the trap of really only having four or so patterns to employ. I haven’t seen any satisfactory fixes for the odd seals yet.