Applied Mathematics and Computation 120 .2001) 211±225 www.elsevier.com/locate/amc Tensional computation: further musings on the computational cosmography q J.F. Nystrom Department of Electrical Engineering, University of Idaho, Moscow, ID 83844-1024, USA Abstract Universe operates in pure principle only, wherefrom we can choose to view Universe as an emergent computation. The proposed method for this most basic of computations is the computational cosmography, wherein a system of interacting deformable poly- hedra utilizing the geometric primitives of Fuller's synergetic geometry coordinate and evolve within an isotropic vector matrix. A gedanken experiment involving gravity on a two-dimensional lattice is also discussed, as are some consequences associated with this computational cosmographical viewpoint. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Cellular automata; Synergetic geometry; Computational cosmography 1. Introduction Universe operates in pure principle only. If successful and accurate micro- scopic models of various aspects of Universe are to be built into a computa- tion, said computations should utilize, or exhibit properties similar to, all the known generalized principles operative in Universe. Herein it is proposed that the term computational cosmography .CC) is an apt description for a computing endeavor aimed at modeling the microscopic .and eventually macroscopic) phenomena found in Universe. Evidently, Webster's New Collegiate dictionary is in congruence with this assertion: q Presented at the Seventh Bellman Continuum, Santa Fe, New Mexico, May 1999. E-mail address: [email protected] .J.F. Nystrom). 0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S0096-3003.99)00248-9 212 J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 Cosmography: .1) a general description of the world or of the universe. .2) the science that deals with the constitution of the whole order of nature. The remainder of this paper outlines the proposal for a geometrically grounded computation style in a sometimes general, sometimes speci®c, and often philosophical manner. Speci®cally, in Section 2, an overview of a speci®c CC is given, one that 1. adopts the viewpoint of Universe as discretely ordered and ®nite, 2. accepts the premise of emergent computation [1] as a way to model and ex- plain how this total functioning of Universe can evolve from basic .micro- scopically) low-level rules, 3. and recognizes that the correct fundamental geometrical model of Universe is given in Buckminster Fuller's synergetic geometry system. The idea of addressing fundamental questions in physics by assuming that Universe is discrete, and that Universe is actually using some type of cellular automata .CA) .to accomplish those tasks we attempt to describe with the physics) goes back at least to Feynman [2,3], and has been carried forward by many .see, for instance [4±7], and references therein). This assumption .of a discrete Universe) aords one the opportunity to make conjectures on the possible relationship between computation and the ways of the natural world. The emergence hypothesis assumes that there are properties of whole system behaviors that are not simply a sum, or superposition, of the be- haviors of the systems parts. That is to say that some complex systems ex- hibit synergy. The basic computer model for a discrete physics is the CA. The most famous CA model is Conway's Life .see [8]). Life is a two-dimensional CA which has many interesting forms and patterns that emerge from simple seed values and simple grid rules. Evidently, the two-dimensional CA is not as well understood as the one-dimensional CA [9]. Roughly speaking, one can view the CC pro- posed herein as an extension to a general CA, where .in the CC) polyhedra replace the more usual bits at lattice points .as is common in most CA), and an active grid is proposed to replace the ®xed background structure of standard CA models. In Section 3 the argument for a particular choice of geometrical basis starts. A type of pre-CC, asynchronous CA model is outlined as a sort of gedanken .thought) experiment. Section 4 includes a discussion and further philosophical inquiry into the consequences associated with the hypothesis that all macro- scopic .and most microscopic) phenomena in Universe are in fact a synergism resulting from some lower-level dance of base forms and structures. The ideas of form and structure in computation, a type of CC morphology if you will, is discussed, and the virtue of basing a computation on .geometric) form is ex- tolled. J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 213 2. Overview of the computational cosmography If we for a moment accept the premise that macroscopic phenomena is an emergent result of lower level `rules'or principles, then as a matter of style one would not directly incorporate gravitational type interactions into the rules, but rather follow the lead given by some results of the general relativity [10]: Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve. To this end we can let the discrete grid of a CA-type computation act as `space'and concentrate on producing rules which describe the interaction be- tween the grid and the objects that live on .or in) it. In [11] this type of active grid is proposed as part of the computational cosmography, an archetype for a new level of CA that actually computes with forms on an omnidirectional lattice that suciently spans the three Cartesian directions. The CC borrows geometrical ideas from the synergetic geometry of Buckminster Fuller [12±15] and proposes a computation that uses geometrical forms .modeled using vol- ume enclosing polyhedron) which interact with, and are de®ned as part of, an overall background structural system .the .active) lattice of space/time/matter). To motivate the choice of a particular lattice for the CC model of space, consider the advantages that might be obtained by having an understanding and a speci®c geometry for the so-called quantum geometry [16] .their citation [3]): ... Riemann suggested [3] that the geometry of space may be more than just a ®ducial, mathematical entity serving as a passive stage for physical phenomena, and may in fact have a direct physical meaning in its own right. General relativity proved this vision to be correct: Einstein's equa- tions put geometry on the same footing as matter. Now, the physics of this century has shown us that matter has constituents and the three- dimensional objects we perceive as solids in fact have a discrete underly- ing structure. The continuum description of matter is an approximation which succeeds brilliantly in the macroscopic regime, but fails hopelessly at the atomic scale. It is therefore natural to ask if the same is true for geometry. Does geometry also have constituents at the Planck scale? What are its atoms? Its elementary excitations? Is the spacetime continu- um only a `coarse-grained'approximation? If so, what is the nature of the underlying quantum geometry. On this topic of a possible underlying geometry of nature, Reichenbach [17] makes it very clear that this decision .concerning the actual geometry of na- ture) is not a mathematical problem: 214 J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 The geometry of physical space had to be recognized as an empirical problem; it is the task of physics to single out the actual space, i.e., phys- ical space, among the possible types of space. It can decide this question only by empirical means: but how should it proceed? As the Greeks knew, the minimum volume enclosing form is the tetrahedron .which is a regular polyhedron). As shown in Fig. 1, if we connect all the adjacent midpoints of the edges of the tetrahedron, we outline the octahedron .another regular polyhedron). We can show from this relationship that the octahedron with unit length edges contains four tetrahedron .tet) volumes. To uncover the vector equilibrium .VE) .a.k.a. the cuboctahedron), we connect the adjacent midpoints of the octahedron edges. This construction is also shown in Fig. 1 .cf. Fig. 3). It can be shown that the VE is composed of eight tetrahedron and six half-octahedron, and then taking the edges of the VE to be of unit length, we ®nd the VE contains 20 tet volumes. Further analysis of the tetrahedron and octahedron identi®es two modules which can be used to build up most of the polyhedra we might ®nd of interest. The A-module is obtained by subdividing the tetrahedron into 24 parts, as shown in Fig. 2 .obtained from the digital archive of Hawkins [18]). To ®nd the B-module, divide the octahedron into eight 1/2 tet volume polyhedra .as shown in Fig. 2). This polyhedron is divided into six equal volume structures, which produces a 1/12 tet volume structure, which is composed of a 1/24 tet volume A-module and a 1/24 tet volume B-module. The volume relationships between various polyhedra will be seen as important when contemplating the type of dynamics that might be available from a system of interacting deformable Fig. 1. The tetrahedron, octahedron and the vector equilibrium. J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 215 Fig. 2. The A-module, B-module and Mite. polyhedra. Fig. 2 .[12], .part of) Color Plate 17) also shows the combination of two A-modules and one B-module within a section of an isotropic vector matrix .IVM) ± which is simply a grid fashioned by interconnecting VE. This com- bination of two A-modules and one B-module is called a mite, which is the minimum space-®ller.