Sphere packing as a clue to "voice hearing" and "voice touching"

Memetic Analogue to the 20 Amino Acids as vital to psychosocial life? (Part #12)

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World views as "voices": The argument of the main paper could be understood as the challenge for "camp-us" of hearing the voice of "camp-them" -- even of being "touched" by others in some fruitful way. The presentation by Ernest McClain of a tonal array as a lattice (as depicted above) can be thought of otherwise when each tone, as a "voice", is fruitfully understood as engendering a world, or a world view (even a universe). In discussion of the array, McClain refers to the early traditional practice of pebble counting. It is from that recognition that star numbers and centered hexagonal numbers arise -- notably highlighting the number 37.

From pebble counting naturally follows the question of how spheres can be packed. Hence the extensive literature on sphere packing, namely with regard to the arrangement of non-overlapping spheres within a containing space. In three-dimensional Euclidean space, a particular concern is with close-packing of equal spheres. With each sphere understood as a world (or as a world view), sphere packing possibilities then offer intriguing pointers as to the extent to which the worlds can be arrayed such that each can "see" (or recognize") the other, or "hear" the other, or be "touched" by the other -- especially when each voice is held to be "equal". Succinctly stated, the organization of global society frames the question as to how many distinct voices can be recognized, how many can be heard, and by how many each is touched.

Packing possibilities and "kissing numbers": With respect to close packing, two simple arrangements correspond to regular lattices:

• cubic close packing (or face centred cubic) -- where the layers are alternated in the ABCABC sequence.
• hexagonal close packing -- where the layers are alternated in the ABAB sequence.

Many other layer stacking sequences are possible (ABAC, ABCBA, ABCBAC, etc.), in order to generate a close-packed structure. In all of these arrangements each sphere is surrounded by 12 other spheres. Especially valuable with respect to sphere packing, is the clarification of Buckminster Fuller's arguments offered by Amy Edmondson (A Fuller Explanation: the synergetic geometry of R Buckminster Fuller, 1992).

Somewhat ironically, given the sociopolitical connotations of "touching", the geometry of touching is defined and termed as the kissing number problem. This seeks the maximum possible kissing number for n-dimensional spheres in (n + 1)-dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space. In one dimension the kissing number is 2 (as characteristic of the quality of thinking of the "us and them" of "camp-us"). In two dimensions it is 6. The Wikipedia description lists kissing numbers up to 24 dimensions -- relating to the problem of hypersphere packing.

Beyond "us and them": Ironically again, this offers the challenging question of the dimensionality of an individual, otherwise too readily framed schematically as a point identity, a circle or a sphere. The sense of higher dimensionality emerges in the arguments of various authors (Ron Atkin, Multidimensional Man; can man live in 3-dimensional space? 1981; Antonio de Nicolas, Meditations through the Rg Veda: Four-Dimensional Man, 1978; Steven M. Rosen, Topologies of the Flesh: a multidimensional exploration of the lifeworld, 2006), as discussed elsewhere (Global Brane Comprehension Enabling a Higher Dimensional Big Tent? 2011; Hyperaction through Hypercomprehension and Hyperdrive: necessary complement to proliferation of hypermedia in hypersociety, 2006).

Framed in musical terms, McClain's argument extends from the 2-dimensional hexagonal array to what he discusses extensively in terms of a "magic mountain" of tones as associated with defining prime numbers. The form of this mountain is a challenge to comprehension, as he presents (and depicts) in 2-dimensions what is effectively (at least) a 3-dimensional array. He makes no reference to polyhedral sphere-packing arrays which could render any such form more readily comprehensible to the musically challenged.

Of particular interest is the 3-dimensional array which would correspond to the larger star numbers and centered hexagonal numbers (as noted above):

• In the case of the star numbers, this takes the form of a centered hexagram (six-pointed star), of which the prime numbers earliest in the series are highlighted in the polyhedral maps above, namely: 13, 37, 73, 121, and 181.
• In the case of a centered hexagonal number, this takes the form of a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The earliest in the series, which are primes, are again highlighted in the polyhedral maps above: 7, 19, 37, 61. They also include 91, figuring there, itself the product of two primes in those maps (7x13).

Global organization as "close packing": With respect to the sociopolitical preoccupation of this argument, a significant issue is whether a polyhedral form would help to clarify the distinction between "seeing" (understood as recognizing, or even "re-cognizing"), "hearing" and "touching" -- in the light of distinct packing patterns. This offers a language for (provocative) discussion of the question as to whether global sociopolitical organization should be explored as one of "closest packing" -- and the nature of the looser arrays to be considered as alternatives (with the criteria of seeing, hearing and touching understood otherwise).

Could this reframe "us and them" situations ("kissing number" of only 1?), degrees of relevance with respect to complex issues ("spreadthink"), or understanding of the constraints suggested by the Dunbar number (circa 150). The latter is the suggested cognitive limit to the number of people with whom one can maintain stable social relationships (as in social media) -- potentially to be generalized to the number of organizations with which any one of them can maintain fruitful relationships? (Maria Konnikova, The Limits of Friendship, The New Yorker, 7 October 2014; Drake Bennett, The Dunbar Number, From the Guru of Social Networks, Businessweek, 10 January 2013; Jacob Morgan, Why Dunbar's Number is Irrelevant, Social Media Today, 25 January 2010). Is this number also a constraint on democratic representation in national and international assemblies?

Hyperdimensional identity: All such constraints would need to be reconsidered in the light of, any "hyperdimensional" understanding of personal identity, as highlighted by Anil Ananthaswamy (The Man Who Wasn't There: investigations into the strange new science of the self, 2015), and as may be imagined (Being a Waveform of Potential as an Experiential Choice: emergent dynamic qualities of identity and integrity, 2013). Such argument can be extended to collective identity -- as with the so-called "international community".

With the cognitive constraint indicated by George Miller (The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review, 1956), is the challenge represented by the 20-fold/30-fold dynamics in the core of the polyhedral maps to be explored in terms of extraordinary forms of packing -- ensuring degrees of mutuality which do not lend themselves to verbal articulation? One such form is the Szilassi polyhedron -- topologically a torus, with seven hexagonal faces, each face sharing an edge with each other face -- perhaps to be recognized as an extreme form of touching. This is a characteristic shared only with the tetrahedron at the centre of the polyhedral maps above.

Animations of Szilassi polyhedron of 7 faces (4 types), 21 edges (12 types), 14 vertices (7 types) [totalling 42=2x3x7; product=2x3273] (paired shapes coloured identically; prepared using Stella Polyhedron Navigator)
Folding together of the two complement nets	Rotation of polyhedron

As an aid to cognitive clarification, the polyhedron can notably be used to configure the relationships between questions, as extensively discussed and illustrated separately (Mapping of WH-questions with question-pairs onto the Szilassi polyhedron, 2014; Potential insights into the Szilassi configuration of WH-questions from 4D, 2014). With respect to looking at an "other" -- namely the "us and them" challenge for "camp-us" -- the Szilassi polyhedron has been used to explore the variously discussed "13 ways of looking" (and listening) framed by the theme Anticipating When Blackbirds Sing Chinese (2014), in sections on:

These notably refer to insights to be derived from the pattern of polyhedra.

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