Knots, Braids and Möbius Strips

The following sections are included:

• Dedication
• Acknowledgments
• Contents
• Preface

In a way, this introduction is sort of a continuation of the Preface so you will be faced with a lot of words. However, where the preface was mostly about point of view, what we have here is mostly about context, like it says, historical and philosophical and by philosophical I mean some epistemology and some ontology — a little dose of E and O every once in a while does no harm I should think…

In this chapter, a small set of particles is defined to serve as the basis for what constitutes a particle model, the Alternative Model (AM) that, eventually, we can compare to the Standard Model of particle physics. In the spirit of the FYS model, these particles, in combination, are relied upon to generate all the rest of the model's taxonomy, interactions and attributes all of which will be seen to be highly visualizable, a factor that lent encouragement to the pursuit of the model's development. In fact, before we describe the model in any detail, it may be revealing to recount an incident, you might say an epiphany of sorts (but without spiritual connotations!) that, for me, initiated the process of model development. In the waning years of the 20th century I was staring at the cover of a small book on recreational topology published in 1964 (by coincidence, you will recall, the same year Gell-Mann and Zweig brought out their quark theory)…

We begin with a sort of birds eye view by considering an abstract, group theoretic approach which bypasses the detail but summarizes the general architecture of the taxonomy. In this (top-level) approach (cited in [10] as per the treatment in [15]), a particle hierarchy is developed as the direct product of vector spin spaces parametrized by spin. Correspondingly, the abstract result is expressed as the direct sum of subsidiary spin spaces, the so-called Clebsch–Gordan decomposition. With the additional recognition that the applicable group structure (see Sec. III) is that of the gauge group SU(2), the result of the direct product of vector spin spaces with spins s₁ and s₂ is

That traverse proceeds one way through the fermion and the opposite way through the antifermion components of the figure-eight configuration of our boson model highlights how yet another historically iconic item, in this case the Dirac theory, has bearing on our alternative model. According to all accounts, the notion that antiparticles exist emerges in the combination of Quantum Mechanics and Special Relativity via the Dirac theory. To set the stage for how that theory relates to our notion of fusion, as implied by that theory, we abstract and compress it as follows: As is well known, the equation for a particle with rest mass m and spin-1/2 is succinctly stated by the Dirac equation

As per the comment made above in connection with Eq. (2-3), we can view second-order fusion as Matrix M operating on basic fermion vector V, that is, in analogy with Eq. (2-1), we can write

The preceding has been quite general in that we have kept explicit mention of the Standard Model to a minimum (for a reason that shall be divulged in Sec. IV). In fact, we could just as well have been talking about the lexicon of a simple language in which the words are limited to one, two or three syllables. The syllables are the four basic fermions A, B, C and D and their conjugate counterparts, each spelled out in terms of three letters, drawn from a two-letter alphabet consisting of the letter u (pronounced “up”!) and d (pronounced “down”) and their conjugate counterparts. As we have seen, the language is so simple that there is only one constraint in its development, namely that juxtaposition (called “fusion” in the preceding) can only take place between a syllable and a conjugate syllable (not necessarily its own conjugate) which implies that letter d can fuse only with d* and u only with u*. Statements in this language are then to be translated into the language of the SM, a task we accomplish in Sec. IV…

In summary to this point, we reiterate: our Alternative Model has promoted the status of the MS/knot genus from that of a device used to demonstrate the phenomenon of spin and the gauge group SU(2) to the ontological basis for a particle model. But from an even more fundamentally ontological level, you may recall the comment in the Introduction to the effect that underlying the attributes that characterize our particles is their toroidal topology. Thus, broadly speaking, spin (more accurately, order of fusion) serves as a top-level, SU(2) discriminant for organizing FMS taxonomy. We have seen that two traversals of the MS are required for all odd values of its NHT (which we recall is equal to the value of the n of the associated knots) but only one traversal for all even NHT and that higher values of NHT assume the form of composites of triangular planforms. In such composites the traversal requirement of each member is now shared with its partners (in effect, the detour implements the second traversal requirement of a single FMS). Note that each member of a composite still retains its toroidal topology so that, in essence, the overall taxonomy described above consists of tori of genus one, two and three which is to say that we are concerned with tori with one, two and three holes…

To this point we have ignored a major mismatch, namely that between the configurational approach and the direct product/convolution approach to tabulating degeneracy and have emphasized the latter. Now, however, in terms of configuration, suppose we consider the taxonomy from the point of view of the availability of choices imposed by the basic topology of each order of fusion. In what follows all “words” (recall the discussion in lexicographic terms in Chap. 7) will be assumed to begin with a basic letter rather than a basic conjugate letter. With that stipulation, the first order fusion of a basic letter and a basic conjugate letter can simply be depicted for our purpose here as in the stick figure of Figure 8-1. We see that the process of fusion has resulted in two quirks, two antiquirks and a junction, five items in total. There is thus the availability of 2exp5 = 32 binary choices, twice as many as we found by considering only combinations and permutations…

Here we develop a relationship between the preceding material, in particular the matter of fusion and fission, and what's known as a Bialgebra and a Hopf algebra, the latter often discussed under the subject of “Quantum Groups”. The notions of fusion and fission being clearly central to FMS modeling, the fact that they take place explicitly between particular quirks and their antiquirk counterparts bears reemphasizing. The subject was treated in Sec. II in terms of first, what was called symbolic convolution, and then a vector formalism, both unique to this book and its predecessors…

This chapter considers the possibility of giving the model a quantum mechanical treatment. Since this book is not concerned with particle kinematics, position/momentum relationships are not an issue. However, there is clearly a statistical element in the way FMS are combined in fusion and fission operations and this should rightly be treated in a quantum mechanical manner. Of interest in that regard is the remark of Sec. II, Chap. 3 that matrix M can be viewed as an operator that converts a vector representing the set of four basic, single component (letter) fermions into a vector of four matrices, one that codifies the set of three letter words representing the set of three-component fermions. In this sense, it invokes the quantum mechanical notion that the outer product, say |β⟩⟨α| of a ket, |β⟩, and a bra, ⟨α|, is an operator that converts a ket into another ket…

We begin this chapter with a quotation: “No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors.” The speaker, who is a mathematician of the highest repute in many areas including the field of particle physics and its relationships shall be nameless, mainly because I seem to have mislaid the reference for this rather gloomy prognostication. Were he to become aware of it, I should hope he would forgive me for saying so, but I think we may be able to do better than that; I don't think I can afford to wait centuries and perhaps the rest of this chapter will indicate why we do not really need to…

The magnetic dipole moment of the electron is the product of its spin and what is known as its gyromagnetic ratio. To explain the results of their historic experiments, Goudsmit and Uhlenbeck found it necessary to assign the unusual value of 2 to that quantity (we recall that it was the subsequent need to refine that number which led to the development of QED). Pursuing the implications of the Dirac theory then provided a theoretical justification for the value of 2. This chapter summarizes a simple calculation first published in [10] that deals with an independent confirmation of that value for our alterative model and an associated item, the dipole magnetic moments of the nucleons. In what follows we shall dispense with the ABCD notation for a while and admit that we are talking here about the electron and the nucleons…

Quaternions are useful for manipulating quantities in four dimensions (actually, in three as well) as are complex variables in two and we shall use them in Sec. VI. In this chapter we investigate how they might relate to the development of particle taxonomy, a very different point of view from the way we proceeded in Sec. II and illustrated with some diagrams you might find pertinent. A quaternion can be expressed as

Before we make an explicit connection to the particles of the Standard Model, here is a topic that has to do with the algebra of (2, n) torus knots, specifically with connections between different values of the parameter n. How it might find application will be discussed later but the relationship of Möbius strips (MS) to (2, n) torus knots or links is well-known and was discussed in some detail in the preceding, a key aspect being that the half-twist (NHT) of a given MS is equal to the n value of the associated (2, n) torus knot or link. Thus, citing the equivalence between, for example, a composite FMS, whose NHT is the sum of two individual NHTs and a knot whose n is a composite number, the sum of the two associated individual n values, allows us to study the latter to learn about the former. When we get to some rather free-form ruminations in the cosmological area in Sec. VII, we shall invoke what this chapter talks about (forewarned is forearmed)…

Full disclosure: The main reason so much space was used to develop an alternative model using a general, lexicographic labeling for particles instead of going directly into standard notation is that our alternative model is inherently ambiguous; a given AM label (one of the capital letters) may represent more than one fundamental particle of the Standard Model depending on the interaction being modeled. The ambiguity is summarized in Figure 15-1…

We now consider the creation of delta particles by the excitation of nucleons operated upon by pions (or conversely, delta decay into nucleons). Since no leptons are involved, these interactions are unambiguous so there is no reason not to express this directly in SM notation. However, for direct comparison purposes, we begin with our model's notation in the associated FMS schematic shown in Figure 16-1…

In the case of Delta creation and decay, translation between the two notations was unambiguous. However, in modeling weak interactions that is no longer the case. In any event, we begin with neutron decay, our original notation and with the excited states shown in the NHT = −1 column of Figure 5-5, noting specifically the terms AC*C and CC*A (which are operationally identical for our purpose) in the uppermost triplet of the NHT = −1 column. Using CC*A, the stick figure schematic of Figure 17-1 below then shows it as the first stage, a folded-over (excited) version of the original FMS configuration of B, the Alternative Model's surrogate for the neutron, n. On the basis of the quirk structure, C and C* translate unambiguously to p and p*, which conserves twist but not baryon number or charge…

In [11] a model was shown of pions mediating Yukawa type exchanges between nucleons to maintain deuteron stability in what may be characterized as (strong) isospin space and we reproduce it here in Figure 18-1 (again from the cited presentation). This is a dynamic process, as summarized in Figure 18-1, (another presentation figure) postulated to maintain that stability. The figure is pretty much self-explanatory: all four pions of the AM are involved, two for the proton and two for the neutron. At each stage of the process what was a free proton becomes a neutron and conversely what was a free neutron becomes a proton. Also, in (what we might arbitrarily call the first stage) what was a π⁻ becomes a π^0R and what was a π⁺ becomes a π^0L. Two fusions and two fissions take place, in each case. The process then reverses to recover the original pair of nucleons…

The symmetry properties of the basic fermions are simple if there are no external influences. Of course the diagram for the electron (e) must be invariant to rotations about an axis normal to the plane of the diagram and, provided it is compensated by parity (direction of traverse) reversal to rotations of 180° about each of the axes a, b, and c as suggested in Figure 19-1…

So far, everything we have discussed pertains to a single generation of the SM, which, as we know, actually consists of a family of three generations of particles. The object of this section is to display a corresponding triplication of our alternative model, in fact a corresponding family structure consisting of three and only three generations. Although most of the argument is demonstrated in some detail in [10], here, while omitting some of the detail, we shall go a bit further in terms of the symmetries inherent to our model…

So, with that to cheer us on let us proceed with the work of the book. The previous sections were primarily concerned with the algebraic geometry of flattened MS and their interactions, that is, with FMS as particles but at this point, we embark upon a different kind of investigation; our particles are presumed to have a life before they become “particles”, so, in this section we will be talking about the nature of these nascent particles which means mainly in terms of their differential geometry in one form or another. In the preceding, there was some fine-sounding talk about Toroidal Topology and about Solitons being, as it were “formed out of the dust of spacetime” (that allusion suddenly emerged unbidden and I could not resist it!)…

Although a torus knot would appear to be topologically ineradicable, it is of interest to explore its explicit behavior from the point of view of its description, as per the Introduction, as a disturbance in the continuum. We begin with the notion of a miniscule, representative disturbance, moving along the knot according to the canonical equation for a geodesic,

Equation (22-3) can also be derived from a Lagrangian but before we do so it is useful to talk about the sine–Gordon equation briefly. It appears in a variety of contexts but the canonical situation is the time-honored case of an idealized pendulum consisting of a stiff, weightless rod of length l and a weight of mass m constrained to rotate in a plane as portrayed in Figure 23-1 under the influence of gravity…

We recall a fundamental attribute of particle physics, namely that to every elementary particle there is associated a conjugate antiparticle. This concept was, of course, the landmark contribution of Paul Dirac, realized as a result of his mating of quantum theory with the generalized energetics implied by special relativity (see Sec. II). Although particles are visualized herein as solitonic disturbances “in and of an otherwise undistorted continuum”, to begin with, there was no mention, to this point, of what it is that does the disturbing. Now we see that it is the traversal by the solitonic torsional distortion around the putative toroidal core that does so. However, note that there are two directions of traverse available in the model employed herein and, in effect, only one has actually been invoked, by implication, traverse to the right (as per Figure 21-1). Thus it seems natural to ask whether our solitonic model can also accommodate the notion of antiparticles by invoking traverse to the left. Although this is the nominal state of affairs in Sec. II, it was occasioned by the requirements of fusion. Here, we ask whether that requirement is satisfied inherently…

The objective here is twofold; first to derive some measurable quantities associated with our particle model but also, to show how a leftward traverse implies a negative mass. We begin by multiplying Eq. (24-5) by Ar(v/c)²(Δℓ) where v = ds/dt is velocity along the knot. The result, again to first order in r/R, is

The previous section considered particle mass only in a relative way in terms of its variation with particle size. Here we are concerned with the actual magnitude of mass and in particular, since our particles are viewed as distortions of the fabric of spacetime, how the nature of that fabric in turn influences particle mass. To begin with, the answer to that question appears straightforward: Eq. (25-7), which we repeat here for reference

In this chapter we follow up on the above suggestion that each basic AM particle experiences “a situation reminiscent of the symmetry-breaking topology that preconditions the Higgs mechanism”. For purposes of comparison we need something definitive but the complete Higgs mechanism is, necessarily, rather complicated; without it the Standard Model would not be able to attribute mass to the elementary particles of the model without violating some of its basic theoretical underpinnings, primarily SU(2) gauge symmetry and the efficacy of renormalization. However, it is not necessary to invoke the entire mechanism here because, in the first place, SU(2) symmetry in the AM is inherent to particle structure and, furthermore, renormalization is not an issue. What is basic to the SM's Higgs mechanism, however, is the need to introduce a symmetry-breaking version of the potential energy that each particle encounters. That potential is also of interest here because of the way in which symmetry breaking was introduced into the AM, that is, by way of the initial implicit assumption of toroidal topology and the reduction, as per the above, of the value of linear energy density, A, of spacetime in the neighborhood of the MS…

Admittedly, what we have described above is a rather unusual approach especially since, we note, not only was the underlying toroidal topology of our AM particles initially simply assumed without explanation or justification, now we are introducing a further hypothetical entity. However, now that we have gone this far, suppose for the sake of argument that we go one step further: suppose that we simply assume the existence of a complete Higgs-like characteristic interior to each elementary particle to begin with. This is essentially analogous to the SM's approach to the Higgs field, eventually justified, it is now claimed, by the discovery of the long-sought Higgs boson. In other words, the novel hill and valley topography of the AM is then to be viewed as a fundamental attribute of each elementary particle…

Going back to Eqs. (28-1) and (28-2) of the previous chapter, we note that since both of these relationships refer to either the electron or a nucleon, we are really concerned with four cases, namely

Benedict (Latinized from Baruch) Spinoza (1632–1677), a contemporary of Leibnitz, was one of the most important philosophers of the 17th century (but, of course, you already knew that). What is of interest here is what may be termed his ontological philosophy of existence, known as Monism, namely that, while it can evidence itself in various ways, there is only one, universal, absolutely infinite, fundamental substance (which he equates with God, a viewpoint whose complexity we need not address here). To which I might legitimately claim a kinship for my (repeated insistence that there is only) one field ontology, spacetime itself. You may recall our discussion of reductionism as an epistemological tenet; here, we have a rather significant reduction (from, say, the field-per-particle ontology of the SM)…

Many papers, essays, tracts, even books have been written about the nature of time and from many points of view, although none quite so lyrical as those few lines by Omar via Fitz. Our concern in this chapter is much more limited: you may recall a statement made at the end of Sec. II to the effect that our FMS are to be regarded as occupying a 2+1 spacetime wherein the out of plane dimension is time. Thus, the discrete charges of the quirks are associated with steps in time. In this section we explore some of the ramifications of that point of view. To begin with, we note that the underlying ontology of our alternative model may be viewed as a kind of Kaluza–Klein theory, especially in terms of Klein's modification of Kaluza's five-dimensional modification of General Relativity that combines it with electromagnetism. Apparently what Klein did amounts to compactifying the fifth dimension of Kaluza's original theory into a circle in spacetime. Correspondingly, we note what amounts to the meridianal compactification of our solitonic knots so as to encircle their toroidal “last”, also in spacetime with time as the dimension normal to the plane of the torus and ultimately of the flattened Möbius strip, the FMS.

What is known nowadays as the Standard Cosmological Model postulates the existence of “Dark Matter”, the source of the gravitational influence that keeps galaxies from flying apart. Most literate people have at least seen or heard of it and are aware that it is quite unlike “ordinary matter” with which it otherwise does not interact. And, perhaps also, that there are numerous candidates for its constituents. We will talk some more about DM when we get to cosmology in Sec. VII, but for now, here is another candidate, admittedly speculative, but emerging from our alternative model in a logical way: Always lurking at the perimeters of our earlier deliberations but never really contributing to them is that enigmatic fourth elementary fermion, the one with a twist of +3 and a charge of +2. Clearly, this FMS as well as its taxonomical combinations have little to do with the alternative model's taxonomy or interactions. Nevertheless, they must appear in our modeling by virtue of symmetry and so they do, at each level of fusion, a kind of super (anti) symmetry…

Connecting the alternative model to “dark matter” would be most gratifying but there are more “shovel ready” topics for additional development. One such is the proposition that our model constitutes a manifestation of Topological Quantum Field Theory (TQFT). According to Wikipedia, “A topological quantum field theory is a quantum field theory which computes topological invariants”. If that is a little too cryptic, it goes on to say that “in a topological field theory the correlation coefficients do not depend on the metric on spacetime. That means that the theory is not sensitive to changes in the shape of spacetime; if the spacetime warps or contracts, the correlation functions do not change. Consequently they are topological invariants”…

This chapter is going to be even sketchier than the previous three; String Theory is a really big subject and there is no way we can do justice to it in summary here. However, we can mention a few items that tend to indicate some commonality with our alternative model. For example, we should emphasize that each of the concatenating torus knots that makes up an MS is explicitly wound around a torus which amounts to what is known in string theory as toroidal compactification. But, of course, there is more to the comparison than that. For one thing, the coefficient of the kinematic term in the Lagrangian yielding the sine–Gordon equation as discussed in Sec. V is identifiable as string tension. Also, string theory is notable in developing a number of versions, each requiring extra spatial dimensions which must then be hidden from view in the Kaluza–Klein tradition. The hidden dimensions then take on what are known as Calabi–Yau shapes (after the mathematicians who invented them) some exotic in the extreme as the number of dimensions increases. However, the simplest such are for two dimensions in which case the Calabi–Yau shape is simply a torus…

Sine–Gordon solitonic behavior is of course not limited to our toroidal model, the definitive example being simply the pendulum of classical dynamics as discussed in Sec. V. Another intriguing example is the Phase-lock loop, a feedback device ubiquitous in the electronic implementation of signal processing systems such as radio, radar, television, etc., for which it is necessary to maintain synchronism between received carrier signals and local references. Such systems are prone to chaotic behavior but their ability to initiate and maintain lock is well known and is herewith anticipated to have bearing on the genesis and stability of the particles of our model. Figure 35-1 shows a prototypical analog phase lock loop (PLL) with input and output V₀ = A cos ϕ₁ and V₄ = cos ϕ₄, respectively. Although the actual implementation is somewhat more involved, as per the feedback path, the two signals are basically multiplied together to generate a signal that encounters the several indicated operations. Thus, LIM is a Limiter that eliminates the input amplitude A. LPF is a minimal type of low-pass RC filter as shown in Figure 35-2 (R and C standing for resistor and capacitor, respectively) with transfer function in Laplace notation V₃ = V₂/(1 + τs), where τ = RC…

The objective of this chapter is indeed to recapitulate and summarize what we have said about our alternative model of the elementary particles — its salient features, how they interrelate and their connections to historical antecedents and to the subject of elementary physics in general — in other words, basically to provide some perspective. In that regard it might be a good idea to resurrect the elementary question: “Just exactly what is an elementary particle?” Or, perhaps “what do we mean when we use the terminology, ‘elementary particle’?” might be a better way to phrase the matter. In effect, the answer in this book has been and remains that we have arrived at particle elementarity when there is no need to continue the search for it by further process of reduction because the consequent relationships and phenomena appear to be those we want…

Remember the chart we showed at the end of the Introduction? Here it is again (Figure 37-1) and at this point we are more or less at its right-hand side (at the top in this display!)…

Our cosmic snake with the enormous appetite is not the only way to portray interdimensional closure. In fact, as mentioned above, it is not exactly the kind of closure I had in mind all that long time ago. Another possibility is in terms of the so-called Klein Bottle, a sort of topological generalization of a Möbius strip (in fact slicing it the right way yields a pair of Möbius strips, with NHT = ±1). It features closure but in four dimensions and, in fact, in three-dimensional depictions, an extension of the entire bottle is inserted into a hole cut into the bottle itself, a kind of surgery not necessary in the actual re-entrant phenomenology of four dimensions. In any event, we are not prepared to talk about that now; maybe next time…

Perhaps you remember what I said in the Introduction about what I did to earn my keep when I was gainfully employed all those many years ago. A lot of it was involved with dynamic systems, most often linear, whose behavior, in one application or another, was described by an impulse response function in the temporal (or spatial) domain or, equivalently, by its spectral behavior, that is the response to each of a spectrum of frequencies usually denoted by its transfer function in the patois of that period, basically the Laplace or Fourier transform of the system's response to an impulse. The phase lock loop I talked about above is one such system but applications can of course be simpler, such as a thermostat or, more generally, a homeostat or even something more complex such as a large system with multiple inputs and outputs…

Perhaps a few words about the relationship of the alternative model to quantum considerations might not be inappropriate at this point. As mentioned in Sec. III, this book is not concerned with kinematics and therefore with the quantum mechanical uncertainty relationships of the associated canonical complementary dynamic variables. Nevertheless, there are some aspects of the model that coincide with the basics of quantum mechanics. To begin with, right up front we can cite the fundamental fact that the twist of a torus knot is quantized in the same sense that the De Broglie–Schrödinger electron's orbital waveform is quantized. Also we recall that, as the outer product of two vectors (whose components are fermions), the boson matrix, M, constitutes an operator that produces a set of states in the next higher fusion process. As we have seen, in detail (see Sec. II), that process is inherently statistical and its analysis calls for a quantum mechanical-like treatment of the associated degenerative states that can result…

In 2005, to commemorate the 100th anniversary of Einstein's “miraculous year”, World Scientific published a book entitled “100 Years of Relativity” with the subtitle “Space-time structure: Einstein and beyond”, ed. Abhay Ashtekar. It will probably come as no surprise that the emphasis on the “beyond” is essentially on the unification of General Relativity and Quantum Mechanics. The book is divided into three parts and a total of 17 chapters each by a different author with seven chapters devoted exclusively to the unification of General Relativity and Quantum Mechanics, often viewed as the piece that would make the jigsaw puzzle of elementary particle physics complete after its partial unification by Yang–Mills theory and its extensions (plus, of course, the Higgs contribution)…

Although we had a chapter (Chap. 34 in Sec. VI) on our model's relationship to String Theory, we probably did not spend as much time on it as we should have so, to make up for it, I would like to begin these concluding remarks with this profound assessment excerpted from a 1989 paper by Edward Witten: “In my opinion, the basic challenge in string theory is not, as sometimes said, to ‘understand nonperturbative processes in string theory’. In fact, the basic problem does not lie in the quantum domain at all. The basic problem, now and for many years to come, is to understand the classical theory properly.” And further: “Understanding the classical theory means above all understanding the geometrical ideas that parallel those of general relativity —”…

What has also emerged but very late in the development of this book appears to be a universally applicable principle which in application may come in more than one form and will take a while to explain. If you recall, we began this journey by talking about (2, n) torus knots and two-strand braids with closure, both of which may be viewed as forming a set of four rudimentary Möbius strips, three of which (according to the thesis of this book) constitute the basis for a taxonomy and, ultimately, of all the atoms in the periodic table, those atoms that underlie the consequent physics, geology and chemistry of Planet Earth and even the biochemistry of life itself, based, as it is on the geometry of those two-strand braids of DNA (Recalling Flapan's “Moebius Ladder”) and the four kinds of molecules that connect them! And another, overlapping, complementary set of three MS that (according to the speculative discussion in Section VI) constitutes a complete, complementary universe identified (in that section) as Dark Matter…

Well, the book finished with a bit of a flourish, uniting the seemingly disparate formalisms of the Alternative Particle Model with DNA, that well known, two-strand format upon which we depend for the perpetuation of our species and making possible the emergence of a unifying principle…

Note from publisher: This paper contains only quotes.

As per the Preface and Introduction, this book is about how the elementary particles constitute localized distortions of Spacetime and we are concerned with the geometric nature of that distortion. Broadly speaking, we can discuss the subject in terms of both the algebraic (essentially discrete) and the differential (continuous) aspects of that geometry. While this section emphasizes the algebraic aspects, as we will see, it is probably just as much about how geometry can be used to elucidate the algebra and vice versa.

Although this section is not absolutely crucial to the main subject matter of the book it provides some useful additional perspective…

Well, finally! I hope the preceding subject matter made the delay worthwhile!

Preamble: William Kingdon Clifford (4 May 1845–3 March 1879)

After I wrote most of this section it occurred to me that it would be fitting to say something here about Clifford beyond the mention in the Introduction; his views on space and matter were so strikingly prescient of what this book is about. Of course, we encountered his Clifford algebra in Sec. II, Chap. 4, but what is more germane here is where in the Introduction we talked about his views on the geometry of space. You may recall how he described matter as “little hillocks” of geometrical distortion that moved within and of it — very much like the basic ontological philosophy of this book and especially of this section; can you not just see those little hillocks wandering around…

Note from publisher: This paper contains only quotes.

A Look Back and Forth, In and Out, Around and About.

The following sections are included:

• Appendices
  • Appendix A: Illustrating the Double Traversal Requirement
  • Appendix B: Torus Knot Connection Terms
  • Appendix C: Curvature for the Lagrangian
  • Appendix D: Contingency in Second Order Fusion
  • Appendix E: The Beta Switch
  • Appendix F: The Alternative Model, DNA and Dark Matter Redux
• References
• Index
• SERIES ON KNOTS AND EVERYTHING