Overview for Ways of Looking at Physics Draft 0.1 Hiveism 2022-02-02 Introduction This is an overview to the ways of looking at physics I’ve begun writing about. As a purely conceptual overview it will be very dense and assumes a lot of prior knowledge, despite its casual language. It’s okay not to be able to follow it. It functions more as a sketch of a map to guide you through the rest of it, so you know where we’re going. It’s missing a lot of details, but there is just so much to talk about. Despite its limitations, this captures the whole story from non-duality, over the requirement of existence, boundaries, the need for quantum mechanics, the stationary action principle, the holographic principle, emergence of spacetime, gravity, particles and their forces and interactions with connection to string theory and other prior work, to the emergence of stable structure that allows for thinking beings able to ask what all of this is about. All seen as one coherent picture. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Unknowable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Octonions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Higher Dimensional Structures . . . . . . . . . . . . . . . . . . . . . . 9 24-cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Numerology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Conclusion and Open Questions . . . . . . . . . . . . . . . . . . . . . . 17 Philosophy of Consciousness and Alignment . . . . . . . . . . . . . . . 18 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1 Unknowable Let’s let go of all assumptions - easier said than done - so we can build up from there. With no place to start we cannot decide between and therefore have to assume all worlds. That means all possible ways something can exist. To exist is to be different from something else. Non-existence is the lack of differentiation - pure symmetry. Any existence, any differentiation is a break away from this pure symmetry. We can talk about difference in several ways. It can be a boundary or a relation - these are dual ways to look at it. Relations can be continuous, so the degree of freedom is also a dimension in some abstract space. Relations between relations span higher dimensional spaces. This gives an infinite dimensional tower of rela- tions (~infinity category) with top and bottom at infinity that can be described mathematically. And the thing that exists is itself located somewhere in this structure. This gives a perspective from which the rest of reality is observed (metaphori- cally speaking, not implying the need for a conscious observer). This perspective is itself a relation and hence a boundary (there is nothing else it could be), so I use these terms almost synonymously. This is related to category theory but more general. In category theory each object is only defined by its rela- tions (Yoneda’s lemma), but category theory does not look “inside” the objects. Here however there is no distinction between objects and relations (any such constraint would already be an assumption), each perspective is defined by its relations, but is itself also one. This gives it the dual property of a boundary - relating or separating one side from the other. As a boundary it separates the information it encodes versus the information it does not encode. In doing so it separates known from unknown, order from chaos, past from future, elliptic curvature from hyperbolic and so on. The boundary itself is the edge between order and chaos, the present, and flat. It is the perspective itself - the nature of being an observation of reality - that sets the scale and ground level (dimension 0). Chaos and order are here not only metaphors. To encode information requires some mathematical structure that captures all of it. For a mathematical struc- ture to be consistent means to close in on itself. But since Gödel we know that no formal system can be consistent and complete. The incompleteness surfaces as singularities - undecidable fixed points. Approaching these infinities, the smooth consistent structure diverges. Representing these infinities in a limited perspective requires some hyperbolic representation, as we will see later. Hyper- bolic is where the structure does not close on itself, elliptic where it does, and flat Euclidean geometry is in between these two - the edge of chaos. The space of relations is itself infinitely complex. The seeming order we observe as “physical laws” comes from the mathematical regularities that come with 2 observation. The probabilistic nature of quantum mechanics happens where the math of observation cannot capture the infinite complexity of relations. This subjective nature of reality also explains the role of observation in QM (related to the many worlds view). Conversely, where regularities exist, they compound when summing over all pos- sible relations from one to the other. Since the perspective encodes all relations at once, the actual observation of it is the superposition of all these paths over relations. This is the origin of the principle of stationary action. Looking “outward” from the perspective means simultaneously going to higher dimensions and tracing a light cone in relational space. But while this is infinite in nature, the regularities of mathematics ensure that there are some properties surviving. Many properties repeat every 8 dimensions (Bott periodicity). Cer- tain objects, like the icosahedron in 3D and the analogue 600-cell in 4D, only exist at a certain number of dimensions. And regularization can make sense of infinite sums (e.g. famously the sum of all integers can be seen as equaling -1/12). Mathematics, in that sense, describes which structures are stable in an infinite sea of relations. These are the properties of infinite series (through e.g. regularization) as well as exceptional structures that don’t fit into infinite series. However, mathematics as a pure platonic ideal existing “out there” wouldn’t explain physics. Everything that exists is located somewhere in this relational space and has a subjective perspective on the whole. This means we see the mathematical structures from the inside and are part of them. Or one might say that mathematics cannot exist without physics (both ways). This subjec- tivity seemingly breaks some of the perfect symmetry (it’s still perfect btw.). What survives this projection from an infinite structure onto a finite perspec- tive is what we see as our universe. This way physics gives a natural map of mathematics, from most to less symmetric, putting all of it in relation. Time The first symmetry break is that between existence and non-existence - the boundary itself. As a one dimensional relation it sets the zero on the real number line, splitting it in positive and negative, left and right part. As it turns out, there are exactly four number systems in higher dimensions that have similar nice properties like the real numbers. These are called normed division algebras and exist in dimensions 1, 2, 4 and 8. Each higher one loses a property that the lower ones have, this gives them distinct roles in physics. Beyond that the division algebra property itself breaks down. The 1D real numbers ( ℝ ) are special for their total order - a clear sense of one thing being before or after another. This allows for a notion of causality and linear time. This again makes sense as a perspective and boundary. It sees what it knows 3 (the information it encodes) as past and what it does not know as future. The linearity of time only exists as a retrodiction onto the known. This asymmetry at the same time also explains the arrow of time and the apparent increase in entropy. Splitting the real line gives a binary contrast, positive and negative as well as the notion of a scalar (magnitude of a number). But notably, in order for something to exist, it has to have a nonzero value on this time axis with a symmetric version on the “other side”. This explains baryogenesis, as matter and anti-matter are symmetric versions of each other, only differing by the time direction. The boundary itself needs a bias into one direction in order to exist and therefore the universe it observes seems to be biased. Waves The complex numbers ( ℂ ) extend the reals into a second dimension but do so at the cost of losing total order. On the real number line plus and minus are separated by zero acting as a hole in the middle. The higher dimension allows to transcend this duality by routing around the hole. The ability to rotate one scalar into another makes them vectors. In higher dimensions this rotation will then show up as waves and U(1) sym- metry. Complex numbers, vectors and waves are the language of quantum mechanics, describing states and probability. With the ability to go around a hole comes the notion of a topological loop. The boundary surrounding the hole is a loop and since all relations to the hole have to go through the boundary, it naturally encodes the information of the hole being there. The ability to go around several times gives integers, quantization and vibrational modes of string theory. As it is often said: a string in string theory is a one dimensional object. The answer to the question: “Made of what?” will become clear later: One dimension. The key property of the complex numbers is that they are commutative. This means the order of operands is interchangeable. This makes them nicely behaved in a mathematical sense, but is incompatible with a causally ordered past. Spacetime The 4D quaternion number system ( ℍ ) loses commutativity (still anti- commutative: ab = -ba) but on the other hand is the last one that is still associative. This brings time back into the picture as it matters what relations in what order are traversed. The associative property on the other hand, still allows causal relations to be chained in 4D - or alternatively: regions to be glued. Beyond 4D this property breaks down, which means that 4D is the maximum number of possible space dimensions. Since total ordering of time 4 is one dimensional we get one time dimension with three orthogonal spatial dimensions - the (3,1) signature of spacetime. Which dimension is the one of time, as well as the notion of locality is imposed by the perspective. The quaternions also double cover rotations in 3D (similar to how a 24-hour day “double covers” a 12 hour clock), which gives us spinors - the building blocks for fermions as matter (left spin) and anti-matter (right spin). Bosons are undecided regarding spin - they are the vectors from above. With 3D space emerging as a “natural” way to look at relational structure this begs the question where the remaining infinite dimensions went. This is where it gets interesting. It essentially requires a projection from some infinite dimensional structure onto a 3D space. All the other dimensions don’t get lost - they instead become the content of spacetime. What is observed is a light cone of 3D space (since the quaternions are associated with the (3,1) Lorentz group) extending back in the time dimension. The boundary as light cone sets a 3D Euclidean reference - space extending out in an, on average, flat spacetime. Everything inside the light cone is “known” and has elliptic curvature, compactified dimensions - matter. What is outside is “unknown” and hyperbolic - dark energy and quantum uncertainty. In the infinite tower of relations, looking out in space corresponds to tracing relations “outwards” (spacetime) and simultaneously “upwards” (past). Rela- tions can be interpreted as transformation of information. So that a constant rate of information change gives the speed of causation and hence light as fun- damental unit. Down in this tower is the light cone of what the perspective will affect in the future. Since the perspective sets its own scale, this future light cone is unknown to it. While the known appears as macroscopic certainty, the unknown appears as microscopic uncertainty. The scale of this uncertainty - set to 1 as natural unit - is the Planck scale. Progression in time means resolving this uncertainty. But with no absolute ref- erence frame, each new moment is re-scaled to these natural units. The content of spacetime are the patterns that are stable under this re-scaling. Primarily: particles and black holes as the fixed points of renormalization flow. More on where the particles come from later. Let’s look at the background first. These fixed points are like defects in spacetime - holes, singularities. Probing or calculating them will result in infinities because they are outside the limited structure of the mathematics of spacetime. When particle physics models them as points tracing lines as they move, then this is an abstraction that implies infinite certainty in their location and nature. This works as long as we don’t look too closely, but when they interact - say an electron and positron meet to form a photon - this means that a transformation takes place. Then we theoretically have to look at all the infinite paths the particles can take. For a transformation to take place, particles need to leave their attractor states and find a new one. Spacetime does not contain the rules for how this happens. It has to occur in the uncertainty of the infinite space of relations, exploring all 5 possible options. It is undecidable, just like the bifurcation in complex systems is chaotic. String theory is the result of physicists finding a novel way to deal with this. Instead of modeling particles as points and lines, they account for the irreducible uncertainty by modeling them as strings and the world sheets they trace out. Knowing of boundaries we can now make sense of this. It means we are looking at a boundary around the particle from the outside. This is a different boundary than the one of the perspective of the observer itself. It happens between subject and object - something that the perspective observes. The boundary necessarily has to be orthogonal to the relations that cross it (since they are dual). So it encodes all the information of these relations. By modeling only the boundary, we only need to consider the relations and do not need to unpack the contents. We do not know the internals of particles for them to act as they do. This way we can realize that two phenomena of string theory that usually are thought of as unrelated are the same thing: Strings and holography. The string is a one dimensional boundary enclosing a disk that holographically encodes the particle content. It’s a way of looking at something fundamentally unknowable, it does not mean that particles are “made of strings” as it is often presented. These boundaries are 2D surfaces in space. But space itself only exists through relations, so everywhere in space one can imagine a boundary encoding the relations of one side to the other. This way space can be thought of as infinite nested boundaries that foliate it (like a volume of a book is made of thin pages). While one can imagine any possible way to draw these boundaries, it makes more sense to only look at those that are orthogonal to the gradient of information flow. If a particle is at a point, then the boundaries form nested shells at equal distance from that point. Relations can be seen as lines going from a boundary of one layer to the one at the next higher or lower layer (a fiber bundle). This gives a beautiful picture where the matter particles (fermions) are the information on the boundary, while the relations between them are the force particles (bosons). The hypothesized graviton (as closed string with zero vi- brational mode) then simply is the nature of boundaries and relations without content. Fields and space arise together. You’ve likely seen images of this be- fore. It’s the same as electromagnetic field lines and equipotential surfaces. The flux through the boundary encodes the amount of electric charge enclosed by it (Gauss’s law). The curvature of the information gradient also gives gravity. This clarifies four different approaches to explaining gravity: the graviton as boundary from string theory, information gradients for entropic gravity, rela- tions for gravity as entanglement and boundaries for AdS/CFT/dS - although this last connection is harder to make sense of. Erik Verlinde, who works on entropic gravity, also says that both area and volume have to be included when calculating gravitational force. One gives the usual inverse square law, while the other gives some correction at large distances that looks like dark “matter” anomalies. By a related reasoning and 6 Figure 1: Draft note: the pictures are currently only placeholders. in conjunction with AdS/CFT he also concluded that space has to expand - i.e. dark energy. This becomes more intuitive in the light of knowing that 3D space is only a projection of a higher dimensional structure. The apparent expansion of space is accounting for the scale-free nature of this projection from hyperbolic relational space onto a flat spacetime. In one way the “big bang” can be seen as the beginning of our universe, but in another sense it is just the horizon of what we can resolve given the limited scale. Then there is no true beginning of the universe as the direction of time is subjective. Given that, we can now understand “where the remaining infinite dimensions went”. The projection forced them to become the information within spacetime. Across boundaries, they surface as microscopic wormholes connecting parts of space which is entanglement (by the ER=EPR result). Within a boundary they are compactified into the strings and hence particles. It’s all relational structure seen through the lens of a perspective embedded in it. Octonions Structure with more than four dimensions cannot be extended in spacetime. They have to be hidden from sight, so that an outside observer only sees a projection or boundary. This is the case for the 8D octonions ( 𝕆 ). While they give us quarks, these have to be confined, forming protons, neutrons and other compound particles. This means they always have to occur in color neutral com- binations and cannot propagate freely alone. This arises from the mathematical 7 property of non-associativity (but still alternative: (ab)c = -a(bc)). Causality cannot be chained as in 4D, so the higher dimensional structure has to resolve before propagating. Trying to separate the quarks anyway will force some kind of tube of higher dimensions (a wormhole) connecting them. These flux tubes are the original inspiration for string theory and are what we now know as glu- ons. Their SU(3) symmetry is a subgroup of the exceptional 𝐺 2 group which is itself the automorphism of the octonions. The octonions are also very special for the fact that left and right spinors as well as vectors have the same dimensionality. This allows them to be exchangeable representations. This unique symmetry is known as triality . This is the seed for many exceptional structures like the E8 Lie group. More on how that is relevant later. Together with the other normed division algebras these combine as ℝ + ℂ + ℍ + 𝕆 or ℝℂℍ𝕆 for short. Such constructions have been explored by many. In particular, Cohl Furey succeeds in deriving the particles of the standard model from this combination. The “frog view” of the perspective explains why we get all of them together (so it does not have to be assumed) and why their symmetry is broken in some cases. The quaternions set the base of 4D spacetime. One dimension is singled out as time. This gives three spatial dimensions which can each pair with the time dimension to give three planes of complex numbers embedded in space. These complex numbers allow U(1) rotation and hence the electric field. The spatial dimensions can also pair up to give three further planes for the U(1) of the magnetic field. This allows the complex numbers to be unbroken and mediate causation throughout spacetime at the speed of light. The binary difference of the real numbers gives chirality of which the neutrinos are the mediators. The spin from 4D allows for electrons and their anti-partner. The confinement of the octonions shows that space and time are macroscopic features. If we zoom in on individual particles and their interactions the distinc- tion stops making sense. It becomes a more pure 4D (and beyond) structure which we only observe through the boundary. This way matter particles look like internally having 4D properties. And this is where we get the Higgs and the weak interaction from. Higgs Mechanism You see, so far I’ve been describing structure that purely derives from consis- tency and the limits of consistency. This would give only one particular way the universe could be. It would be an extraordinary coincidence that this particular structure also allows for all the properties that are relevant for the emergence of life. The anthropic argument expects that there are many universes to choose from of which at least one is suitable for life and that much of the apparent fine tuning in physical constants is just the selection effect. At least that’s how the standard argument goes. The critique is that this explains nothing as it could justify everything. The parsimonious application of the principle on the other 8 hand should assume only the minimal break from this pure structure to allow for life. And this minimal break - as far as I can tell - is the value of the vacuum expectation value of the Higgs field (VEV). This is our multiverse generator. The Higgs field is a scalar field with an SU(2) symmetry. This means it has a non-zero scalar value everywhere in space. Using the normed division algebras again we can also explain this. What the Higgs does is to mediate between left and right chirality versions of particles. Those that don’t alternate (the photon) propagate at the speed of light. Those that switch between both chiralities lose some of the speed in this back and forth movement. They are still moving at c, just that part of it is localized in internal movement. This effectively slows them down and hence looks like them gaining mass (a bit like a 3D ball falling from a height being slowed down when rolling on the wall). If the VEV were zero, then the distance for alternating would also be zero, no mass would exist and no universe to speak of. But as established earlier, existence requires a break in the first dimension - time - resulting in an asymmetry between matter and antimatter. This break results in an overall background of a scalar (the size of the ball in the analogy). This means the Higgs field acquires its “Mexican hat” potential. Moving around it requires a non-zero distance which then slows down particles. The Higgs mechanism also gives rise to the bosons of the weak force. They effectively exist only in the 4D confined version, which is why the weak force has a short range. With this all particles of the standard model have been covered, except the three generations. But the octonions have the triality symmetry which allows to exchange three identical representations of the same structure. This can be a hint. If we would continue the doubling pattern of the dimensions associated with the division algebras, we’d end up with number systems that are no longer alternative and have (nontrivial) zero divisors and don’t give any new interesting symmetries. The octonions are in a way the largest structure that allows for certain properties, but triality also makes them highly symmetric. This gives rise to a variety of 8-periodic patterns in algebra, topology and other areas, called Bott-periodicity, and related phenomena like the Hopf fibrations. Since these properties repeat every 8 dimensions, this makes them stable in the infinite tower of relations. Higher Dimensional Structures I consider my main contribution here twofold. First in clarifying the ontological confusion that physicists have been struggling with. When you understand the structure of perspectives as boundaries and the groundless nature of existence, then you’ll have understood the core of it. Second in pointing out the bridge of this understanding to established physics and mathematics. For example how it allows for a simplified string theory. One that does not involve picking one 9 compactification from a landscape of possibilities and has no supersymmetric partners of the known particles (bosons and fermions are already their super- symmetric partner, but broken symmetry gives them their different appearance). There are several fascinating connections that suddenly start to make sense in light of this. However, it is also a highly complex area of mathematics, I don’t claim to understand all of it and there are still many things in the overall pic- ture of mathematics that we just don’t know. Therefore this part is more about providing an interpretation for what is already known, plus some guesses on how to continue exploring these structures. Building on the octonions there are higher dimensional exceptional structures that become relevant. Above we established that particles move through 3D space, but that their interactions happen in 4D and beyond. To predict anything about these interactions would theoretically require to sum over the infinite relational paths. However, just like the consistency requirements of mathematics allow us to observe regularities in 3D, higher dimensions do, in a similar way, add their own regular structures. This affects not only interactions but also seemingly static properties. For example, computing the magnetic moment of the electron using Feynman diagrams simplifies to a sum of terms that involves the Riemann zeta function and modular forms. That means that the electron encodes an infinite sum over more and more complicated structures that come from number theory. But why should infinite sums define properties of things that we finite beings can observe? The uncertainty imposed by our limited energy scale means that we observe boundaries that hide an infinite fractal of structures. Looking at them with more precision requires higher energies which subjectively transforms uncertain non-existence into certain existence. That is, when we measure particles or the vacuum using high energies, then virtual particles can become physical by using this energy. What differentiates virtual from physical is our subjective level of resolution. As an analogy one can imagine particle interactions like a landscape where stable particles are basins of attraction. To go from one basin to another requires over- coming some resistance. This means, traversing the structure through higher dimensions. Summing over Feynman diagrams is a bit like mapping a mountain range by exploring more and more roads and drawing a network of them, but never leaving the road to look out onto the landscape and see the big picture. The higher the resistance of traversing a mountain, the lower the probability and hence a lower contribution to the result. Highly symmetric structures are like mountain passes that allow this transition. Therefore mathematical regu- larities contribute to the sum the more symmetric and lower dimensional they are. With this in mind it is no longer surprising that some calculations and in particular string theory involve deep and beautiful mathematics. Probing with higher energies reveals more of these structures, but it is infinite and therefore not knowable to absolute precision. These interactions happen “outside” of spacetime. All we can see are the results as a projection onto some 10 boundary. In 3D space this means looking at a 2D boundary. Just like all the information of a black hole is holographically contained on its surface, so is the information (that is relevant to us) of particles and their interaction knowable through their boundaries. Black holes, in this sense, can be seen as a special kind of particle, where gravity overwhelmed the other forces. The idea that a lower dimensional surface can encode the information of a higher dimensional bulk is known as the holographic principle. It is the basis of the AdS/CFT correspondence, but with the picture sketched here it becomes clear that it goes much deeper than that, connecting different areas of mathematics. Topological information like holes can be described as waves on the boundary and nested boundaries form fields. One dimensional strings with their winding modes turn out to also be the simplest instance of this. The different modes that strings can take are related to possible standing wave patterns on a circle (with a duality of winding mode to quantized wave). The dualities of string theory gain a geometric meaning. Since the boundary en- codes the relation from inside to outside, it means that all information inside has a representation outside. So the charge of an electron surfaces as a potential in the electric field outside. This is T-duality. In a similar way one can look closer at a string and imagine another closed boundary that encloses the line. This new boundary subsequently encodes the information of the string to someone looking at it from the side. But this also works in the other direction, resulting in two interlocking loops, forming a torus. This then is S-duality relating, for example, electric and magnetic fields. In 4D two interlocking loops do not need to share any dimensions. They can be in two separate 2D planes (which is why 4D allows for two independent rotations). Embedding a torus with a 2D surface in four dimensions therefore allows the surface to be flat. Just like the surface of a cylinder is flat in 3D. This can be visualized as a torus. On this 2D surface information again is encoded as standing waves which form lattices (e.g. square and hexagonal lattice). A map of all possible 2D lattices and their relations is the j-function. The j-function again is at the intersection of different fields of mathematics: elliptic curves, modular forms and the Monster group. The connection to the Monster was so surprising that it was called “monstrous moonshine” by Conway. The proof by Borcherds involved vertex operators (“vertex” as in Feynman diagrams) and the no-ghost theorem, which come from string theory. The Monster group (please don’t let mathematicians name beautiful things) is the largest of the 26 sporadic simple groups. It is the biggest discrete symmetric structure that does not belong to an infinite series. In a way it represents the maximum discrete symmetry a structure can have. We started with pure symmetry as unknowable undifferentiated potential. Being unknowable it is described by continuous uncertainty. Now we arrived at the other pole of pure discreteness and certainty. Many of the surprising connections between areas of mathematics are between the continuous and the discrete. A 11 physical universe requires both: discrete certainty for structure and continuous uncertainty for freedom to evolve. Understanding this connection therefore can help shed light on some deep questions of mathematics. A way the Monster group can be constructed makes the connection to string the- ory apparent. This involves the Leech lattice which in turn can be constructed from three copies of E8 root lattice which in turn comes from the octonions. The octonions with their triality are the seed of many highly symmetric struc- tures. From them we get the five exceptional Lie groups G2, F4, E6, E7 and E8 with the associated root lattices and algebras. G2 is the automorphism group of the octonions, while E6 and E7 are included in E8. F4 is also directly tied to the quaternions. We’ll revisit it later. The E8 lattice is special in that it is the smallest even unimodular lattice. Uni- modular means it is self-dual and evenness makes it work well for physics by avoiding the appearance of negative signs. Both requirements for high symme- try. Their combination (plus being positive definite) can only exist in dimen- sions that are multiples of 8. In 8D there is only one, the E8 lattice. In 16D there are two, 𝐸 8 × 𝐸 8 and 𝐷 + 16 which are the basis of the string theories known as E8×E8 and SO(32) respectively. In 24 dimensions something interesting hap- pens. Because 24 = 3×8, this means that the internal triality symmetry of the octonions can be externalized. We don’t get three but 24 even unimodular lat- tices of which the Leech lattice stands out as especially symmetric and densely packed (densest sphere packing in 24 dimensions). The internal symmetry of triality becomes a global symmetry and allows for this unique structure to exist. All these higher dimensional lattices have to be confined such that we can only see them through a boundary. In string theory terms the dimensions in which the lattice lives have to be compactified on the boundary in order to be “hidden”. Adding back the two dimensions of the boundary to the above structures (as transverse dimensions) gives us the standard dimensions of the different string theories 8+2 = 10 for superstring and 24+2 = 26 for bosonic string theory. Heterotic string theory has 10D as left moving strings and 26D as right moving. Of these 26 dimensions 16 have to be compactified to arrive at 10D string theory and this works with the two 16-dimensional lattices from above. This also connects to another exceptional lattice with the fancy name 𝐼𝐼 25,1 This is the unique even unimodular Lorentzian (not positive definite) lattice in 26 dimensions and it is used in one construction of the Leech lattice (Conway and Sloane), as it contains it. Lorentzian here means that one dimension is negative, which is why it has a (25,1) signature - not positive definite (25 - 1 = 24 = multiple of 8). This is a time dimension and gives it the notion of light cones, something the pure Leech lattice misses. This makes sense as it means that we have a 2D spacetime surface with a 24D lattice compactified on it. The analogs of 𝐼𝐼 25,1 in 10 and 18 dimensions, 𝐼𝐼 9,1 and 𝐼𝐼 17,1 , correspond to the other mentioned lattices and string theories. The Monster group in turn is a symmetry of the Monster CFT, which is a 26 12 dimensional bosonic string theory compactified on the Leech lattice torus. It also seems to play a role in black hole physics, TQFT and AdS/CFT. 24-cell One mathematical object that sits at the intersection of many of these things is the 24-cell. It is a regular polytope like a cube or an icosahedron, except it exists in four dimensions and has no analogue in any other dimensions. If “24-cell” is too technical you may also call it “hyperdiamond”. I can tell you, it really is beautiful enough to deserve a name like that. It is both self-dual and has triality symmetry. Have a look at this 4D polytope viewer, choose the 24-cell from the dropdown and follow along enabling triality and projection etc. as we go on. It has several rough edges, but it does the job. Regular polytopes are actually lattices on a sphere of one dimension lower. But the 24-cell is also a lattice in another sense. It is possible to tessellate 4D space with these objects, which gives the densest (known) packing of spheres in 4D. Looking at the vertices of this tessellation gives the D4 lattice. Including the dual version gives F4. The D4 root lattice (not to be confused with the dihedral group of order 4) is itself the reason triality exists. Figure 2: Coxeter diagram of the D4 lattice with threefold symmetry on the left. Feynman diagram of boson (vector) and fermion (spinor) interaction. Images taken from John Baez’ blog. As a consequence the 24-cell also exhibits triality while being a 4D object. In particular, three 16-cells (8 vertices each) can be inscribed in it to give the 8 × 3 = 24 vertices of the 24-cell. The 16-cell in itself is the 4D analogue of the octahedron. Just like the three axes of 3D space span the octahedron, so do the four axes of 4D space span the 16-cell. The 16-cell therefore encodes a representation of 4D space. 13 Figure 3: Cross-eye picture of the 24-cell (grey) with three inscribed 16-cells (RGB) Importantly, a perspective in 4D space is part of the structure. When choosing any point in 4D as a perspective, the way to look at the 4D structure is con- strained in the same way we established earlier in the discussion of the normed division algebras. Choosing a perspective therefore is like choosing a vertex of the 24-cell to look at the rest. This vertex is part of only one 16-cell (as they don’t overlap) - breaking triality symmetry. It is part of one axis out of four - choosing time with three spatial axes left over. And it chooses a side on the axis - a direction of time. A perspective effectively sees a lower dimensional projection with three axes spanning the octahedron and hence 3D space. The vertex of the perspective has six edges connecting it to the ends of the three spatial axes. Therefore each edge connects two dimensions, one space and one time, which makes for a complex space that has U(1) rotational symmetry. Since this exists for each spatial axis this allows for oscillations in time throughout space. These are the waves of the electric field. Likewise, the vertices of the spatial dimensions connect to each other for the magnetic part of electromagnetism. The oscillation itself is possible because the uncertainty principle also includes uncertainty in time and energy. The U(1) symmetry can also be viewed as the fourth dimension being com- pactified into a loop. This gives the possibility for a hole inside the loop which becomes a defect in spacetime - the fixed point view of particles. By the relation of electric and magnetic fields we get two interlocking loops, each describing the other. These form a torus in 4D. The projection down to 3D gives spin. Having spin makes these particles fermions. For there to be some particle requires information and hence a break in symme- 14 try. The simplest break is parity again, which gives us the basis for neutrinos as simplest particles. Together with the Higgs mechanism these form a sea of positive Higgs VEV. Spin also allows for charge, which gives the electron and positron. The particles of the weak force bosons, quarks and gluons are higher dimensional and hence confined. They are made of the same relational non-stuff as the other particles, just living in a different space. This way one 16-cell gives us a complete set of spacetime with one generation of fermions and forces. Now recall that the 24-cell allows for three interlocking 16-cells. The other two are alternative frames of reference. These too would result in the same physics. But these alternatives have not been covered yet as none of the vertices of these 16-cells overlap. Interaction in these frames of reference are still possible, but we only see a “sideways” version of them. Seen from a perspective limited to one spacetime, it seems plausible that these second and third sets of fermions then look like three generations of particles. Yet I’m still missing the mechanism and details that would allow one to derive the mass differences. Now, why should it be possible to map the contents of our universe onto the structure of a 24-cell? It’s not like the vertices are located somewhere, they instead represent the discrete symmetries that we necessarily have, which allow for different perspectives. These are given by the normed division algebras and we can count them: • 2 from the real numbers, as th