## Hyperset Networks

### Proposed Mathematical Framework for a Grand Unified Theory of Physics (aka Quantum Gravity, and possibly Consciousness)

**by Paul Hughes**

**January, 2013**

**Note to readers:** *I’ve hesitated to make this public, as it’s very much a work in progress. Below are more summary notes, and various scribblings I’ve made over the past few years. Although I’d been thinking about this domain for much of my life, nothing compares to the “aha” moment I had more recently that has convinced me this theory holds *the key* to solving quantum gravity, not to mention opening up new dimensions we never considered before. After a very extensive search of the scientific abstracts, I was unable to find any reference to “hypersets” in relation to advanced theoretical physics dating all the way back to the birth of quantum physics in the 1920’s. At the very least, I was expecting at least one mention, if for no other reason than to discount it as a valid predicate in building a grand unified theory, or an actual theory of quantum gravity itself. So please dear reader, read the following with that in mind. These are just my thoughts and ruminations on how our universe might surprise us even further.*

A few months ago, while staring at an Amanda Sage painting called *Eggscension*, the original of which I’m lucky to have in my possession, I had a profound experience of something like *quantum* cosmic consciousness. In this experience there was no distinction between the very small and the very large, between a Planck scale quanta and the expanse. All of my perceptions and conceptions of “scale” simply collapsed into an infinitely nested singularity. Every part of the universe contained every other part. Evert point contained the universe, and not in some fuzzy low resolution way like a hologram, but the whole thing! There was no beginning or end, no cause and effect, only a collapse of seeming paradoxes into full unification, where big and small, inner and outer universes became one. Something like this has been the holy grail of physics and science for almost a century. So it was quite a shock when the whole thing finally made sense to me. I sat on this experience for a while, letting it gestate and integrate. I understood it intuitively but had yet come to any mathematical formulation. Then one evening I made the connection to *hyperset theory –* a rather errant branch of mathematics that seems perfectly suited to describing this *grand unification*.

It seems obvious to me that if we are to arrive at something that unites all of these forces, including the “geometry” of gravity, we need to devise something that is *pregeometric* in nature (something the physicist John Archibald Wheeler advocated). *Something that has connectivity (i.e. entanglement) at it’s core, but is independent of topology and dimensionality.*

In math, we often think the simplest concept is a number like 0 or 1, but the simplest concept is something called a *set*. A set is a collection of entities, like a bag of groceries, or even a bag with nothing in it. A set that has nothing in it is called an empty set:

So what happens when we have a set that contains itself as a set? We get something very interesting. If the set is a bag, then we would get a bag that contains a bag, that contains a bag, that contains a bag, and so on to infinity. Since it doesn’t contain anything other than itself, is the bag empty or full?

This is an example of a *hyperset* . Hypersets are sets which contain themselves as sets. They are part of *Naive Set Theory*. Hypersets can result in all sorts of interesting paradoxes such as *Russell’s Paradox*, where you can end up with sets that are both themselves and not themselves simultaneously. This is kind of like saying a donkey is not a donkey. Or you get something like the Liar’s Paradox such as “this statement is false”. These paradoxes bothered some people so they created *Axiomatic Set Theory* to eliminate them. However, many mathematicians prefer keeping them around because they are the *natural* result of valid mathematical reasoning. One of these people was Kurt Gödel, who with his Incompleteness Theorem showed that self-reference cannot be banned from mathematics. And besides, these natural paradoxes seem to explain how the universe actually works – just look at quantum mechanics! Ultimate questions about existence always seem to involve paradox. Examples: Why are we here? Why is there something, rather than nothing? If there was a beginning, what happened before the beginning?

Since sets are the simplest mathematical form, then any theory of “everything” should include them as a foundation right? And since hypersets (as I’ll explain) beget unlimited complexity, a theory of “everything” should include those too right? However, after doing an extensive search in the peer reviewed literature on St*ring Theory* and *Loop Quantum Gravity*, I was unable to find any mention of hypersets anywhere in the search for a unified physical theory. This is astonishing to me! Hyperset’s simplicity and profundity explain more clearly the paradoxes of quantum physics, how we can go from nothing to something, and how the very small world of quantum mechanics and the very large world of space-time can be unified within a single and simplified mathematical framework. If we give mainstream physicists the benefit of the doubt, there should be at least one mention of hypersets in the literature, if nothing else than to discount hypersets as useless in solving the unification problem. Since there is no mention of them at all, the only conclusion I can make is they are missing the obvious. This would not be the first time. How many times has a major conceptual breakthough been “obvious” in hindsight? This is something I have long suspected about the current pursuit of a unified theory of physics, given the overwhelming complexity that makes up contemporary string theory. Clearly they are not right and are in fact down some blind alley, otherwise they would have solved the quantum gravity problem by now. Or at least made serious progress towards testable predictions. They haven’t, which is why some physicists have said that string theory has gone so far off into the weeds that it is not even wrong. Obviously, the final equations of a fully unified theory of nature would have to include all of the unique constants and properties of our particular universe, but those equations should exists within a simple and elegant mathematical framework like hypersets. Since they are also quantum hypersets, they would be *probabilistic* hypersets, as Bucky Fuller might say in summation:

**Universe is infinitely nested hyperset probability networks.**

Which really means it’s an infinitely nested fully unbounded multiverse, of which temporal causation is a step-down from what is fundamentally acausal.

So what happens when we define a set as containing itself? The result is *infinite self-recursion*. Out of nothing, we get the beginning of “something”, in this case SPACE, or more accurately – *super-space* – an infinite dimensional space. On the other side of the metaphysical spectrum, in an idealist metaphysics, such as that of the late great Franklin Merrell-Wolff would say, “Consciousness without an object”

The empty set contains itself. Therfore it contains something, but that something is nothing, or because it’s something with nothing in it, it would more accurately be called empty. Hence we’ve gone from nothingness to emptiness (aka. infinite super-space). Hence it is a non-well-founded set, or hyperset, or empty hyperset. Any parts of the empty hyperset are identical, either a large part (O) or the singleton {O} ; the union of empty sets is also the same:

.

Expanding outwards, these nested hypersets can split into “different” sets, each set containing itself and all other sets. Example:

However, because each set contains the same as every other set they are still the same set, so we’ve gone from emptiness to “somethingness” in the absolute minimal sense of that term, or what G. Spencer Brown’s in his *Laws of Form*, calls the first difference that makes a difference.

The next step is to create sets that contain every other set, except for itself. When you do this, interestingly, you’ve still have a hyperset, it’s just that the original set is one degree down – hidden within one of the subsets, and in this case just one subset away. Mathematically this is what I call 1st degree hypersets – one degree of seperation between the original set and an identical subset. Example:

For the first time we have three unique sets! But this number could just as easily be an infinite number of unique sets. Yet they are still hypersets with a degree of separation of 1. To have unity, you have to go from either A to B back to A. or A to C and back to A.

However in all of this there still is *no time*. Without time, it would be meaningless to say one set gave birth to another set. Since each set contains itself, or is contained at some subset down in the recursive chain, this means there was no initial set! Since Set A contains Set B which contains Set C, which contains Set A, it’s a loop, with no beginning or ending.

These are not closed loops, but open recursive loops, which are both internally and externally expanding and contracting simultaneously. This is an incredible far-out and seemingly paradoxical thing and has **huge** implications, because we are talking about something that is both infinitely within and without simultaneously! Both an incursion and excursion, infolding and exfolding – a full union between inner and outer space. If this is true, and I believe it is as demonstrated by the rigorous framework of Hyperset Theory, then:

**The underlying fabric of existence is an eternal, hyper-dimensional, hyper-connected, synergistic, interpenetrating network between observer and observed, with infinite degrees of freedom and possibility.**

As the nested hypersets get more diverse with greater degrees of separation (Hausdorff distance?), out of this eternal network emerges an endless dance of beautiful and astonishing multidimensional forms and expressions, including the universe we see today. Because every set (quanta) ultimately contains a copy of itself, every quanta in the universe is an instantaneous portal to every other quanta as predicted by Bell’s Theory and quantum entanglement. And because every set ultimately contains itself and all others sets. So instead of saying every part of the universe is connected to every other part, it would be more accurate to say,

**Every part of the universe not only contains every other part, it is every other part.**

We just don’t notice it because it’s happening at the Planck scale. This is very similar to what the *Holographic Paradigm* has been saying for decades, except each part isn’t a lesser detailed version of the whole. Each part *contains* every other part – i.e. every part not only contains the whole, it is the whole.

Our experience of the infinite or the finite depends on how “far up” the hyperset network we are. As I mentioned earlier, this core logic is pre-geometric, therefore it is pre-scale. “Scale” is a higher ordered emergent phenomena that comes from these interacting nested hyperset loops. They are not “strings”.

The trick then is describing how we went from this eternal hyperset network to the specific and more finite properties of our particular universe. The pathway from here to there should entail an increasing hyperset unity towards the “empty” set which contains all sets. And should this unity be described mathematically how it shapes our universe, then a grand unified theory will have been achieved. On the idealist front, mystics and psychonauts have been talking about these insights for ages. Like I said, I’m not skilled at things like *Lie Algebras*, and *Non-Abelian Group Theory*, to develop a more rigorous mathematical formulation, so my hope is someone will take this where I cannot.

To recap:

- Nothingness to Emptiness – We started out with an infinitely self-recursive set of nothing, which gives us both zero (nothing) and infinity (everything) simultaneously, but the results is still 0, with a degree of seperation of 0.
- Emptiness to Something– Then we create many different sets that contained themselves and other sets, giving us sets from 1 to infinity, but the result is still just one identical set, but with degrees of seperation of both 0 and 1.
- Something to Everything – Then we created many different sets that don’t contain themselves, but which contain others sets, which eventually might contain the original sets. So we get sets from 1 to infinity, with degrees of speration from 1 to infinity.

The animation is a simplified visual notion of how self-recursion from a singleton might give rise to beautiful diversity and complexity.

**An Idealist Perspective:**

So we go from nothing to something to everything. From unity to disunity (symmetry breaking), resulting in diversifcation of sets and complexification that starts to look like our early universe. Continue this diversification of sets long enough and you get the complex mathematical structures and hyper-dimensional fractalization, of which one variation set is our modern day universe. This is not at odds with the conscious observer created universe. The first set, the empty set could be consciousness itself – the *void* of Buddhism. The first empty set containing itself, is conciousness turning back on itself, obvserving itself. The *empty vessel* of Buddhism, the place of no form. See John Lilly’s Before The Beginning. From this perspective, set diversification is consciousness co-creating, making differences that make a difference.

Although I still don’t know the precise set complexification that resulted in our particular universal contants, what this does show is how both singleton/singularity/quantum recursions gives rise to space, time, matter and energy – a unification of Quantum Mechanics and Relativity – Quantized Space Time, or the “Theory of Everything” that remains the holy grail of modern physics.

## Further Reading, Links, and References

### Hypersets

- Hyperset Theory – http://tinf2.vub.ac.be/~dvermeir/mirrors/www.cs.bilkent.edu.tr/%257Eakman/jour-papers/air/node8.html
- Hypersets – http://en.wikipedia.org/wiki/Hyperset – Set theory which allows sets to containt themselves.
- Naive Set Theory –http://en.wikibooks.org/wiki/Set_Theory/Naive_Set_Theory
- Kurt Godel with his Incompleteness Theorem showed that self-reference cannot be banned from mathematics in difference to Bertrand Russell and his paradox.
- Well Foundedness and Hypersets
- Huasdorff Distance (or Hausdorff Metric) – Hyperspace and set theory – In mathematics, the
**Hausdorff distance**, or**Hausdorff metric**, also called**Pompeiu–Hausdorff distance**,^{[1]}measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff. - Comparison of Hypertopologies
- Set Theoretic Topology – how to get from hypersets to hypertopological manifolds?
- Impredicativity and Hypersets – concept of impredicativity and the apparently attendant notion of non-well-founded sets, sometimes called hypersets.
- List of Paradoxes – http://en.wikipedia.org/wiki/List_of_paradoxes
- Paradox of set theory – http://en.wikipedia.org/wiki/Paradoxes_of_set_theory
- Russell’s Paradox – http://en.wikipedia.org/wiki/Russell’s_paradox
- Self Referentiality – http://crca.ucsd.edu/~syadegar/MasterThesis/node32.html This statement is wrong, etc.
- Liar’s Paradox
- Recursion – http://en.wikipedia.org/wiki/Recursion
- Infinite Loop – http://en.wikipedia.org/wiki/Infinite_loop
- Fractals generation
- Stuart Kaufmann – Origins of Order – Auto-catalytic Sets – Auto-catalytic Hypersets?
- Self-recursion growth in biology
- Cellular Automata
- Artificial Life
- Stephen Wolfram – A New Kind of Science
- Hypersets and the formation of “mind” and “self”, through self-recursion.
- Books –
*Non-Well-Founded Sets*by Peter Aczel.*The Liar*by Barwise and Etchemendy.*Vicious Circles*by Barwise and Moss - Early/Original Work –
- Within computer science, there is the notion of computability. This is the idea of whether it is possible to calculate something or not. It is provable that there exists a set of problems that cannot be computed See the Halting Problem
*I*f we take the attributes of the traditional Christian god, that he is all knowing, and all powerful, it appears this creates a logical contradiction. Here is an example:- Event A causes event B within the universe.
- God decides that he does not want event B to occur so he stops event A from happening.
- Event A no longer happened and therefore God would never intervene in the first place. Go to 1.

This is a very simplified example, and suffers from the problem that it is grounded in the concept of time, and this would imply God is able to circumvent this paradox because of the fact he is outside of time.

### Unified Field Theories

**String field theory**is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory. This is accomplished at the level of perturbation theory by finding a collection of vertices for joining and splitting strings, as well as string propagators, that g…**Superstring theory**is an attempt to explain all of the particlesand fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings. Superstring theory is a shorthand for**supersymmetric string theory**because unlikebosonic string theory, it is the version of string theory that…**M-theory**is an extension of string theory in which 11 dimensions are identified. Proponents believe that the 11-dimensional theory unites all five 10 dimensional string theories and supersedes them. Though a full description of the theory is not known, the low-entropy dynamics are known to be…- Heim Theory – 12 dimensional unified physics theory gaining some traction.
- D-Branes – In string theory,
**D-branes**are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai,Leigh and Polchinski, and independently by Hořava in 1989. In 1995, Polchinski identified D-branes with black p-brane… - Supersymmetry
**supergravity**(**supergravity theory**) is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry (in contrast to non-gravitational supersymmetric theories,- Loop Quantum Gravity – http://en.wikipedia.org/wiki/Loop_quantum_gravity
- Loop Quantum Cosmology – http://en.wikipedia.org/wiki/Loop_quantum_cosmology
- Spinors, Twistors, Roger Penrose
**Symmetry breaking**in physics describes a phenomenon where (infinitesimally) small fluctuations acting on a system which is crossing a critical point decide the system’s fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations (or “noise“), the choice will appear ar.. – ** These infinitesimally small fluctuations are the differences that make differences in the hyperset network divergence – see also Laws of Form.- In particle physics,
**supersymmetry**(often abbreviated**SUSY**) is a proposed symmetry of nature relating two basic classes of elementary particles: bosons, which have an integer-valued spin, and fermions, which have a half-integer spin. Each particle from one group is associated with a particle from the other, called it… - Holographic Principle – http://en.wikipedia.org/wiki/Holographic_principle –> improvement on this theory is each set contains all the others, rather than just has information about the others connections.
- Holographic Universe – Michael Talbot
- Implicate Order – David Bohm
- Grand Unification Epoch – In physical cosmology, assuming that nature is described by aGrand unification theory, the
**grand unification epoch**was the period in the evolution of the early universe following the Planck epoch, starting at about 10−43 seconds after the Big Bang, in which the temperature of the universe was comparable to the charact… - In theoretical physics, a
**super-Poincaré algebra**is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symmetries), and are Lie superalgebras. Thus a super-Poincaré algebra is a**Z**2 graded vector space with a graded Lie bracket such that the even part is a Lie algebra containing the Poincaré algebra, and the odd part is built from spinors on which there is an anticommutation relation with values in the even part **Planck Epoch**– In physical cosmology, the**Planck epoch**(or**Planck era**) is the earliest period of time in the history of the universe, from zero to approximately 10−43 seconds (Planck time). It is believed that, due to the extraordinary small scale of the universe at the time,quantum effects of gravity dominated physical interactions..**Planck temperature**, denoted by TP, is the unit of temperature in the system of natural units known as Planck units. It serves as the defining unit of the Planck temperature scale. In this scale the magnitude of the Planck temperature is equal to 1, while that of absolute zero is 0.**Planck units**are physical units of measurementdefined exclusively in terms of five universal physical constantslisted below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units have profound significance for theoretical**Yang–Mills theory**is a gauge theory based on the SU(*N*) group, or more generally any compact, semi-simple Lie group. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-Abelian Lie groups and is at the core of the unification of the Weak and Electromagnetic force (i.e. U(1) × SU(2)) as well as Quantum Chromodynamics, the theory of the Strong force (based on SU(3)). Thus it forms the basis of our current understanding of particle physics, the Standard Model.- Particle Physics and Representation Theory – The connection between particle physics and representation theoryis a natural connection, first noted in the 1930s by Eugene Wigner, between the properties of elementary particles and the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give…
- A
**Calabi–Yau manifold**, also known as a**Calabi–Yau space**, is a special type of manifold that is described in certain branches ofmathematics such as algebraic geometry. The Calabi–Yau manifold’s properties, such as Ricci flatness, also yield applications in theoretical physics. Particularly in superstring theory, the… **Triality**– John Frank Adams (1981), Spin(8), Triality, F4 and all that, in “Superspace and supergravity”, edited by Stephen Hawking and Martin Roček, Cambridge University Press, pages 435–445.

### Unification Mathematics/Philosophy of Relativity and Quantum Theory

- Consistent Histories Formulation – Quantum Cosmology’s
*Consistent Histories Formulation*– http://en.wikipedia.org/wiki/Consistent_histories - Topos Theory (Chris Isham) in a Nutshell – http://math.ucr.edu/home/baez/topos.html
- An Informal Introduction to Topos Theory – http://arxiv.org/abs/1012.5647
- Fotini Markipoulou-Kalamara –
**Eternalism**is a philosophical approach to the ontological nature of time, which takes the view that all points in time are equally “real”, as opposed to the presentist idea that only the present is real. Modern advocates often take inspiration from the way time is modeled as a dimension in the theory of relativity, giv

### Famous Theorists – String and Loop Quantum, etc.

- Chris Isham – theoretician, may be the person to contact about hyperset networks
- Ed Witten – Super smart guy at IAS at Princeton
- Paul Townsend
- Roger Penrose
**Charles W. Misner**is an American physicist and one of the authors of*Gravitation*. His specialties include general relativity andcosmology. His work has also provided early foundations for studies of quantum gravity and numerical relativity.

### General Relativity

- The
**Friedmann–Lemaître–Robertson–Walker****metric**is anexact solution of Einstein’s field equations of general relativity; it describes a homogeneous, isotropic expanding or contractinguniverse that may be simply connected or multiply connected. The general form of the metric follows from the geometric properties of hom… - Twistor Theory – In theoretical and mathematical physics,
**twistor theory**maps the geometric objects of conventional 3+1 space-time (Minkowski space) into geometric objects in a 4 dimensional space with metric signature (2,2). This space is called twistor space, and its complexvalued coordinates are called “twistors.”

### Quantum Mechanics

- Superposition Principle – two states simultaneously – is and is not simultaneously – hypersets can have sets that are not themselves.
- Double Slit Experiment
- Heisenberg’s Uncertainty Principle
- Einstein Rosen Podolsky Experiment
- Bell’s Theorem
- In quantum mechanics, the
**Wigner–Weyl transform**or**Weyl–Wigner transform**is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operatorsin the Schrödinger picture. Often the mapping from phase space to operators is called the**Weyl transform**whereas the mapping fr

### Quantum Field Theory (QFT, QED, QCD)

*QFT – Quantum mechanics at relativistic speeds (i.e. particle physics).*

- Constructive Quantum Field Theory – In mathematical physics,
**constructive quantum field theory**is the field devoted to showing that quantum theory is mathematically compatible with special relativity. This demonstration requires new mathematics, in a sense analogous to Newton developing calculusin order to understand planetary motion and classical gravi… - Non-Commutative QFT – Heisenberg was the first to suggest extending noncommutativity to the coordinates as a possible way of removing the infinite quantities appearing in field theories before the renormalization procedure was developed and had gained acceptance. The first paper on the subject was published in 1947
- The quantum electrodynamic vacuum or QED vacuum is the field-theoretic vacuum of quantum electrodynamics. It is the lowest energy state (the ground state) of the electromagnetic field when the fields are quantized. When Planck’s constant is hypothetically allowed to approach zero, QED vacuum is converted to classical vacuum, which is to say, the vacuum of classical electromagnetism.

### Sub Topics

- Geometrodynamics – http://en.wikipedia.org/wiki/Geometrodynamics

**S-duality**(also a**strong–weak duality**) is an equivalence of two quantum field theories or string theories. An S-duality transformation maps states and vacua with coupling constant*g*in one theory to states and vacua with coupling constant*1/g*in the dual theory…**compactification**means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic.**Positive Energy theorem**– in 1984 Schoen used the positive mass theorem in his work which completed the solution of the Yamabe problem.**Tolman-Oppenheimer-Volkoff limit**- Renormalization – In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures,
**renormalization**is any of a collection of techniques used to treat infinities arising in calculated quantities. - The positive mass theorem was used in Hubert Bray’s proof of the Riemannian Penrose inequality.
- In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons.
- The quark model was originally just a very good classification scheme to organize the depressingly large number of hadrons that were being…
- Two Time Physics – 2 dimensions of time.

### Mathematical Domains

- Hyperset Theory
- Set Theoretic Topology – Wikipedia, Math at Auckland U, Handbook of,
- Topos, Topi
- Algebraic Geometry and Topology
- Differential Geometry and Topology
- Lie Sets, Lie Groups, Lie Algebras
- a
**Lie algebra**is an algebraic structure whose main use is in studying geometric objects such as Lie groups anddifferentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term “Lie algebra” (after Sophus Lie) was introduced by Hermann Weyl in the **Lie group**is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups.- Non-Commutative Geometries
- Spin Networks – a
**spin network**is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of matrix groups. The diagrammatic notation often simplifies calculation because simple diagrams may be used to represent complicated functions. Roger Penrose is credited with the invention of spin networks in 1971, although similar diagrammatic techniques existed before that time. Spin networks have been applied to the theory of quantum gravity by Carlo Rovelli, Lee Smolin, Jorge Pullin and others. They can also be used to construct a particular functional on the space of connections which is invariant under local gauge transformations. - In mathematics, the
**representation theory of the symmetric group**is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum mechanics for a number - Echochamber – great forums for math/science/etc – http://forums.xkcd.com/index.php
**H-cobordism theorem**– Before Smale proved this theorem, mathematicians had got stuck trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were even harder. The*h*-cobordism theorem showed that (simply connected) manifolds of dimension at least 5 are much easier than those of dimension 3…- In mathematics and physics, in particular in the theory of the orthogonal groups (such as the rotation or the Lorentz groups),
**spinors**are elements of a complex vector space. Unlike spatial vectors, spinors only transform “up to a sign” under the fullorthogonal group. This means that a 360 degree rotation transforms - In mathematics, a
**Lie superalgebra**is a generalisation of a Lie algebra to include a**Z**2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the*even*elements of the superalgebra correspond to bosons and*odd*elements - Non-Abelian Gauge Theory – In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations. The term gauge refers to redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group which is referr…
- In geometric topology, the
**Clifford torus**is a special kind of torussitting inside**R**4. Alternatively, it can be seen as a torus sitting inside**C**2 since**C**2 is topologically the same space as**R**4. Furthermore, every point of the Clifford torus lies at a fixed distance from the origin; therefore, it can also be viewed as **4-manifold**is a 4-dimensional topological manifold. A**smooth 4-manifold**is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure and…- Exotic Sphere – In differential topology, a mathematical discipline, an
**exotic sphere**is a differentiable manifold*M*that is homeomorphic but notdiffeomorphic to the standard Euclidean*n*–sphere. That is,*M*is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar on… - K3 Surface – In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle. In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0. Together with two-dimensional complex tori, they are the Calabi-Yau Manifold.
- In algebraic geometry, the problem of
**resolution of singularities**asks whether every algebraic variety*V*has a resolution, a non-singular variety*W*with a proper birational map*W*→*V*. For varieties over fields of characteristic 0 this was proved in, while for varieties over fields of characteristic*p*it is an open problem in dimensions at least 4… **Quadrics**(as opposed to conics?) in the Euclidean plane are those of dimension D = 1, which is to say that they are curves. Such quadrics are the same as conic sections, and are typically known as conics rather than quadrics. In Euclidean space, quadrics have dimension D = 2, and are known as quadric surfaces. By ma…- the
**Riemann hypothesis**, proposed by, is aconjecture that the nontrivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. - the
**Poincaré conjecture**is a theorem about thecharacterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states: An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence… - the
**Cayley–Dickson construction**, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as**Cayley–Dickson algebras**. They are useful compositio… **Octonion**– Octonions and the Fano Plane Mnemonic (video demonstration)- Quaternion algebra was introduced by Hamilton in 1843. Important precursors to this work included Euler’s four-square identity (1748) and Olinde Rodrigues‘ parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra.
- The Hopf construction, viewed as a fiber bundle
*p*:*S*3 →**CP***1*, admits several generalizations, which are also often known as**Hopf fibrations**. First, one can replace the projective line by an*n*-dimensional projective space. Second, one can replace the complex numbers by any (real) division algebra, including… - Seven-Dimensional Cross Product – In mathematics, the
**seven-dimensional cross product**is abilinear operation on vectors in seven dimensional Euclidean space. It assigns to any two vectors**a**,**b**in ℝ7 a vector**a**×**b**also in ℝ7. Like the cross product in three dimensions the seven-dimensional product is anticommutative and**a**×**b**is orthogonal to both**a**a - Homotopy groups of spheres – In the mathematical field of algebraic topology, the
**homotopy groups of spheres**describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise ge… - In mathematics, a
**convex regular polychoron**is a polychoron(4-polytope) that is both regular and convex. These are the four-dimensional analogs of the Platonic solids (in three dimensions) and the regular polygons (in two dimensions). These polychora were first described by the Swiss mathematicianLudwig Schläfli in th… - In finite geometry, the
**Fano plane**(after Gino Fano) is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point… - In mathematics, the projective special linear group
**PSL(2, 7)**(isomorphic to**GL(3, 2)**) is a finite simple group that has important applications in algebra, geometry, and number theory. It is theautomorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements PSL(2, 7) is the s… - John Frank Adams (1996), Lectures on Exceptional Lie Groups (Chicago Lectures in Mathematics), edited by Zafer Mahmud and Mamor
**Chirality**– In geometry, a figure is**chiral**(and said to have**chirality**) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise…- A
**tetromino**is a geometric shape composed of four squares, connected orthogonally. This, like dominoes and pentominoes, is a particular type of polyomino. The corresponding polycube, called a**tetracube**, is a geometric shape composed of four cubes connected orthogonally… - In knot theory, a
**prime knot**is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be**composite**. It can be a nontrivial problem to determine whether a given knot is prime or not… - In mathematics, the
**symmetric group***Sn*on a finite set of*n*symbols is the group whose elements are all the permutations of the*n*symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself. Since there are(n factorial) possi*n*!

### 7 States of Matter Chart

- Quark Matter – Quark-Gluon Matter or Plasma or QCD matter refers to any of a number of theorized phases of matter whose degrees of freedom include quarks and gluons. These theoretical phases would occur at extremely high emperatures and densities, billions of times higher than can be pr
- Plasma
- Gas
- Liquid
- Solid
- Bose-Einstein Condensate
- Fermion Condensate

### Items and Tidbits

- Negative energy/mass, exotic energy/mass, differences?
**Absolute hot**is a concept of temperature that postulates the existence of a highest attainable temperature of matter. The idea has been popularized by the television series*Nova*. In this presentation, absolute hot is assumed to be the high end of a temperature scale starting at absolute zero,**Krasnikov tube –**A**Krasnikov tube**is a speculative mechanism for space travel involving the warping of spacetime into permanent superluminaltunnels. The resulting structure is analogous to a wormhole with the endpoints displaced in time as well as space. The idea was proposed by Serguei Krasnikov in 1995.- Exotic Stars
- Preons
- Gluons
- Glueball – In particle physics, a
**glueball**is a hypothetical compositeparticle. It consists solely of gluon particles, without valencequarks. Such a state is possible because gluons carry color charge and experience the strong interaction. Glueballs are extremely difficult to identify in particle accelerators, because they… - Gravitons and Gravitinos
- Quark Matter
- Quark Nova – A
**quark-nova**is the violent explosion resulting from the conversion of a neutron star to a quark star. Analogous to a supernova heralding the birth of a neutron star, a quark nova signals the creation of a quark star. The concept of quark-novae was suggested by Dr. Rachid Ouyed (University of Calgary, Canada) and Drs. - Strange Matter
- A
**strangelet**is a hypothetical particle consisting of a bound stateof roughly equal numbers of up, down, and strange quarks. Its size would be a minimum of a few femtometers across (with the mass of a light nucleus). Once the size becomes macroscopic (on the order of metres across), such an object is usually called - Lattic QCD –
**Tachyon condensation**is a process in particle physics in which the system can lower its energy by spontaneously producing particles. The end result is a “condensate” of particles that fills the volume of the system. Tachyon condensation is closely related to second-order phase transitions.- Imaginary time is a concept derived from quantum mechanics and is essential in connecting quantum mechanics with statistical mechanics.
- Imaginary time can be difficult to visualize. If we imagine “regular time” as a horizontal line running between “past” in one direction and “future” in the other, then imaginary time wo
- Multiple Dimensions of Time – The possibility that there might be
**more than one dimension oftime**has occasionally been discussed in physics and philosophy. ***An Experiment with Time*by J.W. Dunne (1927) describes anontology in which there is an infinite hierarchy of conscious minds, each with its own dimension of time and able to view even **Photodisintegration**is a physical process in which an extremely high energy gamma ray is absorbed by atomic nucleus and causes it to enter an excited state, which immediately decays by emitting a subatomic particle. A single proton or neutron or an alpha particle is effectively knocked out of the nucleus by the.**R-Process –**The**r-process**is a nucleosynthesis process, that occurs in core-collapse supernovae (see also supernova nucleosynthesis), and is responsible for the creation of approximately half of the neutron-richatomic nuclei heavier than iron. The process entails a succession of*rapid*neutron captures (hence the name r-process) b..- A
**pentaquark**is a hypothetical subatomic particle consisting of four quarks and one antiquark bound together (compared to three quarks in normal baryons). As quarks have a baryon number of +, and antiquarks of −, it would have a total baryon number of 1, thus being classified as an exotic baryon. - The
**skyrmion**is a hypothetical particle related to baryons. It was described by Tony Skyrme and consists of a quantum superposition of baryons and resonance states… **Metalogic**is the study of the metatheory of logic. While*logic*is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves. According toGeoffrey Hunter, while logic concerns itself with the “truths of logic,

Exotic Mesons – Non-quark model mesons include

**exotic mesons**, which have quantum numbers not possible for mesons in the quark model;**glueballs**or**gluonium**, which have no valence quarks at all;**tetraquarks**, which have two valence quark-antiquark pairs; and.. In particle physics a tetraquark is a hypothetical meson composed of four valence quarks. In principle, a tetraquark state may be allowed in quantum chromodynamics, the modern theory of strong interactions. However, there has been no confirmed report of a tetraquark state to date. Any established tetraquark state would**hybrid mesons**, which contain a valence quark-antiquark pair and one or more gluons