Act 1
In the year 2000 I began thinking seriously about the geometry that could pass all my requirements and subsequently explain in a flawlessly coherent language each and every little thing that is a coexistent part of the universal reality. Two years later I became convinced that I had achieved that goal, and as a celebratory testimony of that belief I made a 90 minutes video I called "Let there be light". And if that wasn't dramatic enough, to top it all off I commissioned a professional to make 1000 DVD copies of it and launched it promptly on the Internet.
End of Act 1
Act 2
Let me begin this second act by asking you one question. How many corners do you think a cube has? If your answer is eight I'm sorry, but you are wrong. The cube, I say, has a total of four corners, and that fact is easily demonstrable. For a quick proof take a good look at the collage of pictures below.
A cube is a compound of five tetrahedra of two different kinds. One of those two kinds is that of a regular tetrahedron, which you can see in the pictures above as the brown coloured one, and the other four are of the kind that is formally named as a trirectangular tetrahedron, coloured white in the pictures above.
About those four white tetrahedra I want to tell you right from the outset, however, that from this point on I will never call them by their formal name. For two reasons. One, because when I discovered them I had no idea that they were even known--let alone formally christened--and two, because I truly believe that the name I had given them is much more adequate to the roles they play and is fundamentally quintessential to the greater story as such. From this point onward I will therefore call them as I always have: goniahedra (goniahedron). (Gonia means corner in Greek; moreover, in a loose translation it can also mean cornerstone, which has an obvious biblical connection.)
Now, if you have noticed that I said above "when I discovered them", and if you wonder what exactly did I mean by that, then let me make it crystal clear now that what I meant by that is exactly what the connotation of the word "discovery" says.
It all started with an educated hypothesis that at a fundamental level the geometry of spacetime had to be cubical, not spherical as one is most inclined to believe. Why cubical? For two good reasons, according to my mind. First, because of observations like the one pictured below.
I'm not referring here to the number of spikes (which usually amount to six) seen in photo and video recordings of the sun and other luminous objects, even though I have great doubts about the conventional view of them. What I'm referring to instead is that hexagonal shape the sun appears to reveal more often than not in photographs and videos, even though it undoubtedly is spherical in reality.
Then the second reason is the fact that the hexagonal shape is an extraordinarily common and versatile attribute of all material objects--more so than any other we've seen thus far.
Now, of course, the hexagon is a two-dimensional shape, a polygon not a polyhedron. On the other hand, however, of course a hexagon in two dimensions is a cube in three, and I do not think there's any need of me to say more than that on the subject.
So my journey began with an assumption that at the most fundamental level the universe had a cubical geometry. But there was also another factor working in concert with my initial assumption that eventually led me to the discovery I mentioned earlier, and that was the idea that the fundamental matter of my universe had to be created through a process of compression, or contraction, if you will, rather than one of expansion, as in the conventional understanding. Stay with me, everything will get much clearer in a jiffy.
In the beginning there was nothing, and that nothing was a void of infinity. No one knows for how long that situation persisted, but one thing has since become a real part of the universal history. Essentially, through the power of reasoning I can quite safely assure you that the initial period of nothingness lasted exactly until the time One realised that a state of infinite nothingness is a logical impossibility, which thus conclusively means that the state of nothingness lasted for a period of exactly zero time.
A state of infinite nothingness is a logical impossibility because in effect that state was spatially an infinite cube. And a cube is totally unstable, whichever way you choose to look at it. Consider, for instance, just the fact that in a cube with a volume equal to 1 (which of course means that each of its sides also equals 1) the distance from one of its corners to the centre is equal to half of the square root of 3 (which has a numerical value of 0.86602540378443864676372317075294...) while the distance from one of its faces to the centre is equal to 0.5. That situation in itself is more than enough a reason for immediate instability. But you may argue then that perhaps that state of nothingness had a spherical spatial extension, which would therefore circumvent the instability of the cube. Alas, a state of spherical extension demands the existence of a relative motion and a gravitational force, which are impossible phenomena in a state of absolute nothingness.
But the cube's main reason for being highly unstable is due to its being not one object per se, but an aggregate of different objects. To build a cube in your mind, for instance, just picture a regular tetrahedron (which should be quite easy to do) and then imagine laying on each of its four faces a goniahedron like the ones you have seen in that collage above. Easy, isn't it?
Well, the truth is that in spite of that apparent simplicity picturing that aggregate of tetrahedra in one's mind is nowhere near as easy as it seems at a first glance. In any event, it certainly wasn't easy for me when I first began thinking about it. But I'm digressing.
Due to its inherent instability imagine the contraction of the infinite cube of nothingness not as a real movement of objects, but simply as an increase in the potential pressure levels within the cube that occurs along the direction indicated by the red lines in the picture below.
Let us pause here for a few moments in order to take stock of how far we got thus far and also to keep you up to date in the order of the events that happened to me all these years ago.
As soon as I became aware of how a cube was an aggregate of two different kinds of tetrahedra I started thinking about what kind of quantifiable relationships existed between them. In doing that I did not want to follow the advice of a friend, who was rather proud of his apparent mathematical abilities and who--upon seeing real models of all five tetrahedra, which were beautifully made out of transparent acrylic marked geometrically with black lines along every intersection of faces--suggested, most pragmatically--I guess--to measure the relevant variables and then simply calculate what I wanted by using the relevant formulae. Instead I chose to try to deduce those relative relationships entirely abstractly from first principles. And in the end that turned out to be a relatively simple and straightforward endeavour, not to mention a much more satisfying one.
Thus, taking as the first base a cube with a side length = 1 and therefore with the same numerical volume, in short time I found that the regular tetrahedron at the centre of the cube accounted for 1/3 of the entire volume and that each of the four goniahedra accounted for 1/6 of it. Taken together, then, the four goniahedra had a total volume of 2/3 of the cube's volume, and the volume of the central tetrahedron was exactly twice that of one goniahedron.
Now that is neat and beautiful in some ways, but let me tell you that in fact is merely a minuscule side of the entire story. To show you why that is truly the case let us leave aside those numerical relationships and continue the discussion from the paragraph highlighted above in green.
It turns out that the contracting stage we discussed, and illustrated above, is just the first step in a literally infinite number of subsequent stages. To put things into the right perspective let us from this point onward describe in some detail what happens in each subsequent event to the first. Let's also convene to name each step in the contraction process as a relevant Level.
Thus, at Level 1 the initially infinite cube (with a volume of 1) decays into one central tetrahedron (with a volume of 1/3) and four goniahedra (each having a volume equal to 1/6).
At Level 2 the tetrahedron from Level 1 (which had a volume of 1/3) decays into four regular tetrahedra (each with a volume of 1/24, or 0.04166666666666666666666666666667) plus 8 goniahedra (each with a volume of 1/48, or 0.02083333333333333333333333333333) while the 4 Level 1 goniahedra decay into 4 regular tetrahedra (each with the same volume of 1/24) plus 24 Level 2 goniahedra (each with the same volume of 1/48).
Now, in order to get a visual understanding of what I was saying in the paragraph above I will first show you pictures of each decaying act and I will also explain in words what exactly happens in each particular case.
First let me show you what is probably the easier part, in which the Level 1 central tetrahedron decays into four regular tetrahedra plus eight goniahedra. See the two pictures below.
A regular tetrahedron, regardless of its particular level within the cube's infinite number of levels, is also an aggregate of the same two different kinds of tetrahedra that form a cube. In the two pictures above the four regular tetrahedra are conspicuously evident, but those eight goniahedra are not immediately so. Nevertheless, you can rest assured that those eight goniahedra are there, right in the centre of the tetrahedron, even though in the two pictures only three of them are visible. The fact is that those eight goniahedra are congregating into another type of polyhedron--an octahedron. See the two pictures below, with the first one showing a regular octahedron by itself and with the second showing how octahedra are distributed inside a Level 3 tetrahedron.
And now let us proceed to the next step, which is to explain how a goniahedron is also an aggregate of the same two kinds of tetrahedra. First let me drop below a couple of pictures depicting the aggregate of tetrahedra that makes up a goniahedron.
A good look at the first picture should be enough to give you a pretty solid perspective of how goniahedra and a regular tetrahedron combine to make up a goniahedron. Obviously two goniahedra are missing from the entire aggregate, but I am sure you have noted why I had chosen to omit them. In the second picture, on the other hand, I left out the regular tetrahedron and one goniahedron, giving you a perspective of the empty space that otherwise the two missing items would occupy. To give you a full perspective of how the six goniahedra plus one tetrahedron combine to make up a goniahedron have a good look at the two pictures below.
Then the second reason is the fact that the hexagonal shape is an extraordinarily common and versatile attribute of all material objects--more so than any other we've seen thus far.
Now, of course, the hexagon is a two-dimensional shape, a polygon not a polyhedron. On the other hand, however, of course a hexagon in two dimensions is a cube in three, and I do not think there's any need of me to say more than that on the subject.
So my journey began with an assumption that at the most fundamental level the universe had a cubical geometry. But there was also another factor working in concert with my initial assumption that eventually led me to the discovery I mentioned earlier, and that was the idea that the fundamental matter of my universe had to be created through a process of compression, or contraction, if you will, rather than one of expansion, as in the conventional understanding. Stay with me, everything will get much clearer in a jiffy.
The universal genesis, according to my own reasoning and understanding
In the beginning there was nothing, and that nothing was a void of infinity. No one knows for how long that situation persisted, but one thing has since become a real part of the universal history. Essentially, through the power of reasoning I can quite safely assure you that the initial period of nothingness lasted exactly until the time One realised that a state of infinite nothingness is a logical impossibility, which thus conclusively means that the state of nothingness lasted for a period of exactly zero time.
A state of infinite nothingness is a logical impossibility because in effect that state was spatially an infinite cube. And a cube is totally unstable, whichever way you choose to look at it. Consider, for instance, just the fact that in a cube with a volume equal to 1 (which of course means that each of its sides also equals 1) the distance from one of its corners to the centre is equal to half of the square root of 3 (which has a numerical value of 0.86602540378443864676372317075294...) while the distance from one of its faces to the centre is equal to 0.5. That situation in itself is more than enough a reason for immediate instability. But you may argue then that perhaps that state of nothingness had a spherical spatial extension, which would therefore circumvent the instability of the cube. Alas, a state of spherical extension demands the existence of a relative motion and a gravitational force, which are impossible phenomena in a state of absolute nothingness.
But the cube's main reason for being highly unstable is due to its being not one object per se, but an aggregate of different objects. To build a cube in your mind, for instance, just picture a regular tetrahedron (which should be quite easy to do) and then imagine laying on each of its four faces a goniahedron like the ones you have seen in that collage above. Easy, isn't it?
Well, the truth is that in spite of that apparent simplicity picturing that aggregate of tetrahedra in one's mind is nowhere near as easy as it seems at a first glance. In any event, it certainly wasn't easy for me when I first began thinking about it. But I'm digressing.
Due to its inherent instability imagine the contraction of the infinite cube of nothingness not as a real movement of objects, but simply as an increase in the potential pressure levels within the cube that occurs along the direction indicated by the red lines in the picture below.
Let us pause here for a few moments in order to take stock of how far we got thus far and also to keep you up to date in the order of the events that happened to me all these years ago.
As soon as I became aware of how a cube was an aggregate of two different kinds of tetrahedra I started thinking about what kind of quantifiable relationships existed between them. In doing that I did not want to follow the advice of a friend, who was rather proud of his apparent mathematical abilities and who--upon seeing real models of all five tetrahedra, which were beautifully made out of transparent acrylic marked geometrically with black lines along every intersection of faces--suggested, most pragmatically--I guess--to measure the relevant variables and then simply calculate what I wanted by using the relevant formulae. Instead I chose to try to deduce those relative relationships entirely abstractly from first principles. And in the end that turned out to be a relatively simple and straightforward endeavour, not to mention a much more satisfying one.
Thus, taking as the first base a cube with a side length = 1 and therefore with the same numerical volume, in short time I found that the regular tetrahedron at the centre of the cube accounted for 1/3 of the entire volume and that each of the four goniahedra accounted for 1/6 of it. Taken together, then, the four goniahedra had a total volume of 2/3 of the cube's volume, and the volume of the central tetrahedron was exactly twice that of one goniahedron.
Now that is neat and beautiful in some ways, but let me tell you that in fact is merely a minuscule side of the entire story. To show you why that is truly the case let us leave aside those numerical relationships and continue the discussion from the paragraph highlighted above in green.
It turns out that the contracting stage we discussed, and illustrated above, is just the first step in a literally infinite number of subsequent stages. To put things into the right perspective let us from this point onward describe in some detail what happens in each subsequent event to the first. Let's also convene to name each step in the contraction process as a relevant Level.
Thus, at Level 1 the initially infinite cube (with a volume of 1) decays into one central tetrahedron (with a volume of 1/3) and four goniahedra (each having a volume equal to 1/6).
At Level 2 the tetrahedron from Level 1 (which had a volume of 1/3) decays into four regular tetrahedra (each with a volume of 1/24, or 0.04166666666666666666666666666667) plus 8 goniahedra (each with a volume of 1/48, or 0.02083333333333333333333333333333) while the 4 Level 1 goniahedra decay into 4 regular tetrahedra (each with the same volume of 1/24) plus 24 Level 2 goniahedra (each with the same volume of 1/48).
Now, in order to get a visual understanding of what I was saying in the paragraph above I will first show you pictures of each decaying act and I will also explain in words what exactly happens in each particular case.
First let me show you what is probably the easier part, in which the Level 1 central tetrahedron decays into four regular tetrahedra plus eight goniahedra. See the two pictures below.
A regular tetrahedron, regardless of its particular level within the cube's infinite number of levels, is also an aggregate of the same two different kinds of tetrahedra that form a cube. In the two pictures above the four regular tetrahedra are conspicuously evident, but those eight goniahedra are not immediately so. Nevertheless, you can rest assured that those eight goniahedra are there, right in the centre of the tetrahedron, even though in the two pictures only three of them are visible. The fact is that those eight goniahedra are congregating into another type of polyhedron--an octahedron. See the two pictures below, with the first one showing a regular octahedron by itself and with the second showing how octahedra are distributed inside a Level 3 tetrahedron.
And now let us proceed to the next step, which is to explain how a goniahedron is also an aggregate of the same two kinds of tetrahedra. First let me drop below a couple of pictures depicting the aggregate of tetrahedra that makes up a goniahedron.
A good look at the first picture should be enough to give you a pretty solid perspective of how goniahedra and a regular tetrahedron combine to make up a goniahedron. Obviously two goniahedra are missing from the entire aggregate, but I am sure you have noted why I had chosen to omit them. In the second picture, on the other hand, I left out the regular tetrahedron and one goniahedron, giving you a perspective of the empty space that otherwise the two missing items would occupy. To give you a full perspective of how the six goniahedra plus one tetrahedron combine to make up a goniahedron have a good look at the two pictures below.
End of Act 2
Before concluding this first part about the fundamental geometry of my own universe let me share with you a handful of other issues that have been closely related to the crux of the matter.
For quite some time after my discovery of this amazing bit of three-dimensional geometry I was unable to personally establish if there had ever been anyone else before me aware of its existence. Today, with the benefit of hindsight, I'm pretty sure that to a large extent that happened because of my lack of formal education and experience in the field. Take just one simple and plain reality, which was the fact that for a good period of time I was blissfully ignorant of what those polyhedral objects were even called, so in many ways my searches online were entirely relying on looking for some rather accidental stumbling across pictures and images of them. Despite all those handicaps, however, one day I came across the work of a man called Bucky (Buckminster) Fuller, and it was in his book, titled Synergetics, that I saw for the first time someone else's dissection of a cube in exactly the same manner as mine. I then read Bucky's book with a great deal of difficulty (for reasons that anyone familiar with the man's work would readily understand) and with a significant pleasure I eventually realised that Bucky's personal approach and treatment of the subject led him onto a path completely different to mine (even though the great man, like myself, firmly believed that that was the geometrical road that had to lead toward a better understanding of the physical reality). For instance, in total contrast to my own view, Bucky Fuller took as the fundamental basis of his spatial geometry only the decaying aspects of the regular tetrahedron, not that of the entire cube, as I did. Moreover, to the best of my knowledge he never even discussed (or perhaps knew) the fact that the goniahedron--just like the cube itself, as well as the tetrahedron--was also an aggregate of those two kinds of tetrahedra.
Interestingly, as an additional page to my discovery of the three-dimensional cubic geometry, it so happened that a short time after I made yet another discovery that was related to the first one. Specifically, by using the two different kinds of tetrahedra that are the core of the cubic geometry I managed to construct a polyhedron which, unlike in the previous case scenario, to this day I haven't been able to find out if it had been discovered before me. (Believe it, or not!!)
In any event, below there are two pictures of the polyhedron in question, and to that end I take this opportunity to ask anyone who might be able to shed more light on this issue to drop me a line. The gesture would be greatly appreciated and certainly acknowledged publicly.
That's all for now. Take care. Hooroo.
