Configuring the 64 disciplines of mathematics as a 64-edged drilled truncated cube

Meta-pattern via Engendering and Navigating Pantheons of Belief? (Part #9)

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The Mathematics Subject Classification (MSC) at its highest hierarchical level has 64 mathematical disciplines labeled with a unique two-digit number (as noted above). This could indeed be understood as framing the pantheon of mathematical experience. The preference for a pattern of 64 would seem to be as unexplained as that for other checklists, whether more or less fundameental in implication.

Perhaps only coincidentally, the 64-fold organization is especially suggestive in mathematical terms of possibilities of experimenting with more appropriate configurations of the disciplines of mathematical experience. The 64-fold pattern is of course fundamental in the following respects, as noted by Wikipedia:

• It  is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers. It is also a dodecagonal number and a centered triangular number.] 64 is also the first whole number that is both a perfect square and a perfect cube.
• 64 is the number of codons in the RNA codon table of the genetic code
• There are 64 convex uniform 4-polytopes (namely 4-dimensional polyhedra) clustered into 8 types, including the 6 regular convex 4-polytopes (but excluding the infinite sets of the duoprisms and the antiprismatic prisms)
• The current information-based civilization is now primarily dependent on 64-bit computing, namely a 64-bit central processing unit (CPU), as well as arithmetic logic unit (ALU) architectures based on processor registers, address buses, or data buses of that size.

In the quest for polyhedra suitable for mapping a 64-fold set of distinctions, it is therefore somewhat curious to note that the 64-edged drilled truncated cube is unique in enabling such a mapping in 3DÂ (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015). Other polyhedra have that characteristic but are either complex compounds of simpler polyhedra or 3D aspects of 4D polytopes -- both posing challenges to their comprehensibility for mapping purposes.

The drilled truncated cube can therefore be used in a simple exploration of how the realm of mathematics can be coherently configured in a manner distinct from a checklist of disciplines which does so little to honour the fundamental significance attributed to the mathematical experience.

Understood as a pantheon of a particular form, associating the articulated disciplines of mathematics with the features of that form is then helpful in recognizing how other patterns of cognitive modalities of similar complexity could be ordered in this way.

Exploratory mapping of 64 mathematical disciplines onto 64-edged drilled truncated cube (original discipline names slightly edited to reduce length in order to facilitate mapping)
32 faces rendered transparent	Only 4 octangular faces rendered transparent
Animations developed using Stella: Polyhedron Navigator

No attempt has been made in this preliminary exercise to position the 64 mathematical disciplines on the polyhedron in a manner which might reflect to a higher degree their relationships. Various visualization techniques could be considered for that purpose, including colour and animation. As noted above with respect to citation links between papers in different disciplines, the framework could be used to explore the connectivity of those disciplines in quest of the nature of a pattern that connects -- potentially in dynamic terms.

Logical implications? Exploration of meta-mathematics, and the mathematics of mathematics (as mentioned above), have tended to highlight the role of symbolic logic. In this sense the form of the drilled truncated cube is itself interesting in its resemblance to the structure of the 4D tesseract of significance to the configuration of the sixteen Boolean functions of logic, especially with respect to studies of oppositional geometry -- presumably of relevance to relations between modalities in any pantheon.

Suggestive visual correspondences to configurations of relevance to logical connectivity
The Logic Alphabet Tesseract - a four-dimensional cube (see coding). by Shea Zellweger	Tesseract animation	Topologically faithful 4-statement Venn diagram is the graph of edges of a 4-dimensional cube as described by Tony Phillips	Embedding of the Borromean ring logo of the International Mathematical Union within a drilled truncated cube
Diagram by Warren Tschantz (reproduced from the Institute of Figuring) .	by Jason Hise [CC0], via Wikimedia Commons	A vertex is labeled by its coordinates (0 or 1) in the A, B, C and D directions; the 4-cube is drawn as projected into 3-space; edges going off in the 4th dimension are shown in green.	See Wolfram Mathematica animation of the logo

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