My New Dodeca-Toy and How It Potentially Aligns My Futures and Pasts

           As I stated in RCP2, when I was around 4 or 5 years old I had a way of looking at objects dually, both from the outside, as we all do, and from the inside at the same time, or as inverted negative space. This made doing a 3D puzzle I had from any angle as easy as breathing. It was not literally 'seeing through' things, but understanding positive and negative aspects of objects equally.
           Flash forward some 30 some odd years later: while writing 2D 3D 4D 5D Thinking Made Simple, I had written myself into a box. The discussion of the characters came around to how from a 4D perspective it would appear to be able to 'see through' 3D objects, (seeing all six sides of a cube for instance, equally distant away in front of you, and having that portrayed as somehow seeing through them), as several books on the subject wrongly infer. Yet it made no logical sense, seeing 'through' solid matter. However, I could not pinpoint the cause of the 'paradox' of how both points of view were seemingly correct, how it would and would not be 'seeing through' 3D objects, both at the same time. That is, until I thought of inversion, or curved spaces.

           It is possible to see all 6 sides of a cube at once, without seeing 1mm through its walls, and we do it everyday, from the inside. Sit in the corner of any room and you can see all 6 sides spread before you. 4D sight, in thought examples, is like bending slightly 3D space, or as stated in the book, from slightly outside of it like tilting a 2D plane to see inside the insides and outsides of squares at the same time which you cannot do while 'in line' with that plane, and 3D objects would disappear from such a perspective, tilting the view out of a 3D triplane, if they did not possess some 4th dimensional value.

           So it would be from such a tilted perspective looking at what would have to be 4D objects differently than we see them, our seeing only the 3D aspects of them, but not in any sense would it be 'seeing through' matter. It was all so simple at that moment and the 'paradox' just vanished. This does not get into all of the details, though that chapter you can see by clicking here. But it occurred to me then thinking over the problem again and again while sitting at my “thinking spot,” on Maui that I suddenly realized this, or sort of, without exactly thinking of it in these terms yet.

           “OK, I will need a cube and a ball,” I thought excitedly, and literally jumped up and raced to buy ones as quickly as possible. I even tried to buy some on the way home on my bike, but was disappointed at each store I stopped at, and had to wait until I could get to my car. I had to drive all the way to Kahului (very fast, and without delay, and to buy toys! :-) because I could not find a ball the right size which I could draw on, nor a cube either.

           The only cube shaped object I could think of was a Rubik's Cube, which I bought for about $5 at K-Mart. That was very good for conceptual purposes because it had different colors on it for each of its sides. For the ball, I bought a blue Squash ball and proceeded to draw 6 lines on it, without really understanding why at the time. I only knew that I had to try to put the cube on the ball, which was actually how I was trying to look at it. The first line was drawn from an arbitrary point marking a north pole to what would be the equator, 1/4 of the diameter in length. Then another quarter of the diameter along the equator, and making a 90 degree corners all around until coming back to the top. I knew that was what I had to do, but not why. Then I studied it to understand how that related to the problem.

           My journey to Europe to University studies and to a political asylum hearing, both equal reasons for going before I left, began about a month after that. The original intent was to do the asylum hearing off the bat, when it actually was possible to be granted, and then go to school after that. But that would have delayed my studies for at least a year. While in Europe, I then saw that it was possible to go to school first, though unlikely, but workable, so the asylum thing did not come until the end. I only mention that here because all through my travels across Europe from France to Russia and back, probably the most cherished possessions I had with me, and always nearby in sight, were a ball with lines drawn on it and a Rubik's cube. I added a third toy to complete the trio, a pencil sharpener globe I bought in Lithuania for about a dollar. (There actually was fourth toy, a larger very heavy ball I used to draw shapes on and measure distortions in angles caused by the curvature but it was too heavy to bring along with me as my luggage was already too expensively overweight.)

           With my thinking still stuck on inverting 3D space in Lithuania, I 'discovered' the idea of the 'anti-pole' though it was already known to many others. It is even possible I may have been exposed to the concept before and had forgotten about it, but my thinking at that time necessitated understanding the concept and would have come up with it even if no one else had before. Once understanding the anti-pole concept, the Dual-Earth sculpture idea was pretty much immediately self-evident.

           The easiest way to understand it, curved 3D space, would be to have one earth you would see normally, plus another outside of it, the same one of course, inverted backwards opposed to it, and every single point on a 'ball' or curved 2D plane of points half way in-between the Earths would be exactly the same spot. I was pretty full of myself for figuring that out, and since when leaving Lithuania I would be walking into a relatively unknown and potentially very precarious situation, I posted everything I figured out, all of what would later be called the 5D notes done up until that date (Part 1), to two physics newsgroups in January 2004 before I got on the plane, and figured anyone trying to figure out what I was doing would at least have something interesting to read. If any of it made any sense, someone somewhere would figure it out. Much of what I was doing was inter-connected, at least to me at the time.

           I also made a model at that time in Lithuania of the Dual-Earth sculpture with a fishbowl turned upside down and over a small glass plasma globe in the center, and I painted the forward and backward Earths on the glass surfaces of each. Since I had to leave behind a large globe I had bought to map them out, the first globe I had bought there to visualize the idea, the small pencil sharpener, became my third small toy memento of this dimensional quest for knowledge.

           So with my 3 'toys', a Rubik's Cube, a blue Squash ball with 90 degree lines drawn on it, and a small globe always prominently displayed on my desk or on a shelf in my dorm rooms (I still had the fishbowl and the plasma globe Earths but they were usually out of sight in my closet), I studied them, played with them, and delved more and more into the curved space ideas which were necessary to be thought out first before getting into the 4D parts of the book.

           When I had time many months later, over the summer of 2005, I finally wrote those curved space parts as well and tried to wrap the whole thing up as best I could. Literally in a sense besides trying to accomplish much more important things, what I also was doing was trying to buy more time to finish up that stupid book and get a better understanding of 4 dimensional thinking. It was intuitive both in wanting to know it, and in learning or teaching myself about it, at the same time. It had to come out. Life simply was just trying to find the right circumstances to have time to 'learn' about it, though I knew I already knew it. Writing the 5D notes actually let me keep working on the same ideas without obsessing on the project as a whole, while doing the other more important things.

           My new latest 'toy' could not be more perfectly suited to my thinking along such lines, a clear solid crystal dodecahedron. Being clear, you can see through it to the opposite and opposed sides. Because of that, it is easy to imagine seeing it from the outside and inside at the same time, perfectly suited to how I might need to think to continue the project if the world, and my world, settles down to be less confrontational and longer lived. Playing with shapes and words is a luxury at the moment while the more important things are still a priority.

           While working as a non-scientist and non-mathematician on Fermat's Last Theorem, I kept coming across relationships all dealing with the numbers 3, 4, and 5. Mathematics is the ultimate seeing different parts of the Elephant, as the blind men example goes with each feeling a different part of the Elephant and each thinking it is something completely different. Mathematics is as complicated as you want to make it to come up with the most ridiculously complicated ways of saying the simplest of things (often the same relationships in different ways), especially when you don't know what you are really doing.

           To understand Fermat's Last Theorem my way, I made a list of all the possible squares that fit into the pattern, 'A' squared plus 'B' squared equals 'C' squared. Then I saw these fit into a 2 dimensional pattern or array, what my sister who is a math professor referred to as a 'table of squares'. Then I tried to look for relationships in the numbers in-between the ones in the table, or the negative spaces of ones that did not fit into such patterning, the gaps. Then I thought to make a three dimensional model with a similar 'table of cubes', but found you could not do that. Then I tried to figure out why it can work, but seemingly only for 3, 4, and 5. (3 cubed plus 4 cubed plus 5 cubed = six cubed.) (I don't remember if I found any others, but believe there were not any others an could not be any others, though it came close sometimes. Finding another 'A' cubed + 'B' cubed + 'C' cubed = 'D' cubled would have made for a new pattern to overlay the table of squares, and thus a new pattern of the relationship between the 2 and 3 dimensional arrays). It (3 cubed plus 4 cubed plus 5 cubed = six cubed) overlaid the table of squares perfectly in a 3 dimensional sense, (3 squared + 4 squared = 5 squared, again 3, 4, and 5, just some of the many ways 3, 4, and 5 kept coming up) but you can't build a whole new table from one example, though it could be related to the table of squares perfectly. Since Fermat's Last Theorem is also trying to explain a negative, far beyond my abilities then and now, why nothing else can work, I limited myself to trying to see and understand why it did work when and how it did, for different dimensional levels, as represented by the numbers of integers or axises in the question, how many you were trying to compare or add up.

           Though if I remember correctly that there are 5 Platonic solids, 3, 4, and 5 are the ones I am most interested in. 3 as represented best by the tetrahedron, or 4 sided pyramid; 4 as represented by a cube; and 5 as represented by a dodecahedron. But then it stops. Why?

            I have spent a lot of time pondering the tetrahedron and find it a much more logical way of dividing up 3 dimensional space than the 'quadrants' or squared based ones we use, just as triangular grids make for much better maps or ways of dividing up curved 2D planes such as the Earth into equal sections, as has been shown by Buckminster Fuller.

           But as I have noted, we live in cubes and are most comfortable thinking in terms of poles and 2 axises, North-South and East-West, for 2D maps, not three axises on 2D surfaces, which is a shame as it is more developed, more easily triangulable, and more fertile soil for an easier understanding of topological concepts and relationships.

           So other than going always 'down' to 3 (triangular) for insights, which many more are to be found I have not considered yet, since I myself have dealt mostly with cubical and tesseractal modeling for 2, 3, and 4 dimensional thinking, the other 'direction' to go is obviously 'up' to the last 'discovered' Platonic solid, dodecahedral relationships. Obviously yet again I lag behind everyone else in the mathematical and topological fields because I have read articles speculating that the universe instead of being evenly curved as a hypersphere, some speculate instead reverses itself in a dodecahedral image.

           Why they think that, I have no idea or have forgotten. Perhaps some repeating numbers in mountains of data. I prefer to deal as the Ancient Greeks did first and foremost in simple logic as manifest in simple repeating geometric shapes. Why only 3, 4, and 5? What is the most significant relationship between those numbers and 3 dimensions? There may be an infinite number of answers to those questions or only one. But then, if only one answer, then an infinite number of answers to that question: why only one answer? Such is the problem with trying to understand numbers. They can seem to explain everything while in the end saying absolutely nothing.