About the fundamental geometry of my universe. Part 1.

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.

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.

About the refraction and dispersion of light in my own universe. Part 1.

Let me begin this post by showing you the first out-of-the-blue email I received from Dr. Markus Selmke on December 5, 2018.

Dear Remus,

I can’t resist. I just read your latest blog post out of a mixture of sheer incredulity and fun.
https://remusporadin.blogspot.com/2018/11/on-rainbows-part-7.html

You make a point about the “claim” of most people who have spent some thoughts on the rainbow phenomenon that the rays hitting a raindrop are almost parallel. Now, you see, any good textbook will have this limitation, i.e. referring to rays which are parallel for all intents and purposes / for all relevant calculations. It is a simplification that is justified by the fact that its incorporation would not alter the result in any meaningful way. As for every problem in physics for which an understanding is sought after, some simplifications are required. A study of the rainbow will not start with the nuclear fusion providing the energy for the light emanating from the sun. 

No person in his right mind would state that the sun’s rays are perfectly parallel. After all, the sun is a light source of finite extent (roughly a spherical surface, the sun’s photosphere). Roughly, seen from a distance, it emits like a point source in all directions. It is the distance of the sun relative to the lateral extent of a raindrop which leads a mathematically-versed person to the conclusion (via basic trigonometry) that the maximum angle subtended by two rays will be about 2*ArcSin((R/2) / d), i.e. 4 x 10^(-13)°, i.e. less than a millionth of a millionth of a degree.

https://www.wolframalpha.com/input/?i=(180%2Fpi)*2*ArcSin(1mm+%2F+(2*(distance+earth+sun)))

I, and most other people, feel comfortable calling that practically parallel. If you should be able to measure the non-parallelism of that order of magnitude please file a patent. The reference to the railroad is indeed inadequate if one tries to explain or compute this (minute) non-parallelism or practical paralelism via an argument based on perspective. But it is appropriate in the typical context, which is to explain the apparent everyday experience of crepuscular rays which seem to radially diverge from the sun despite the small non-parallelism of the actual rays (this time, set R = distance observed at the horizon, i.e. a few tens of kilometers, which is small compared to d=distance sun earth = 1.4*10^8 km).

Also, you seem puzzled for not directly seeing dispersion in a sphere. In fact, if you look close at the pictures you posted, you do see dispersion: the edges of the ray bundles show reddish and bluish colors. They occur because of incomplete addition, while the complete superposition causes the inner of the refracted bundle of rays to be white (=well, whiteish, i.e. the color of the sun). It is a typical phenomenon also observed in lateral dispersion, e.g. es seen for refraction through planar wedges of glass.

BTW: Transparent balls are lenses. Raindrops are lenses. Glass-Spheres are commercially available and used in many applications, https://www.edmundoptics.com/f/N-BK7-Ball-Lenses/12436/. The physics, including the paraxial focal length, is fully compatible with classical rainbow theory. So I’m not sure why you are eager to construct the next conspiracy here? Also, lenses do show refraction and dispersion. Please take a close look at any given picture of a sharp edge taken with a digital camera, best at low f-number (large aperture = far from paraxial).

Please, read a physics book in full. Other people have spent time thinking about nature as well, it is not just you. In fact, as I have pointed out before, the detailled understanding they have developed in a community effort and method called “science” has brought you the very laptop / PC you sit in front of.

Then, an hour or two later I received the message below from the man.


...damn, I should have spent two more minutes on my quick mail… my mistake indeed. But the main point of course remains:

the finite but small non-parallalism is described in both situations by the same geometry, with R=radius of the sun, d=distance sun to earth, max angle 2*ArcSin((R/2) / d). I should have drawn the text-book sketch I had in mind and I would have avoided my blunder. My bad. Back to the point: The parallelism is negligible for the main characteristics of the phenomenon. The fine details do require consideration of the angular diameter of the sun (0.5°) which smears out any parallel-ray bundle computation. Alternatively, Monte-Carlo simulations like those done by MiePlot (vectorial EM wave theory-based), if I remember correctly (http://www.philiplaven.com/mieplot.htm), do allow this details incorporation. But understanding the rainbow does not require non-parallel rays to be considered, parallel ray bundles work just fine to produce the rainbow caustics (i.e. the various orders). In fact, using widened collimated (arbitrarily parallel, again not perfectly, though, since there is nothing like a perfect parallel beam in nature, just like there is no perfect electromagnetic plane wave) laser beam, you could get the caustic as well...

Before anything else let me say that there are a number of very good reasons for which I chose to show you in full the rather long and 'slippery' email above. Additionally, I ought to also mention that all of those reasons will become manifestly apparent by the end of this post (albeit, not in the order that they've been laid down by Dr. Selmke in his email).

Now, let me first direct your attention to the following paragraph from Dr. Selmke's email.

Also, you seem puzzled for not directly seeing dispersion in a sphere. In fact, if you look close at the pictures you posted, you do see dispersion: the edges of the ray bundles show reddish and bluish colors. They occur because of incomplete addition, while the complete superposition causes the inner of the refracted bundle of rays to be white (=well, whiteish, i.e. the color of the sun). It is a typical phenomenon also observed in lateral dispersion, e.g. es seen for refraction through planar wedges of glass.

Starting with the first sentence in the paragraph I would like to confess that I was, and still am, in fact, puzzled indeed. However, not for the reason contained in the sentence. Not at all, let me make that abundantly clear. Instead, the real reasons for which I was/am puzzled are, firstly, the obvious refusal--or perhaps omission--of Dr. Selmke to either see or make any mention at all that the truly crucial matter of fact is the conspicuously evident reality that a beam of light is basically 'sharpened' inside a sphere, not 'blunted' as the conventional understanding undeniably proclaims. (And that is the only thing I addressed in the post mentioned by Dr. Selmke, BTW.) Secondly, what truly puzzled me was the blatant 'spin' that Dr. Selmke chose to use in a premeditated attempt to probably appear unfazed by my conclusions and also, possibly, to somehow show that the conventional belief is still in just as much control as ever in the matter at stake.

For those who'd perhaps want to get a clear and objective picture of what I have thus far discussed above I'd recommend a re-visitation here. For anybody else, and also for the benefit of all, I will display below a couple of pictures that are most relevant to the issues concerned.

In fact, if you look close at the pictures you posted, you do see dispersion: the edges of the ray bundles show reddish and bluish colors. They occur because of incomplete addition, while the complete superposition causes the inner of the refracted bundle of rays to be white (=well, whiteish, i.e. the color of the sun). It is a typical phenomenon also observed in lateral dispersion, e.g. es seen for refraction through planar wedges of glass.

I brought forward the next part of the paragraph we're scrutinising at the moment for your convenience. (There's almost nothing more annoying to me than the manner in which all conventional papers are published, meaning that invariably you are consistently forced to leave the page you're reading in order to look at some diagrams or figures, then forced again to return for a little while until the next piece of reference comes into play...a.s.o.a.s.f. to the last word. Ha!) I'll ask you now to read it once again, with care, and then to take once more a close look at the beam of light that runs inside the 'sphere' on the refracted (or bent) pathway it has been forced to follow (see the picture below).

Now, if you have looked at the part of the beam of light carefully you would have been most likely able to discern (albeit, barely) that indeed the two edges of the refracted beam appear to be formed by some very thin lines of two different colours: one "blueish" (running along the upper edge) and the other one "reddish" along the lower edge of the beam. To help somewhat in making the whole issue a little clearer I have added to the original picture the two-coloured arrows on display. (Some of you may have noticed however that the upper arrow seems to be rather more 'violetish' than "blueish". I did that for a very simple reason, which is that the actual colour of the edge itself is rather more 'violetish' than "blueish". This fact will become more evident shortly.)

I must tell you now that when I first read the 'explanation' that Dr Selmke spat in my direction--with the apparent conviction of either an absolute prophet or that of an undeniably complete moron--I found it pretty much impossible to believe. Are you going to ask me why? Really? Okay.

Firstly, because even if one were to accept the explanation given without any questions (and I can assure you that the explanation given is very far from being in any way thoroughly evident and truly unquestionable) the simple and clearly obvious fact is that the so-called dispersion colours that are edging the beam of white light are displayed in the wrong order! In effect the reddish edge is where the blueish one should be, and vice versa.

Secondly, because those thin coloured edges of the beam of white light that is bent inside the 'sphere' are inherited attributes from the incident white light that strikes the refracting 'sphere'! Look again at the picture in question, if you need to.

But, by a long shot, the most far-reaching aspect of what we've seen and discussed on this topic is the reality that in absolute spite of the fact that both those coloured edges of the white beam are clearly in the wrong place to give the results that the conventional theory ascertains, the incredible thing is that those results still appear to be thoroughly fulfilled nonetheless! Think about that, carefully, for believe me it is worth doing it.

At this point I'd like to show you a few screenshots I have taken of the paper from which the picture above, and a host of others that I have shown in some past posts, have been extracted. As you will see I have highlighted some of the more significant parts of it and I hope that some of you will take the time to read them. Following that I will also show you enlarged pictures of the four different experiments that were carried out by the authors of the paper (with the fourth one providing a most interesting perspective into one of the conventional tenets of the rainbow theory and mentioned by Dr Selmke in the email we've been discussing in this post).

Let me now show you two enlarged pictures of the the experiments depicted in pictures 13 and 14 above. You have already seen the enlarged rendition of the first experiment, which is shown above in figure 12, and we have discussed the issues raised by Dr. Selmke. As you will see below the same state of affairs is conspicuously evident in the second and the third experiment.

Finally I want to direct your attention to the fourth experiment, which is shown on the relevant screenshot in Figure 15. In that particular experiment you would have seen (if you read the highlighted segments in my screenshots) that the two authors had used direct sunlight, instead of those tungsten light bulbs they had used in the previous three. According to the authors the divergence of the solar rays (which has a value, according to the reigning theory, of 32') had been carefully measured and monitored in all experiments, including the one in which real sunlight was used. In regards to this fourth experiment, for example, they specifically mention the following:

Rotating the apparatus in a way that the light does not enter the acrylic cylinder the path of the beam is clearly visible and its divergence can be measured. The values obtained varied between 28'±10' and 36'±10', in agreement with the accepted value of 32'.

Now, before getting into the matter that I want to conclude this post with I must tell you once again that when I read this paper for the first time, I was even more shocked than in the other incident that I mentioned earlier about Dr. Selmke's pitiful remark about those reddish and blueish edges that 'proved' my 'obvious' misunderstanding of dispersion. I was even more shocked because what to me was a very simple, a most obvious and an embarrassingly monstrous blunder, to those who have been given the role of teaching and leading the humanity to new levels of progressive development, understanding and intellectual evolution the frightening reality of not being aware of even some of the simplest and easiest bits of knowledge that one could possibly become aware of today must invariably be shocking to any living soul of this world! But let's see what you think after reading the last bit of this post.

Let me first show you an enlarged view of that fourth experiment, which was shown in Figure 15 of the last screenshot above.

Now, can you see what is the most obvious difference between this picture and the other three you have seen? It's not the number of extra rainbows, of course. It's that somewhat triangular protrusion that extends from the centre of the acrylic cylinder towards the rim of the apparatus. Do you know what that is? You should, if you're a physicist. Do you know how it got there, how it came to be? You should, if you're a physicist. Does it bother you that it has no explanation whatsoever in the paper that shows it? It should, regardless of what your occupation may be.

That 'protrusion' is a visual manifestation of the extension of the optical field of the acrylic cylinder of the apparatus described in the paper, which for all intents and purposes is a converging lens. At the tip of the 'protrusion' is the focal point of the lens. There are many more important aspects of this optical feature of a lens and in due time I will discuss them further. For now, though let us direct our attention to how this particular feature of a lens came to materialisation in the picture above.

There is one way and one way only in which the particular optical field of a lens that is seen in the picture above can become visible: by passing of a beam of light through the centre of the lens. And that's not all either, for there is another uncompromising requirement that needs to be working at the same time with it: the beam of light that passes through the lens along its central line must be highly divergent. By "highly divergent" I mean a beam with a much greater degree of divergence than that of the conventionally accepted value of 32'. This fact is easily demonstrable, and I will do it in a moment. For now though let us remember a couple of very important factors that are of relevance to the matter at hand.

First, let us not forget that according to the authors the experiment had been conducted in such a manner that no light was allowed to enter the acrylic cylinder. Second, let us remember that according to the authors--and the conventional understanding--the divergence of the lights used in all four experiments was equal to, or less than, the conventionally accepted value of 32'. It is also worth remembering that those conventionally accepted divergent beams are in all cases the incident beams that extend in all four cases from the slit denoted S to the particular point where they enter the acrylic cylinder in each experiment.

How could one then explain the strange display seen in the picture above? you may ask.

Easily, I say in reply. Have a look at my rendition of the picture above, first.

Observe how the black line that I inserted in the original picture extends in a direct line from the tip of that unwanted 'protrusion' to the middle of the same slit that was used to direct the conventionally divergent beam of sunlight towards the top of the acrylic cylinder. The conclusion therefore is unavoidable: the conventionally accepted value of the solar ray's divergence must be wrong. (As I personally believed, btw😉) I will refrain from saying any more than that now, but I can promise you that I'll revisit this topic at some future point.

Finally, let me show you three pictures of mine that are highly relevant to the topic we've been discussing above. The first of the three shows what the optical field of a spherical lens looks like when it is created by what I'd called earlier a highly divergent beam of light. The second shows what the same field looks like when it is created by a beam of light with a divergence similar to that that is conventionally accepted. The third one shows what the field looks like when it is created by a beam of light with a divergence similar to that that is conventionally accepted when it grazes the top of the lens, like in the picture above.

That's all for now. Take care and think carefully before believing anything of any body, at any point in time.