Magick Summoning Device

This endeavor combined perhaps three of my main artistic interests: difficult mathematical puzzles, quirky engineering and occult oddities with undertones of the macabre.  This is a very long article so if you are a TL;DR sort of person you can scroll down to the bottom for a video of the mechanical contraption in action!

A couple years back I was working on my sigil project; As a reminder, this was on the computer-assisted design of magic squares.  Magic Squares are the arrangement of a range of numbers such that the rows, columns and diagonals all sum to the same number.  Here is a 3×3 magic square:

Besides the obvious mathematics interest in studying these, magic squares have a very long history in the occult and have been found across the ancient world in spiritual contexts.  They were commonly used to create the sigil of some thing by converting the name of that thing to numbers, often via hebrew which is convenient, and tracing the path of those numbers over a magic square of significance.  These sigils were often used to conjure spirits and imbue objects.  You have probably seen sigils created in this fashion and never knew their source, here is the masonic symbol for instance:

For this project I wanted to create a magic square but in reverse; I had a particular name and symbol already in mind and wanted to see if I could devise a magic square that satisfied it.  The name and symbol are another story based on a different set of research that perhaps I will write-up some day.

When I developed software to solve this problem I found these two solutions below.  After drawing them out I realized that that they were related to one another in a very interesting way (one that I quickly realized was shared by all 5×5 magic squares I should add):

What I saw was that one was a complex 3D rotation of the other.  I saw it as the image below where if you could rotate the numbers in the green section about the horizontal axis between 25 & 15 and rotate the numbers in the blue section about the vertical axis between 8 & 9 that you could convert the two squares back and forth.

I woke up one morning and realized that I could actually build this mechanically, so I made a little proof of concept to test it out, here is a quick video of that.  It is a little clunky because I pulled it out of storage and just hot glued down on my work bench to show it off; notice how the arrow rotates 180 degrees in effect:

As I contemplated how to take this proof of concept and make a finished mechanical representation my design evolved and I began to think more about the aesthetics of an oddity showpiece and how to potentially mechanize this magic square even more.  This led me to thinking about the rotation not as a pair of 3D rotations but rather as a five separate 2D rotations where the three numbers on each of the sides and the center 3×3 box would all rotate about their center number 180 degrees.

With that I started to envision a final mechanical design to really embody this mathematical model in a device with the intent of 3d printing it.  The following images are all from the resultant CAD work in Autodesk Fusion 360.  I have dabbled in a couple CAD tools over the years, but I really like this one and I highly recommend as this was my first time using it and this is a very complex design.

So the build starts with the concept of making the four 3-number side pieces and a center 3×3 piece all sitting within a frame with the corner numbers.  The side and center tiles will be rotated by an internal gearing mechanism allowing it to convert back and forth between the configurations by rotating the gears.  Here is that mechanism:

Here are the gear layout, gear final, side and center pieces that will automate the motion of the side titles and center tile:

Here is the frame and you can see how the gears, side and center tiles fit into it.  I am only showing one side tile throughout because I only designed one in CAD as they are identical for printing:

Now for the tiles I created decorative embellishments upon which will be painted the numbers.  The main reason they are separate pieces is so that I could 3d print and glue together to end up with fine printing on both sides of the tile that would have not have been possible to print otherwise;  This also allows me to hide the rivet head attaching it to the base.  Here are those embellishments:

 

And now all of those together:

Clearly that center tile can’t rotate in place, so it needs a mechanism to lift it up and over the rest of the frame.  As you can see the gear shaft extends down from the bottom of the center tile through a hole in the frame which allows for it to connect to a plunger:

This plunger slides within a base that connects underneath the frame and shapes the bottom into a hermetically sealed pyramid structure.  The pyramid structure I thought was a nice touch to the theme of the device and in fact the angles of this pyramid precisely match those of the great pyramid.

Here is a cross section that is helpful.  You can see the gap between the top of the plunger and bottom of the lower frame assembly, this is where a spring goes.  You can also see the complexity of the fit including the holes for the rivets and how the embellishment side title tops cover these rivets (only one shot to get it right!).

As an aside, I want to point out that when viewed from the side, the height of the frame with respect to height of the tile is sized to match the golden ration and furthermore the title being comprised of two layers, the tile gear base and tile top embellishment, are also proportioned by the golden ratio.  Or visually expressed by the arrows below, green is to blue as blue is to red:

For display purposes, this entire finished assembly then neatly tucks away into a stand  that is comprised of a frame and a bottom.  The bottom is inscribed with the symbol and holes at the points on the symbol reveal the numbers that are painted on the surface of the device inside and remain unchanged in either rotation configuration:

And the whole thing together now:

And a final cross section:

Now while I have shown you the final design here, there were some intermediary printed prototypes that I gleaned a number of learnings from to get to this point.  Here are some in progress pictures, prototype work is white printings and final is in black:

For the plunger I had trouble finding a spring, so I ended up making one from a piece of springy steel rod used in landscaping flags I have and wound it with a contracting diameter base to tip so that when compressed it lays flat.

First movement of final:

And with a bit of paint and a coat of sealant, here is the final product.  Note I added feet which are scarabs that I printed from museum 3d scan of ones carved many millennia ago; such a crazy thought to imagine the transcending of time from the original carver to me 3d printing them:

And lastly a final video of the device in action: