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# An eccentric view

One of the problems with combining the real world with information theory is that the real world gives the impression that it is continuous, while information must be granular. Of course quantum theory encourages us to view the world as actually being quantized, but there are always those nagging irrational aspects of mathematics to deal with. One of the things that really bugged me was the fact that no cubic mattrix can be constructed to hold the vertices of a dodecahedron. This not only seems odd, but it is really irritating when you try to incorporate the vertices of a dodecahedron into discrete mathematics, or combinatorics. This is exactly what I tried to do in finding a symbolic system to tease out the information in the genetic code. It just wouldn't work. Rather than give up, I tried to build the discrete system around the dodecahedron, one that only allowed quantized angles - no in-between angles. This is the eccentric view I created, one that I call quanum geometry.

Traditional geometry is the basis of all mathematics. It is an axiomatic system of logic using the concept of point space. Recursion of points produces more points, lines, planes and solids. If we conceptualize this system as the nesting of a single function, we can describe it as follows:

x = point
f(x) = line
f(f(x)) = plane
f(f(f(x))) = solid

However a point is seen as a dimensionless entity, and this is an exceptionally hard concept to illustrate. How does one graphically demonstrate a non-dimension? Of course years of casual usage makes us entirely comfortable with the concept, and we’ve become accustomed to seeing “points” on a printed page.

x = point f(x) = line f(f(x)) = plane f(f(f(x))) = solid This system quantizes distance, but does not quantize angle. Consequently, a continuum of angle is required, which produces a complimentary continuum of distance. The only angle intrinsic to the system is a right angle, so it is natural that the cube is the fundamental solid in traditional geometry. Points generate planes, so planes cannot have thickness. The quantum dimension of distance is derived from the difference between locations of two points, or two planes. If we adopt a protocol that only allows discrete points in three-dimensional space, then only discrete angles between lines can be generated. We know that certain angles can never be generated because certain solids cannot be generated in this space. For instance, a dodecahedron cannot be generated, because discrete points can never perfectly align; therefore, the angles in a dodecahedron will not be discretely quantized in this space. However, a cube is dual to an octahedron and contains two tetrahedrons, so they all can comfortably exist within the system. If points in such a way fill space, then these shapes should also fill it, and the duality of the tetrahedrons in the cube should be apparent. In fact, it is quite common for periodic crystals to use this type of symmetry as a blueprint for growth. However, it is not even possible to construct a dodecahedron in such a system. We could continue to add points to the lattice, and get closer and closer to a fair approximation of a dodecahedron, but this would hardly qualify as a “perfect” solid. This begs the question of how a logical system of cubic points could even deal with a dodecahedron, since it is now an imaginary object. How does an information system make imaginary choices based on imaginary logic? This suggests we try a slightly different approach.

Quantum geometry, like traditional geometry, is an axiomatic system of logic that is based on recursion. The fundamental element is a plane, and a recursion of this element produces a dodecahedron as the fundamental solid. Conceptualization of this system by nesting functions can be described as follows:

x = plane
f(x) = angle
f(f(x)) = point
f(f(f(x))) = shape
f(f(f(f(x)))) = solid
f(f(f(f(f(x))))) = all solids

In this system, all regular and semi-regular solids are generated; therefore, all angles within them are quantized. A non-imaginary logic can now be constructed around any solid. The first iteration quantizes distance, and the second iteration quantizes angle. Operations and notations that embody one will embody the other. No continuums are required, so only quantum operations are permitted. A complete re-examination of the geometric elements allows old terms to acquire new meanings.

x = plane f(x) = angle f(f(x)) = point f(f(f(x))) = shape f(f(f(f(x)))) = solid f(f(f(f(f(x))))) = all solids The most notable difference between the two systems of geometry is the inversion of points and planes in the hierarchy. In point space three points are required to form a plane, but in plane space three planes are required to form a point. Points continue to be dimensionless objects, but they no longer underpin the dimensions of length and angle. The next most notable difference is in the concept of angle. In point space, angle is not iterated; it is derived from lines. Specifically, the relationship between two lines in a plane generates an angle. In plane space the intersection of two planes generates an angle, and the intersection of two angles generates a relative angle in a plane.

Despite the length dimension of a plane, we can still conceive of a point as a dimensionless entity, and in the case of six intersecting planes there are 20 points. To help visualize the concept of a point in this system we will imagine the solid made by the intersection of the six iterated planes. Let’s call this the quantum dodecahedron; it is the atomic level of the system, and is itself not dimensionless. The distance between parallel faces is one plane thickness. These illustrations have reduced the thickness of planes relative to the quantum dodecahedron to merely demonstrate relative orientations not relative thickness. If we remove all but one plane from the intersection we can see the contribution of a single plane to the intersection. Adding a second plane back into the intersection produces the following relationship of two planes. The third plane in the intersection produces a point. There are two discrete segments to this intersection: an upper and a lower. In isolation there is no real need to differentiate between the two segments. However, if we imagine the system as comprised of not just single planes but infinite layers of planes, then there is a need to differentiate between the upper and lower portions of the intersection. For instance, lets imagine the purple planes that lie above and below this intersection. The purple plane above the intersection will see the point defined as Red – Yellow – Purple, and the plane below the intersection will see it as Red – Purple – Yellow. Therefore, the system presents a logical way to determine position of a plane relative to a point. The other issue that is less intuitive is the status of the remaining three planes in the intersection relative to the point. It is as if these three planes are not present at this particular point; however, they are present in the quantum dodecahedron. Similarly, these “missing planes” at a point maintain a logical relationship to the planes participating in the defining intersection of three planes. Therefore we have identified three logical qualities of every plane at a point. There is an upper quality, a lower quality and an absent quality. We can assign the following symbols to these qualities.

“+” = upper
“ -“ = lower
“ 0” = absent

Using these trinary symbols we can now construct a concise description of the twenty discrete points in a quantum dodecahedron. We can provide all of the relevant information in a table with 120 cells – a truth table if you will.  We can now use this information as a basis for building virtually any quantum shape by infinitely layering all of the planes. It is curious that dodecahedrons cannot fill space, but dodecahedral planes can. All the Platonic and Archimedean solids can be easily defined within this system. For instance, we know that eliminating points from the above truth table can create five cubes and ten tetrahedrons. We also can organize symbols to create an icosahedron and octahedrons. Scaling is also a simple matter of adding a global quantifier to the table, opening the door for combinations of solids of varying sizes.

There are many disorienting consequences of these changes in geometry, but all operations of point space can be performed in plane space. The opposite is not true. The operations of plane space are all performed with real numbers. In point space the construction of simple elements, such as circles and dodecahedrons cannot be achieved with real numbers alone. point space requires a continuum of numbers brought about by the absence of quantized angles. Numbers not included in the set of real numbers do not exist in quantum geometry. They are achieved through relative properties of the system. Even Pi becomes a real number, one that is relative to scale.

Because point space is an axiomatic system of logic, it has been used to produce matrices of logic. Quantum geometry can likewise be used to generate logic matrices different from traditional geometry. The matrices of plane space embody the matrices of point space; but again, the opposite does not hold. This becomes apparent when we translate the trinary information in the above table into binary information. To do this, we will map the above six-color quantum dodecahedron onto the four-color “genetic” dodecahedron that we developed earlier. It is readily apparent that the translation involves the addition of six new color combinations to the six that we have been using. Therefore, a translation of the above dodecahedron truth table will require an addition of 120 cells, bringing the total to 240. One might conclude that a binary view of trinary information would neglect exactly half of the information. A view of the genetic code as built around quantum logic would build shapes in plane space, as opposed to point space. Quantum logic can be viewed as a twelve-symbol binary system. This logic simultaneously delivers information about position and angle, because the two forms of information are inseparable in quantum geometry. Our application of the logic of point space has effectively hidden from us half of the information in the genetic code – the stereochemistry half. Here’s what the assignments look like when they are translated into a quantum geometry twelve-symbol format. The linear model views this table as a 4-64 relationship. From the context of a dodecahedron the code becomes a 12-120 relationship. If each of the four nucleotides is taken within context and its positional value is considered, then the code can distinguish between twelve distinct nucleotide symbols. From this perspective codons become syllables in words that mean peptide bonds, and there are potentially more than 64 syllables. This kind of thinking is not without precedent. Harold Morowitz, a master bio-physicist and expert in emergence, suggested to me a parallel to the Pauli exclusion principle in quantum physics. I had to go back to freshman chemistry to see what he meant. Pauli postulated that no two electrons could share the same quantum numbers. This means that each electron carries a unique quantum value, and in this way each electron is informative. In fact, according to Dr. Morowitz, matter itself is informative due to the pruning nature of Pauli’s exclusion principle. I can’t argue with that.

Rafiki is proposing a Pauli-like exclusion principle of its own. In the Rafiki code no two nucleotides can share the same value; therefore, twelve are required. Since only four brands of nucleotide are used in the actual code, the context of each nucleotide becomes essential, and the only context available is the nearby body of nucleotides. This is a model of overlapping contexts – it is a network model. The full set of codon permutations takes on a new hue, as illustrated in the following table: I have barely started playing with the logic of quantum geometry, but I am optimistic that further investigation will yield a more accurate logical mechanism of the genetic code, and therefore allow us to translate all of the information therein. It seems as though certain relationships are a natural fit, and a simple language is behind it. This language gives a better distribution of relationships than the second order binary approach currently in vogue. For instance, the data shows that only ~1.5% of peptide bonds are in the cis configuration. A quantum code would better support that language, as there is a one in twenty chance of two codons being total opposites in their planar configurations. Could this be the signal for a cis bond, and all the rest are trans by default?

Other patterns are intriguing, such as the relationship between the start signal Methionine, , and the three signals for STOP This cluster is directly opposite the “middle” of the code in the area of proline , which is the king of cis bonds. Could there be a correlation between mutations attempting cis bonds and translation termination? It is also interesting to note the non-Gamow-like distribution of the planes within each assignment cluster. This is another striking illustration of the spreading of assignments. Is there a primitive arithmetic utilized by this spreading?

I do not have definitive answers to these, or the more specific question, “what precisely is the language of the genetic code?” More data is needed. However, I feel that this symbol translation will be helpful in finding those answers. More importantly, we should begin to recognize the abilities inherent in a dodecahedral system. Not only does a dodecahedron tessellate in a sequential pattern, thus supporting the storage mechanism of a double helix, it also perfectly carries the leverage mechanisms for information translation. Therefore, the dodecahedron is a great choice for both genetic information storage and translation.

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