Tesseract Meditation

There is a valuable series of meditations on the basic objects of geometry that is a progressive series of meditations on the point, the line, and the cube.  Most people stop here, if they even truly take it that far.  However, for those interested, there is certainly more that can be done in this series.

Charles Hinton (a late 19th century philosopher, Theosophist, and mathematician) coined the word Tesseract, the term that refers to the four-dimensional analog of a cube.  In other words, a Tesseract is to a cube what a cube is to a square.  I prefer to use his word as opposed to hypercube, as hypercube doesn't have as exact of a definition.  A Tesseract is not exactly a hypercube, it is more specifically a four-dimensional hypercube.  There are hypercubes of any number of dimensions, real or imagined.

I developed a model loosely inspired by Hinton's work (a warning that some supposedly went insane meditating upon Hinton's model), and decided to actually build it out of wooden cubes, as two-dimensional diagrams are missing two whole dimensions and make it more difficult.  It stands around 6 inches tall.  Here is a photo of the model, which I've enhanced with white lines:

A Colored Model of an Unfolded Tesseract

The shape of the model is an unfolded Tesseract.  In a similar sense, if you were to unfold a cube into two dimensions, you would have a cross of six squares.  Unfolding a Tesseract into three dimensions produces a sort of cross of eight cubes.  This is only one of many ways to unfold it, but the most appealing and simple.

The system is based on color.  The model represents an inner mentally-produced world.  The colors are set up in pairs.  There are four pairs of flashing colors: yellow and violet, red and green, orange and blue, white and black.  The black cube is represented by the hidden cube at the center.  If you were to travel from the center of any cube in any of the six directions corresponding to the faces of the same cube the distance of two cubes, you will arrive in the cube corresponding to the complimentary color.  Travel two more cubes in the same direction (or any for that matter) and you will arrive at the cube where you began.  The model is to help keep track of the cube you're in and the other colored cubes relating to your place in space while you are training yourself.

Below is an animation I made in PowerPoint demonstrating the connections between edges in folding up a cross of six squares into a cube.  It is followed by lines showing the connecting faces when folding up the Tesseract into the fourth dimension in space.  Understanding what faces are connected is crucial to success with this model.

One goal of being able to relate to such geometric figures as the point, line, cube, and tesseract in this context it the expansion of consciousness.  As there is no four-dimensional imagery for the eyes to look upon, the mind is not accustomed to perceiving a four-dimensional world and is unlikely to be able to readily imagine such a figure.  However, training oneself to observe a four-dimensional figure such as the Tesseract is a readily observable expansion of consciousness: perception of the previously imperceivable.

At first, this is not a four-dimensional experience at all.  It is easiest to start out imagining it like a 3D version of PacMan, that somehow you magically arrive at the other end of the cube after traveling through the opposite end.  After spending lots of time connecting with and developing the inner model, it can then develop into a four-dimensional experience.

This is but one method of approaching the creation of a mental model of a Tesseract.  I recommend utilizing many.

Printable Dodecahedron

I had been meaning to make an easily printable version of the Dodecahedron of Space, and today Andrew from the blog Wanderings in the Labyrinth rekindled my interest in doing so.  For anyone interested, though I still suggest making one from twelve pentagons as it is a valuable exercise in and of itself, here it is:

Printable Dodecahedron - Click to Enlarge

Note that I did not draw the usual flaps that these paper models usually have for joining adjacent faces.  You'll have to do that much all by yourself (it's simply a matter of leaving some extra paper on some of the edges).  I would highly recommend printing to something more sturdy than your usual paper, such as card stock, so that it hold up through all of the folding and taping or gluing.  You might also consider laminating the whole thing to increase its durability, but in that case normal paper would probably be better.

Cheers

Constructing a Dodecahedron

So one of the basic tenants of sacred geometry meditations is actually constructing the figures.  There is no better way to instill their symbolism and mathematical qualities into the mind. In the context of two-dimensional polygons and polygrams, construction is only done with a compass and ungraduated straight edge. Having been working with and meditating upon the three-dimensional geometric figures, I decided to apply this practice to constructing platonic solids. That, and having Ptah as a patron, I like building things.


For constructing such a thing out of wood, you will need:

• Wood
• A band saw
• A disc sander
• 5-minute, 2-part epoxy
• Wood putty
• Sand paper
• A utility knife
• Your favorite Bob Vila flannel shirt (optional).

There are variations on the tools you can use. For example, since I didn't have access to a band saw at the time I cut the twelve pentagons using a laser, from a vector file on my laptop.

You will need to begin by drawing 12 congruent pentagons onto a piece of wood, or multiple pieces of wood more likely. Ideally create yourself a perfect pentagon using a rule and compass or computer software can also do this for you. In either case, you will need a template to keep your pentagons consistent. To minimize the amount of wood needed, arrange these pentagons in a way that minimizes empty unused space on the wood. Make sure your lines are dark enough to see, and then take your piece of wood to your band saw. Be sure to wear safety goggles.

Take your wood to band saw and cut out your twelve pentagons. Error on the side of cutting outside the lines. You can sand these down later, but you can't add more wood back on. Again, these can be cut out by other means as well, such as with a laser.

Once you have your twelve pentagons, you will then need to take these pentagons to the disc sander. A disc sander is needed over any other type of sanding tool, because you can change the angle at which you are sanding, and the angle is the important part. We are now sanding the five edges, so that they will perfectly line up with the other faces of the dodecahedron when we get to the gluing process.

A dihedral angle is the angle between to joined faces of a polyhedron. The disc sander needs to be set to half of the dihedral angle, as the joining face will take up the other half of the angle. For a dodecahedron, the actual dihedral angle is about 116.6 degrees. Half of that would be 58.3 degrees, and this is what we need. However, the sander needs to be set to the complementary angle of 121.7 degrees (180 - 58.3). This can be set with a protractor. I personally cut out a piece of wood on the laser with the proper angle as a I didn't have a protractor handy at the time. Either way, it will not be exact, but you should get it as close as you can.


Set one of your pentagons on the disc sander, the side with the pencil marks facing you. Make sure you are wearing eye protection and begin sanding. Sand until you have sanded right on the pencil line. Keep the edge of the pentagon even with the face of the sander. Repeat the process five times per pentagon for all twelve of them.

11 out of 12 Faces Pieced Together

Then comes the fun, yet potentially frustrating, part. I like to use 5-minute epoxy, as it dries fast and holds very strong. If you can, I would use a dremel or some other tools to roughen up every edge that will be glued to another piece of wood, to give it some tooth to hold onto. Take three pentagons. Make sure they fit together nicely (you may need to do some adjusting here and there with the sander to get all the pieces to fit together). Mix up just enough epoxy to glue just these three pieces together. I recommend a plastic knife or a wooden popsicle stick for mixing and applying the glue. Keep in mind this kind of adhesive is serious, and will stick to things. Do not do this on your nice table. Do it on some newspaper in your garage. Make sure to follow the instructions. Usually you need to mix the two parts of the epoxy together for about 30 or so seconds. Apply the epoxy only to the parts of the wood you are currently gluing. The hard part is holding the three pieces together and constantly adjusting them to keep them in line until the glue dries. Making sure the glue is mixed together for the proper amount of time can make this process easier and quicker.

Those first three pieces are the hardest. After that, I recommend gluing one piece on at a time, adjusting each single piece with the sander as needed.

When you are done, it is very possible that there may be a gap between one or more of the edges. Use some wood putty to fill in any such gaps, and let it dry over night. Then sand down the wood filler with some sandpaper and use the utility knife to cut off any excess epoxy (cut away from your face, man).

Put on a coat of primer and then do with it what you will.

I recommend a series of meditations with the thing just as a plain white symbol. At some point you may wish to paint it in appropriate symbolism. This could be to paint it black or white, pertaining to the quintessence and the universe. You may also elect to paint the twelve faces in symbolism and/or colors of the zodiac.

The Easy Way Out

Or you could just use some sturdy, easier to use, material such as corrugated cardboard.  Cut out twelve congruent pentagons, tape together using your favorite tape.  For the dodecahedron, just know that three pentagons meet at every vertex (not that you could make it any other way).

Be warned that masking tape doesn't hold up very well under paint.  I made a set of all five Platonic Solids this way and had to use a layer of gesso on top of the paint to keep the tape from peeling.

The Five Platonic Solids Painted in Elemental Colors

Have fun.

The Dodecahedron of Space

We are all familiar in this tradition with the concept of the Cube of Space, which maps the astrological forces upon the faces and edges of a cube for a system of meditation.  I have been mapping out astrological forces upon the other four Platonic Solids.  The first of which is what I will be referring to as the Dodecahedron of Space (12 pentagonal faces).  I initially did some research on how others may have arranged these, but I was entirely unsatisfied by any attempts because of a lack of symmetry and harmony of the energies upon it.  I then sought to arrange them based upon my own meditations and direction.

I was inspired to arranged the zodiacal figures upon the dodecahedron so that the complimentary signs (like Aries and Libra) would be on the opposite faces and also that a symmetrical looping pathway was present, such that the Aries pentagon is adjacent the Taurus one, which is adjacent the Gemini one, etc, looping all the way back to Aries from Pisces.  The remaining symmetry I discovered through meditating upon the symbol, having them arranged as such.

Earth Elemental Signs Coming Together, Active Air Signs Surrounding

Water Elemental Signs Coming Together, Active Fire Signs Surrounding

The first attempt at laying out these figures resulted in a certain pattern emerging.  The three fire signs all met in one vertex, as did the air signs; the passive signs were symmetrically distributed.  It shifting the signs along the pathway I adjusted them so that the passive signs would collect together in one vertex and the active groupings would be equally dispersed.  The logic behind this decision is that the passive elements tend to collect together while the active elements are in motion.

The Two Signs Ruled by Mars Connected by the Highlighted Line

Directly Opposite the Mars Line is the Line Connecting the Two Signs Ruled by Venus

The next thing I noticed relates to the planets.  If we consider the line that connects the two faces referred to the signs ruled by Mars (Aries and Scorpio), it is directly opposite, parallel and coplanar with the line that connects the two signs ruled by Venus.

The Two Signs Ruled by Jupiter Meet at the Highlighted Line

The Two Signs Ruled by Mercury Meet at a Line that is Directly Opposite the Jupiter Line

The same occurs with the pair of Jupiter signs and the Mercury signs.  Also the same occurs between the two signs ruled by Saturn and the two luminary signs ruled by the Sun and the Moon - Leo and Cancer.  The results of these relationships relates to the Tree of Life, the middle pillar and the other two pathways that cross through Tiphareth, and these three planes intersect in the center of the dodecahedron.  The sun lies at the center of the dodecahedron with the twelve signs revolving around it.

The Two Signs Capricorn and Aquarius Ruled by Saturn

The Two Luminary Signs Directly Opposite Those Ruled by Saturn

Additionally, there are also nice repeating symmetries for the groupings of cardinal, fixed, and mutable signs.  This particular pattern can really only be seen in three dimensions, so in the example of the Kerubic signs below the signs are labeled per their location, though the faces are hidden from sight.

Symmetrical Arrangements of the Kerubic Signs, only Visible in 3D

Another interesting pattern is found by analyzing the colors on the dodecahedron.  Look at the primary colors: red for Aries, blue for Sagittarius, and yellow for Leo.  If you look at the Aries pentagon, all the adjacent faces have red in them.  The same applies to the other two primary colors.

Aries and Red, Surrounded Only by Colors with Some Amount of Red in Them

In a future blog I will provide instruction on how to create this and other Platonic Solids.
Happy Meditating!