Tensional Computation: Further Musings on the Computational Cosmography Q J.F

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 polyhedra 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 microscopic models of various aspects of Universe are to be built into a computation, 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).

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 behaviors of the systems parts. That is to say that some complex systems exhibit 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 proposed 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 macroscopic .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 between 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 volume 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 equations 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 underlying 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 continuum 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 nature) 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., physical 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 combination of two A-modules and one B-module is called a mite, which is the minimum space-®ller. From this viewpoint within the lattice, we begin to view these polyhedra as `states', or objects on the lattice. With the mite as the base example, we note that A-modules and B-modules can be combined to ®ll space. Furthermore, we ®nd that there is always a 2:1 ratio of A-modules to B-modules for any all-space ®lling arrangement [15]. The possible physical signi®cance of identifying these polyhedra .the A-module and B-module) is discussed by Edmondson [15] .where the included reference is to [12, Section 920.01]):

Impressed by the geometric signi®cance of these modules, Fuller proposes that somewhere within this discovery lies secrets with far greater appli- cability than just to geometry: The A and B Quanta Modules may possibly quantize our total experience. It is a phenomenal matter to discover asymmetrical polyhedral units of geometry that are reorientably compositable to occupy one asymmetrical polyhedral space; it is equally unique that, despite disparate asym- metric polyhedral form, both have the same volume.... Their unit volume and energy quanta values provide a geometry elucidating both fundamental structuring and fundamental and complex intertransfor- mings, both gravitational and radiational .920.01). From their energy associations to their remarkable symmetry, these modules synthesize much of Fuller's research. Signi®cant relationships to physical phenomena may well reward continued investigation, for nature also deals with discrete quanta, creating endless variation through synergetic recombinations. Fuller reasoned that his geometric quanta ± the end result of a systematic and logical progression of steps ± must relate to physical phenomena. The approach is typically Fuller's: assume signi®- 216 J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225

cance until proven otherwise. In essence he suggests that tiny whole or discrete systems should replace irrational unending digits ± somehow pro- viding a comprehensive rational coordinate system.

The use of polyhedral primitives to form the basis of objects is a fundamental part of the CC. Furthermore, the problem of computing with structure is somewhat advanced by having an embedded relationship among polyhedra. Further comments on how the dynamics might arise in the CC is contained in [11], and in Section 3.2 after the ubiquitous jitterbug has been introduced.

3. Tensional computing: a proposed model computation

While it is very popular to concentrate only on local interactions when building rules for CA models, it is not yet suciently clear that this bias towards enforcing a ``... spatial locality of physical law [3]'' could allow for the emergence in computation of Universe's most in¯uential participant, the so- called gravitational attraction [10] that exists between each and every energy event in Universe. Smith [19] discusses the overall problem of incorporating forces into a CA, with particular attention given to attractive forces .of which gravity is the quintessential example). In this section gravity is not treated as if it were a radiative phenomena .i.e., massive objects interacting through exchange of gravitons), but rather viewed as an emergent result of the interplay between objects and the grid they live in. On a two-dimensional plane, if a phenomena similar to gravity is to be sim- ulated, a hexagonal grid would be preferable over a Cartesian grid. This would give `moving objects'in the computation a better chance to exhibit either circular or elliptically discrete orbit phenomena. The interaction of objects with the grid might be modeled, on ®rst blush, as the familiar balls on the rubber sheet example popular with those explaining general relativity. From the lattice point of view, local slope and concavity information .of this virtual rubber sheet) can be viewed as a type of tension in the chords of the lattice.

3.1. Tensional computation on a plane

Consider separating our model system into two separate CA calculations. One CA system .the grid CA) computes values for the chords of the lattice. The second CA .the object CA) handles the object location, direction and speed. The two CA use information produced by the other, but need not be time synchronized per se. The rules for the grid CA will use object locations .obtained through asynchronous message channels [20] from each object in the object CA) along with previous chord values. Some mechanism should be set-up so that an J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 217 object does not feel its own tensional contribution to the grid. A snapshot of the systems chord values at any time during the dynamic evolution of the system should resemble some potential map .or a rubber sheet with objects distributed throughout). For the object CA, we will not require that an object always reside at a lattice point. At any given instant in the computation, an object most probably will be traveling along a chord in the lattice. The rules for the object CA manage what direction an object will take when it gets to a lattice site. The rules use both chord value information from the grid CA and some momentum information associated with the object. When an object reaches a lattice point, the next chord of its journey is chosen, the .linear) momentum is updated and the next location point is communicated to the grid CA. A single object on the grid with non-zero momentum should move con- stantly forward when it arrives at a lattice point. Only when other objects enter the grid is there a possibility of the object ever `turning'. Extensions to this base model include spinning objects .requiring rules for how the grid responds). The presence of spinning objects might require the emergence of new nodes in the lattice .a type of virtual hexagon) to accommodate independent rotations of each of the grids hexagon centers .compare to comments in Section 4.1 on the lattice jitterbug).

3.2. From tensional computing to a computational cosmography

We now extend the two-dimensional grid to three dimensions. The basic cell of the three-dimensional .in this case it is also omnidirectional) grid is the VE, shown in Fig. 3. To produce the overall grid from this basic cell, make every point of the VE the center of its own VE, which after one application gives the two-frequency VE shown in Fig. 4. Note that for each and every .interior) point of an n-frequency VE, four separate hexagonal planes converge. These four separate hexagonal planes of this omnidirectional closet packing arrangement are arranged so that the intersection of any three .of the four) planes produces outlines of two separate tetrahedron, which have a common vertex at the intersection point of these three planes. Furthermore, there are four unique pairings of three elements out

Fig. 3. The vector equilibrium. 218 J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225

Fig. 4. A two-frequency VE. of four elements .i.e., four choose three), giving rise to a total of eight tetrahedra with a common vertex at the center, with each tet edge-joined symmetrically to three other tetrahedra. Solid state physicists and crystal- lographers will recognize this grid as the face-centered cubic, also known to many philomorphs as the isotropic vector matrix, or as a framework of possibility [15]. The polyhedral aggregates .or more speci®cally, the objects of the computation) will cause deformations, or changes in the tensional characteristics of the grid. Then the grid, in turn, will act back on the objects. From this interplay .between the grid and low-level objects) we expect larger forms to develop. In CC the forms .i.e., polyhedral aggregates) are supported by tensegrity ±a combination of the words tension and integrity. In Fuller's words [14]:

In a tensegrity structure, radiation/matter is modeled by the discontinu- ous struts, and gravitation is modeled by the continuous network of wires underlying the structure. This model reconciles these two disparate elements into a single uni®ed ®eld. No other known model does so.

In reference to the CC model, we ®nd an ability to relate tensegrity to the actual tension built up in the various chords of the lattice due to the presence of objects. In this way, we ®nd the lattice tells polyhedra aggregates how to transform, while the polyhedra aggregates in turn tell the lattice how to tense .i.e., how to adjust it's overall tensegrity). Recent popular accounts of tensegrity can be found in [21,22]. I have .and others may have also) pos- tulated [11] that tensegrity is the primary agent of the so-called phase tran- sition, a subject area of much interest to the theoretical community. Or maybe tensegrity is even more pervasive, as Ingber suggests at the end of his ®ne article [21]:

Finally, more philosophical questions arise: Are these building principles universal? Do they apply to structures that are molded by very large scale J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 219

forces as well as small-scale ones? We do not know. Snelson, however, has proposed an intriguing model of the atom based on tensegrity that takes o where the French physicist Louis de Broglie left o in 1923. Ful- ler himself went so far as to imagine the solar system as a structure composed of multiple non-deformable rings of planetary motion held together by continuous gravitational tension. Then, too, the fact that our expanding .tensing) universe contains huge ®laments of gravitation- ally linked galaxies and isolated black holes that experience immense compressive forces locally can only lead us to wonder. Perhaps there is a single underlying theme to nature after all. As suggested by early 20th-century Scottish zoologist D'Arcy W. Thompson, who quoted Ga- lileo, who, in turn, cited Plato: the Book of Nature may indeed be written in the characters of geometry.

If tensegrity is so pervasive, is this then the principle that should dictate the evolutionary dynamics of the CC's polyhedral aggregates and the changes in- duced in the grid? To contemplate this question correctly, it is appropriate to point out that the background lattice can not deform in just any given way. That is to say, there are restrictions on how the grid deforms. Recall that when con- structing the IVM, every lattice point is eectively the center of a VE. In the CC model, every VE .of the grid) is restricted to jitterbug deformations. The jitterbug is Fuller's term for a transformational model that in the `rest' state appears as the outer shell of a VE. The model then can transform from the VE stage, through the icosahedral stage, and eventually to an octahedron stage. This part of the transformation is shown in Fig. 5 .[12, Fig. 460.08]). From the octahedral stage, through another set of transformations the model can be put into a tetrahedron form. Many interpretations of the jitterbug are available. The ®rst thing to notice is that going from the VE to the octahedral stage, all the vertices contract somewhat concentrically towards the center of the total system at an equal rate ±

Fig. 5. Various phases of the jitterbug. 220 J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225

Fig. 6. Synergetic fractals. which would be quite an engineering achievement if the model did not actually incorporate in its essence much of nature's actual design. The next thing that is most noticeable is that at the three canonical stages; icosahedron, octahedron and tetrahedron, the structure is respectively: single-bonded, double-bonded and triple-bonded .i.e., in the octahedral stage, each edge of the octahedron is actually composed of two edges from the original structure). An example of the utility of the synergetic geometry and tensegrity principles is the exact two-dimensional real-space renormalization group calculation .i.e., an Ising model type calculation), which uses what the author calls a tetrahedral gasket [23]. The tetrahedral gasket and the newer octahedral gasket are shown in Fig. 6. Both of these fractals belong to a larger class of what I am calling synergetic fractals [24]. It is an elementary exercise to show that the tetrahedral gasket has a fractal dimension [25] of exactly two, while the dimension of the octahedral gasket '2:585 .i.e., 1 + the dimension of the Sier- pinski gasket). An extension of this work allows one to write a spin Hamiltonian for all the canonical forms .i.e., VE, icosahedron, octahedron and tetrahedron) found within the jitterbug. This type of approach could provide a technique with which to encode polyhedra forms within an IVM lattice, and a way to model the jitterbugs of lattice sites.

4. Concerning the extent of the premise of emergent phenomena

Form and structure in the physical world and in computation exhibit long- range correlations, and just as we ®nd the surface tension to play an important role in forming bubbles, tensional characteristics undoubtedly play an im- J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 221 portant role in the existence of emergence of physical objects and the associated interaction of same. The alchemist's `as above, so below' is appropriate in light of the fact that many local pattern integrity .e.g., forms) are in fact just parts of higher level patterns .i.e., molecular structures use lower level atomic forms and biological patterns use lower level molecular forms). When contemplating form and structure in Universe, even though we have a vague notion .and recognize it when we see it!), what is it that we really mean by this? Consider the case of electromagnetic .EM) energy, and now ask: does the EM energy have some quality of formness .sic) associated with it? If so, how would this form best be viewed? The easy answer is to view the form of EM energy in terms of its usefulness to man .i.e., an anthropocentric viewpoint), and let its form by described by the standard vector calculus and understood in terms of the electrical engineer's devices [26] .i.e., antennas, waveguides, transmission lines, etc.). The form of the EM energy might be found in the fact that ®elds are actually composed of individual photons, or is it more appropriate to view the form of the EM energy in terms of tensor formulations, and point to the an- tisymmetric ®eld tensor [27,28] as a description for the EM form. Or better yet, we could be convinced to view EM energy in terms of dierential forms [29,30], and look to the exterior calculus for insight into the true nature of EM energy. In none of these aforementioned viewpoints of how to describe the form of EM energy is there a technique available for deriving the Maxwell equations. When the correct geometric viewpoint is established, and more generalized principles concerning the nature of EM energy are discovered, the Maxwell equations should appear as special case .and therein an EM form could be established). Summarizing the importance of information, form and structure, Young [31] provides testimony for a natural philosophy grounded in geometry:

There appears to exist a massive and fundamental involvement of various manifestations of form and form characteristics in all universal events, objects and processes ± in the large-scale geometrical ®eld identity of the space-time continuum, in strings, in particle/wave systems as forms of a given ®eld, in atomic patterns, in the structure of molecules, in the widespread form-dependent activities of cells, organisms, nervous systems and brains, and ®nally in mind and mental forms. If we want to un- derstand the universe as a total system, we must come to terms with this fact ± that there appears to exist a morphological or geometrical aspect to its nature that plays a dynamic and powerful role in the identity and behavior of all objects and processes. This view has spawned a great deal of thought, but very little systematic analysis since Plato.

Even if we accept the need for a geometrical foundation, at what level do we begin the implementation .in computation)? It seems the rules of interaction of the systems objects need know nothing of the ®nal characteristics of the desired 222 J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 emergent behavior. In order to obtain some type of correspondence principle, some resultant behaviors .of the computation) should have similarity to the fundamental properties of matter. These properties .of matter) should not be programmed into the rules or the objects. Also, the behaviors of some of the resultant higher level objects should have similarity with .accepted) fundamental particles. Earlier the bias towards encoding only local rules was high- lighted, but we must give Margolus his due, he does recant somewhat towards the end of [3]:

At this point, we might also begin to question our basic CA assumptions. We introduced crystalline CA's to try to emulate the spatial locality of physics in our informational models, but we are now discussing modeling phenomena in a realm of physics in which modern theories talk about ex- tra dimensions and variable topology. Perhaps whatever is essential ®ts nicely into a simple crystalline framework, but perhaps we need to consider alternatives. We could easily be led to informational models in which the space and time of the microscopic computation becomes rather divorced from the space and time of the macroscopic phenomena.

It appears that not only do we need to give up on enforcing a spatial locality in the rules for CC-type computations, but we should also resist encoding emergent properties directly into the computation .this is in direct opposition to some of the ®ne detail proposed in the digital mechanics [4]). We should program for asynchronous system evolutions as opposed to the .Turing limit) restricted synchronous system updates .so popular with reversible computation). Simply stated, instead of programming in the properties of Universe we are most confused about, we must try to build computations that will have those aforementioned properties emerge as a result of some other .geometrically based) rules.

4.1. Listen to what the computation has to say

It seems abundantly clear that Universe is an emergent computation, and our analysis techniques should treat it as such. When we accept the premise of Universe as an emergent computation we must resist the temptation of programming higher level relationships or properties into the rules of the CA-type system, but rather let the results of the computation tell us, for example, what an electron really is .an emergent form, a big bit [32], a certain type of polyhedral aggregate possibly intimately connected to the photon via some type of pho- tonic phase .hmmm...), an actual point entity, or something currently un- foreseen). Whatever the electron is, if it emerges out of the computation, the computation should also give insight into what mass, charge and spin represent. J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 223

Imagine extending the two-frequency VE in Fig. 4 to some very large frequency. One might be tempted to view this snapshot of the computational grid as some large neural network [33], with each node connected to 12 others by links with varying weights. In actuality, this connectionist viewpoint might be useful initially in establishing techniques for updating the grid tensegrity values, but it is misleading. When introducing the jitterbug model, it was noted that this mechanism .the jitterbug) places a restriction on how the grid deforms, but since the program should allow every VE center to its own jitterbug, we must now note that in order to accommodate these local deformations in the grid, we will require the emergence of virtual ± VE at practically every node of the lattice. Thus, local deformations of the grid cause the background lattice space to expand from within, or rather from with ± everywhere .sic). This might now appear as some type of `foamy'agglomerate [34], and might even be a candidate for a place where Bohm's philosophy [35] could reside. The generalized principles that are operative in Universe operate at the very lowest level of local interaction and at the higher levels as a result of the interplay between local events and a global .universal) grid. While Young [31] talks of `mass±energy laws'when it might be more appropriate to use the term generalized principles, he does sum up the situation well:

Every set or system of relations between events in the physical universe is a manifestation of all the known and possible mass±energy laws applicable to that region of space at which such events occur. In other words, for every set of relations in the universe, there exists an aggregate of all the laws of space-time .known and unknown), which are its essential ingredients and control factors. More simply still: All sets or systems of relations in the universe are the physical manifestations of all applicable mass±energy laws or principles.

Although much work needs to be done before the computation style outlined herein can be realized .if it can be realized at all), it does appear that the real- ization of a CC-type computational artifact could be just one of the steps in some overall evolution of computation: e.g., Computer as calculator, Computer as toy universe, Computer as teacher, Computer as pure mind-extension tool.

Acknowledgements

I would like to thank Rick Bono, John Ulinder and George Patsakos for their willingness to discuss many of the arguments contained herein. I also appreciate the very direct comments of one of the anonymous reviewers of a 224 J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 companion submission [11]. Lastly, I am very appreciative of the travel assis- tance provided by the Department of Electrical Engineering and the MRC Institute at the University of Idaho, and Jim and Carol Nystrom.

References

[1] S. Forrest .Ed.), Emergent Computation, MIT Press, Cambridge, MA, 1991. [2] R. Feynman, The Character of Physical Law, MIT Press, Cambridge, MA, 1967. [3] N. Margolus, Crystalline Computation, in: A. Hey .Ed.), Feynman and Computation, Addison-Wesley, Reading, MA, 1998. [4] E. Fredkin, Digital mechanics, in: H. Gutowitz .Ed.), Cellular Automata: Theory and Experiment, MIT Press, Cambridge, MA, 1991. [5] E. Fredkin, A new cosmogony, in: PhysComp'92: Proceedings of the Workshop on Physics and Computation, IEEE Press, New York, 1992. [6] C.G. Langton, Life at the edge of chaos, in: C.G. Langton, C. Taylor, J.D. Farmer, S. Rasmussen .Eds.), Arti®cial Life II, Addison-Wesley, Reading, MA, 1992. [7] S. Rasmussen, Aspects of information, life, reality, and physics, in: C.G. Langton, C. Taylor, J.D. Farmer, S. Rasmussen .Eds.), Arti®cial Life II, Addison-Wesley, Reading, MA, 1992. [8] J.L. Casti, Alternate Realities: Mathematical Models of Nature and Man, Wiley, New York, 1989. [9] N.H. Packard, S. Wolfram, Two-dimensional cellular automata, J. Statist. Phys. 38 .5) .1985). [10] C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, New York, 1973. [11] J.F. Nystrom, In search of the precepts for a computational cosmography, working paper. [12] R.B. Fuller, Synergetics, Macmillan, New York, 1982. [13] R.B. Fuller, Synergetics 2, Macmillan, New York, 1983. [14] R.B. Fuller, Cosmography, Macmillan, New York, 1992. [15] A.C. Edmondson, A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller, Birkhauser, Boston, MA, 1987. [16] A. Ashtekar, J. Lewandowski, Quantum theory of geometry: I. area operators, Class. Quantum Grav., 1997. [17] H. Reichenbach, The Philosophy of Space & Time, Dover, New York, 1957. [18] R. Hawkins, Digital Archive, access via http://www.cris.com/rjbono/index.html, R.J. Bono's website. [19] M.A. Smith, Cellular automata methods in mathematical physics, Ph.D. Thesis, MIT, Cambridge, MA, May 1994. [20] G.R. Andrews, Concurrent Programming: Principles and Practice, Benjamin/Cummings, Menlo Park, CA, 1991. [21] D.E. Ingber, The architecture of life, Scienti®c American 278 .1) 1998. [22] R. Connelly, A. Back, Mathematics and tensegrity, American Scientist 86, March±April 1998. [23] J.F. Nystrom, An exact two-dimensional ®nite ®eld real-space renormalization-group calculation, MS Thesis, University of Idaho, December 1996. [24] J.F. Nystrom, An exact ®nite ®eld renormalization group calculation on a two-dimensional fractal lattice, Int. J. Modern Phys. C 11.2) .2000). [25] J.T. Sandefur, Using self-similarity to ®nd length, area, and dimension, The American Mathematical Monthly 101.2) .1996) 107±120. [26] D.K. Cheng, Field and Wave Electromagnetics, Addison-Wesley, Reading, MA, 1989. [27] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. [28] A.O. Barut, Electrodynamics and Classical Field Theory of Fields and Particles, Dover .edition), New York, 1980. J.F. Nystrom / Appl. Math. Comput. 120 .2001) 211±225 225

[29] H. Flanders, Dierential Forms with Applications to the Physical Sciences, Dover .edition), 1989; Wiley, New York, 1998. [30] W.L. Burke, Applied Dierential Geometry, Cambridge University Press, Cambridge, 1985. [31] P. Young, The Nature of Information, Praeger, New York, 1987. [32] P.H. Huyck, N.W. Kremenak, Design and Memory, McGraw-Hill, New York, 1980. [33] J.L. McClelland, D.E. Rumelhart, Explorations in Parallel Distributed Processing, MIT Press, Cambridge, MA, 1988. [34] I. Peterson, Loops of Gravity: Calculating a foamy quantum space-time, Science News 153 .1998). [35] D. Bohm, Wholeness and the Implicate Order, Routledge .reissue), London, 1995.