Potentially indicative patterns of prime numbers associated with polyhedra

Memetic Analogue to the 20 Amino Acids as vital to psychosocial life? (Part #5)

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The concern in what follows is the cognitive role of prime numbers in relation to patterns of psychosocial organization. How may they have a vital role in the emergence of patterns felt to be meaningful and integrative?

Prior to further discussion of polyhedra, within which the potential role of 37 may be further clarified, there is a case for recognizing the curious fact that 37 is the 12th prime number and therefore can be represented with the others either on an icosahedron or on a dodecahedron -- recognizing the morphing possible between them as duals, as illustrated below.

Mapping of first 12 prime numbers onto polyhedra
Dodecahedron (faces)	Icosahedron (vertices)	Morphing between duals

Wikipedia offers a List of prime numbers, acknowledged to be incomplete. Of particular interest is the variety of so-called "primes" of many named forms and types -- of which 100 are identified.

It is intriguing to note the continuing interest in whether one is a prime number. It is unclear whether this derives from the manner in which a prime number is defined, for purposes of mathematics, rather than any more fundamental criterion. It can be conventionally defined as:

• a number that can only be divided by one and itself with no remainder. When we talk about the divisors of a prime number, we are always talking about natural numbers (whole numbers greater than 0).
• a positive integer which has no factors other than 1 and itself. 1 itself, by definition, is not a prime number.

From this perspective, 2 is the first prime number -- although this is not the case in some varieties and types of primes. As such it is the only prime which is an even number. Whatever the degree of perceived ambiguity, this relates to the fundamental issues of the relation between 0, 1 and 2 from a cognitive perspective. It can also be seen as contributing to reflection on the ambiguity regarding numbers which are a "near" miss of some form of set completion (as with 63) or proximity to a regular prime, as can be seen below.

Within the set of primes, it is appropriate to note recognition by mathematicians (and others) of many curious facts regarding the number 37 as the 12th prime (see notably lists of 37 factoids and prime curios. There are so many such references to patterns associated with 37 that a major issue is how to distinguish those that are of relevance as being potentially indicative of fruitful pathways of transformation -- rather than verging into the obscure, recreational mathematics, number games or numerology.

Any method of detecting significance is subject to criticism. The process as a whole is somewhat reminiscent of what was termed moonshine theory by mathematicians through which vital correspondences were found enabling discovery of the so-called Monster Group of symmetry theory, as discussed separately (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007; Theories of Correspondences and potential equivalences between them in correlative thinking, 2007).

The indications, variously clustered, include:

• Uniqueness:
  • 37 is the fifth good prime the fifth lucky prime, the first irregular prime, the third unique prime and the third cuban prime of the form.
  • 37 is the smallest prime that is not also a supersingular prime.
  • Since the greatest prime factor of 372 + 1 = 1370 is 137, which is obviously more than 37 twice, 37 is a Størmer number.
  • 37 appears in the Padovan sequence, preceded by the terms 16, 21, and 28 (it is the sum of the first two of these).
  • The least number of 5th powers needed to represent every possible integer
  • 37 is the only two digit number in base 10 with the following property: The difference between the two digits equals the square root of the difference between the number itself and the least common multiple of the two digits.
  • 37 is the initial prime which is not a factor of the order of the Monster Group having cardinality:
      808017424794512875886459904961710757005754368000000000=
      2⁴⁶ x 3²⁰ x 5⁹ x 7⁶ x 11² x 13³ x 17 x 19 x 23 x 29 x 31 x 41 x 47 x 59 x 61.
    This finite sporadic simple group has generated much interest and constitutes a group of symmetries on 196883 dimensions.
  • The maximal number of regions into which a circle can be divided by eight lines.
• Sums and products:
  • 37 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 9 = 37. Note that 12345679 x 3 = 37037037.
  • 37x03 = 111; 37x06 = 222; 37x09 = 333; 37x12 = 444; 37x15 = 555; 37x18 = 666; 37x21 = 777; 37x24 = 888; 37x27 = 999.
  • 3x7x37=777
  • The sum of the first 37 primes is a Fibonacci number
  • 37 = 33 + 3 + 3/3.
  • 37 is the sum of the first five consecutive composite numbers; 4 + 6 + 8 + 9 + 10
  • 37 is the sum of the prime digits appearing in all previous primes.
  • 37=33+3+3/3
  • 37/3.7=3+7
  • 37=33+(3x3)
    
• Significance is attributed to reversal of the digits in a number, especially that related to primes -- therefore termed emirps, when the reversal gives rise to a different prime:
  • 37 is the fourth emirp, as indicated by the sequence: 13, 17, 31, 37, 71, 73, 79, 97 [of relevance to the discussion which follows]
• 37 is the first of three consecutive primes that have the same sum as their reversals.
• 37 and its reversal are the only primes formed from a pair of cousin primes
• 37 is the only known number n such that the reversal of n^prime(n) is prime.
• 37 is the only two-sided emirp whose reversal is also a two-sided emirp
• The smallest emirp p such that reversal(p + reversal(p)) is prime.
• The smallest emirp p such that reversal(p²) is prime.
• The prime p = 37, and its reversal q = 73, are the only known emirp pair such that p! + 1 and q! + 1 are both primes.
• pi(37) = 12 and pi(73) = 21.
• 3! * 7! minus 37 is a palindromic prime.
• 37 is the only two digit number in base 10 whose product, when multiplied by two, subtracted by one, and then read backwards, equals the original two digit number: 37×2=74, 74-1=73, 73 backwards is 37.
• Biology:
  • Mitochondrial DNA commonly found in most animals contains 37 genes.
  • Normal human body temperature is 37 in degrees Celsius
• Symbols
  • 37 can be represented as 100101 in binary notation. Note that the 0s are in prime positions and the 1s are in non-prime positions when read left-to-right.
  • The most well known multiple of 37 is the beast number (666). The first occurrence of 666 in appears on digits 2441, 2442 and 2443, when including the initial 3 before the decimal point, i.e. the whole series of 's digits. The sum of these is 7326 or 11 * 666.
  • 37 minutes is about a Golden Section of an hour.
• Games and packing
  • The French version of solitaire uses a board with holes for 37 pegs.
  • European Roulette is played using a wheel containing 37 numbered slots (1 to 36, plus a 0).
  • There are 37 nonominoes (a polygon in the plane made of 9 equal-sized squares connected edge-to-edge) with holes.
  • 37 is the number of centered hydrocarbons with ten atoms.
  • 37 is the number of hydrocarbon structures that can be drawn (excluding stereoisomers) for 4 carbons
  • 37 is a centered hexagonal number (a "hex number"), and a star number (as noted below)

As a star number, 37 is a centered figurate number that represents a centered hexagram (six-pointed star), such as the one that Chinese checkers is played on.

Star number (reproduced from Wikipedia)

A centered hexagonal number is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice.

Centered hexagonal number (reproduced from Wikipedia)

Which of these factoids are of potential significance, and which are indeed mere curiosities, if not trivia?

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