Five Orthogonal Subspaces II | 12 Complex Dimensions, Quantum Reality & Enlightenment

Hinzugefügt: 2026-05-27

[00:00:14]So, we're back and we're now going to continue with our discussion of these five orthogonal subspaces of the arena that makes up our universe. So, here we go.

[00:00:31]We were talking about David Bow.

[00:00:34]David Bow uh is no longer with us, but Steven Wolfframe is. And Steven Wolffra is pursuing uh a physics project tackling the exact same mystery that we've been tracking through complex analysis and the philosophy of David Bow. But Steven Wolffrram is approaching it from an opposite direction. Instead of starting with a continuous elegant mathematical space like the complex plane, Wolram starts with something totally discreet, what's called the hyperraph, a collection of abstract points that are connected by relations.

[00:01:13]And and there's no notion here of any predefined space, time, or matter. is just simply a hyper graph, a collection of points collect connected by some sort of relations. So by applying simple local computational rules to this graph over and over again, the system grows.

[00:01:32]The profound realization of Wolram's physics project is these microscopic localized computational steps inevitably lock themselves into a vast global immutable structure that perfectly mirrors the law of physics. And that's what's truly amazing here. That's that's why it's interesting to and this is being approaching now from the point of view of uh modern computation uh and uh modern information technology.

[00:02:04]So we end up first of all with causal invariance which is the discrete version of the Koshi Reman rule that we touched on earlier. That was if you recall the kosher reman uh things involved the relationship between the real and imaginary parts of a function and their partial derivatives that they had to be appropriately lined up. So here's how the local to global emergence in wolram's universe connects to our thread of infoldment and rigidity. In complex analysis, we saw the strictness of the Koshi Reman equations that ensures that no matter what direction or path you take to approach a point, the derivative is always the same. This path independence gives the complex plane its rigid crystalline harmony. And so in the wolf from physics project, the equivalent concept is what's called causal invariance. When a local rule is applied to a hyperraph, there are often many different places on the graph, but that rule could be applied first. This generates a massive branching tree of possibilities called a multi-way graph.

[00:03:18]Each branch represents a different historical path of choice of the update order. So if a position possesses causal invariance, it means that even though the paths branch out into different histories, they're mathematically guaranteed to eventually merge back into one another. That means that that when we do something that the end result is invariant, we could go over this way, that way, the other way. Uh you could think of this as for example as timelines going in different timelines but at the end of the day we all end up at the same point the same timeline.

[00:03:55]So these paths that lead to these identical causal futures, we we start off with an initial hyperraph state.

[00:04:02]Then we take a path from path A that's taken or we take path B, two different paths. We have local branches there, but at the end of the day, we end up at the same place, the identical causal future.

[00:04:15]So what happens here in this causal graph is no matter what sequence of local choices you make the network of cause and effect relationships or the causal graph remains identical. So just like a holorphic function preserves angles and local geometric identity regardless of direction, a causal invariant rule ensures that the lo global largecale structure of spaceime remains stable, objective and immutable and it's unaffected by the chaotic microchoices made at the computational level. That's the key thing here is all all of these micro choices wash out causal invariance.

[00:04:57]So what in the world is space here?

[00:05:00]Well, space is now an emergent holographic crystal. In standard physics, we treat space as a smooth continuous background canvas. In Wolram's physics model, space doesn't exist. It's purely emerging out of this hyperraph. When you run a simple rule trillions of times, the sheer density of the abstract points and connections and the hyperraph begins to behave like a continuous space. You start to fill out everything because you're running this simple rule this time, that time, that time. You do it trillions of times. What started out as discrete, everything seems to get filled in. And therefore, it looks just like a continuous space.

[00:05:43]And if you zoom out far enough, you get a geometry that satisfies Einstein's equations of general relativity. So this mirrors the transition from local to global geometry. And the local rules we're dealing with here are the DNA or algorithm of the universe. And the global structure we're dealing with here is the smooth unyielding geometric fabric of spaceime. So just as the global behavior of the holomorphic function is rigidly bound to its local derivatives via analytic continuation, the large scale layout of wolram's emergence space is tightly bound to its local graph rewriting rules. You cannot alter a local computation rule in one corner of the hyperraph without fundamentally rewriting the entire global dimensionality and curvature of the resulting universe. There is a huge constraints here and he calls this Wolf takes this concept of its absolute philosophical limit with a construct that he calls the rule. If you consider not just one rule but the formal object created by running every possible computational rule simultaneously, you get the rulad, the rule of all rules, the ultimate implicate order. And this is the entangled totality of all possible computations. This rouad is the ultimate mathematical realization of David Bow's implicate order and its completely unified infinite unbroken computational matrix. Truly amazing to come at this from a totally different way.

[00:07:19]So the rule versus our physical universe, we have this ultimate implicate order in its entirety of all possible computational rules. And this leads to perception that observers have of our physical universe which gives us the explicate order which we call spaceime general relativity and quantum mechanics.

[00:07:41]So there's a deep connection here uh between these polymorphic functions the implicate order and the rule of wolf ram. Inside the rule, everything is non-locally entangled because every rule and state is connected. Space, time, particles, and even our own consciousness as observers are just localized slices or perspectives carved out of this single immutable global object. So we perceive separate objects moving through independent space which is the explicate order only because we're computationally bounded observers capable of seeing only a tiny facet of the grand interconnected mathematical crystal. There's a very deep connection here. Whether you look at it through continuous complex calculus which is holorphic functions whether you look at it through quantum metaphysics which is BM's implicate order or you look at it through discrete computation which is wolram's rouad the core insight remains the same separation is an illusion of scale the local fragment is never isolated it's an active expression of a local of a global immutable whole that is locked into place by a profound found structural harmony.

[00:08:58]Oops. I have to get the battery going here. Hold on.

[00:09:08]So, we go back here to our complex plane now. And here we've got a typical function Z, which is 4 + 4 I. So, we go out here four units on the real side, four units on the imaginary side, and we come up with Z right here. And there's an angle here of roughly 45°.

[00:09:28]So here here we go with a representation in the complex plane. So now let's take a look. Let's take a simple example here. Here's here we start off in in the top here and we have something that's simply the number one with no imaginary part one. Now let's just multiply this by the square root of minus1. Well, if we do that, that takes us up to here.

[00:09:51]Well, if we look at this, this simply took this point and it rotated it up here by multiplying by minus1. Let's take this point and multiply by minus1 again. That takes us over here. We've gone from here to here to here around the circle. Now, we're here at minus one on the real axis. Let's do it one more time. That takes us down here where we're again 90° going south. And so every time we multiply by the square root of minus1, we we end up going in some direction that's perpendicular to where we were before. So this is how multiplication by the square root of minus1 is going to make one subspace orthogonal to another subspace.

[00:10:35]So let's dig into this.

[00:10:41]We're going to go back to what we said at the beginning. In the beginning, Einstein in 1915 showed that gravity uh in his theory was not a traditional force but rather the bending and warping of four dimensions of spaceime, three dimensions of space, one of time.

[00:11:03]So Guza uh took a look at what Einstein did and said, "I wonder if electromagnetism is also just a warping of space time, but it's in a dimension we can't see." So he rewrote Einstein's gravitational field equations using four dimensions, four spatial dimensions plus one of time. And when he solved this then he came up with uh a a match here where he had Einstein's standard uh four-dimensional gravity equations but the other part that he came up with because of this extra dimension was the Maxwell's equations for electromagnetism.

[00:11:43]So by adding this fifth dimension Kuza had unified gravity and light.

[00:11:47]Electromagnetic waves like light were suddenly explained as ripples of vibrating in the fifth dimension.

[00:11:56]Truly amazing. Now Einstein looked at this and said, "Well, there's nothing wrong with what you did, but what in the world's going on here? Where is this other dimension?" So Oscar Klene, the physicist in 1926 uh provided an answer. He suggested that dimensions can come in two varieties.

[00:12:16]extended varieties which are large those are like the three dimensions that we see around us and then compact dimensions which are curled up dimensions. So think of a garden hose from a long distance away the hose looks like a one-dimensional line. Uh but when we get when we move forward or backward along its length uh that's all you can do. It's one dimension. But if you zoom in closely enough you can see that the hose has a second dimension. You can you can run around the diameter of the of the hose. So an ant walking on the hose can can walk around the length of of it, but it can also walk in a circle at at any point around the hose. So Klene argued that our familiar three spatial dimensions are like the length of a hose. They're vast and extended. But this other dimension, however, is compactified. It's curled up into a microscopic circle so imaginably small that we can't perceive it with our senses. Uh and every single point in our 3D space actually contains this tiny loop. So what client is saying here is let's look at a point in our 3D space.

[00:13:29]We go there and we zoom down really close and we we have a little XYZ three perpendicular axes there. When we actually look at it, we see that each of the three axes has a little piece of hose wrapped around it and that there's actually these other dimensions here because I can only run around in a circle um around the hose. He's saying that each one of these only adds one dimension.

[00:13:58]Now, the the problem with this is it's an interesting exercise, but uh is there anything really special that's going on here? like what which which of these dimensions is the one that has the the curled up or compactified dimension? So here here is just a different way of saying it here's the the the axis that we're looking along say it's the x- axis and the the hose is what's wrapped around it. So if we take take this and and unroll it, we see that we sure we can run around this this way. But in fact, we could run around it this way and this way. So actually what Klein did was he had two dimensions wrapped around the the hose. Uh he he was just thinking of one dimension as you'd go around there. But I I can run in a plane. And so every real dimension that I have say an x dimension I have two dimensions that are uh that are compactified that are associated with this. So if that's the case then that leads to a very different sort of thing uh but which we'll come to in a second. But the original theory uh was was was attacked because it couldn't accomplish everything. The first problem that they had here was when physicists tried to apply quantum mechanics to the theory.

[00:15:22]Uh there were all kinds of strange infinities and uh predictions of mass and so forth that uh weren't coming out of this and the uh Kuza climb k theory was designed to unify gravity and electromagnetism.

[00:15:40]But physicists later discovered two more fundamental forces. the strong nuclear force which holds positively charged atomic nuclei together and the weak nuclear force which is responsible for radioactive decay and a single extra dimension wasn't enough to count for that. This is actually wrong because electromagnetism in fact is what generates the strong force and the weak force. When you in electromagnetism, opposite charges attract. Positive and negatives attract. Except when you get very close, when you get uh uh very close, then then opposites uh repel and likes attract. And that's not obvious at all, but that's what happens.

[00:16:29]So when we go back uh uh to what Kuza Klein had done, it it never really died.

[00:16:36]It was it suggested to people there was something here and maybe we can cook this in some way. So the Italian American mathematician Eugenio Clabi was exploring uh things that are called uh Kalor manifolds. They're complex geometric spaces that blend complex numbers, calculus and topology. So instead of having a simply a a axis uh a dimension which is a a real axis we have a complex axis. Now this gets hard to visualize because uh as we just saw the complex plane has two two as dimensions to it but that's what these scalar manifolds are. So Kabi asked something that was rooted in Einstein's general relativity. Can a closed complex geometric space exist that features gravity which involves curvature but contains absolutely no matter or energy?

[00:17:37]That's called a vacuum. So in mathematical terms he conjectured a manifold certain topological conditions that must emit a unique metric here. And his intuition was that this this space that he came up with is a perfect vacuum. And he's betting you could have an incredibly intricate multi-dimensional space that curves into itself purely due to its own topology without needing any mass or energy to pull it into shape. So for over two decades almost the entire mathematics community including Clavi himself at time believed this conjecture was too good to be true and put all their efforts into trying to prove it was wrong.

[00:18:20]All right so we enter Shing Tong Yao in the early 70s. Yao was convinced that Kabi's conjecture was false. He spent years constructing elaborate counter examples to disprove it. Even presented one to Kabi himself at a conference. But afterwards he realized when Kabi asked for clarification on a detail Yao realized that his arguments had a flaw and this made Yao start to think that maybe this conjecture might be true. So he completely reversed his thinking and spent the next several years tackling the problem uh and converted this into a geometric question uh which was incredibly uh complicated technically.

[00:19:03]And in 76 he successfully proved what Kabi had said. And he showed that these flat vacuum spaces absolutely do exist.

[00:19:12]They're probably on the order of 10,000 of them. Uh and this proof earned him a feels medal. Uh so because of this the the two people here Eugene Kabi and Shung Yao they're called Cababial manifolds.

[00:19:28]So what in the world does a Cababial manifold look like? So by definition clavio manifolds are complex multi-dimensional spaces. The ones used in physics are six dimensions, six real dimensions or three complex dimensions.

[00:19:43]So here's a picture of one here, but this is just a snapshot. This thing is evolving over time and you can see it has holes and it has all kinds of curvature and structures to it every which way there.

[00:19:56]Look on the web, you'll find all kinds of animations of these. So, because human perception is limited to three spatial dimensions, we can't really visualize them. But we can look at 2D or 3D cross-sectional slices. So, if we look at the projection, you can see how the surface isn't really smooth like a simple sphere or flat like a sheet of paper, but it's an integrate web of interlocking loops, holes, and handles.

[00:20:23]That's what we have here all over the place.

[00:20:27]And so in 1984, while Yao had proved the conjecture and was doing pure math, he had no idea that this would have anything to do with physics. But across the hallway in physics departments from the math departments, there was a crisis and physicists were developing something called super string theory. And in order for this uh to work, the universe required 10 dimensions. four dimensions of the uh familiar spaceime up, down, left, right, forward, backward in time.

[00:21:00]But then there were another six extra spatial dimensions that had to be compactifified or curled up at every single point in space exactly like the Kuza mechanism. And in 1984 two uh these physicists Andrew Stronginger, Edward Whitten, Philip Candelis, and Gary Horowitz realized that in order to preserve a crucial physical property called super symmetry, uh these six hidden dimensions couldn't just be any shape. They had to be precisely six-dimensional uh colaba manifolds. So he searched the mathematical reg literature and realized that Yao had already proven the existence of the exact shapes they needed. So the extra six dimensions of the universe are now believed to be cla manifolds. So why does this topology matter?

[00:21:56]In the string theory, the fundamental building blocks of nature aren't zero-dimensional point particles, but they're actually tiny vibrating loops of energy called strings because they vibrate. Energy can resonate inside the loop and that can uh the energy can be uh sufficiently dense that it looks like a jelly or it looks like a particle. So the vibrating loop can in fact create particle. Uh but it it's really all energy. Uh and so as these strings move through the six hidden dimensions of the clavio manifold, the number of holes and the precise way the manifold folds dictate exactly how these strings can vibrate. So the number of holes in a clavio manifold directly determines the number of generations of particles that can exist in our macro world. So what we'll see here in a second is for the for our world of of matter we the electron actually has two other generations of particles uh that are related to the electron that the physics has found and that suggests that the glabial manifold associated with uh our our space-time dimension uh since there's three of these total the electron and two two other generations.

[00:23:22]There's probably three holes in this globial manifold that couples to our space. And the way that the manifold curves determines the masses, the charges and the constants of all the coupling forces. So this all all the juice is buried in uh the the curvature of the manifold and how it curves. So ultimately, Kabi's vision and Yao's rigor provided the landscape upon which modern uh physics is actually built. Now there's a mirror symmetry here which is an interesting thing to keep in mind. Uh it's one of the most interesting and profound discoveries that emerges from the intersection of physics and mathematics.

[00:24:08]uh and it was it's the discovery that two completely different calabio manifolds can describe exactly the same physical universe. So to a mathematician these shapes could look entirely different. Different geometries, different number of holes, different dimensions but to a string in string theory the universes they create are completely indistinguishable.

[00:24:30]So let's take a look at how how this works. Uh in classical physics, a point particle can only explore space by moving through it. If a space has a large hole or a tiny radius, particle is only sensitive to the immediate coordinate it occupies. But strings, however, are strings. They're extended objects. There's a tiny loop of a string. It's moving through a curled up colab manifold. It can it can interact with the geometry in two entirely distinct ways. It can vibrate. there's momentum and the string can move and vibrate through these extra dimensions just like a point particle but because it's a loop it has can actually wrap around the holes or the cycles of the clav manifold just like a rubber band can be wrapped around a cylinder and because of this dual behavior if you shrink if you shrink a dimension down to a tiny radius the energy costs swap moving around a tiny circle takes a lot of momentum energy wrapping it around takes a very little winding energy. So, physicists realize that a string moving in a manifold of radius R is physically identical to a string moving in a completely different manifold of radius 1 / R. And so this is known as duality. This mirror symmetry is essentially t duality applied applied to the sixdimensional uh real or three three complex dimensional uh landscaped of clavio manifolds.

[00:26:03]So let's take an example of this for swapping holes. When you look at a clavio manifold, its internal structure is defined by topological holes. In higher dimensions, mathematicians count these holes using technical numbers called Hodgej numbers. and they're denoted for example by H11, H21 and so forth. So one type of hole uh controls the size of the manifold and the other type of hole controls the shape the complex structure of the manifold. So the mirror symmetry states that for every clabial manifold there exists a mirror manifold where these two numbers are exactly flipped that is the mana for one uh the manifold m it gets flipped over to the one for two. So the size property in manifold n becomes a shape property in manifold w.

[00:26:55]If you calculate the physical properties of a universe using manifold, then calculating how strings vibrate and interact and then perform the same calculations using the mere W the resulting physical laws, masses, forces, particles are all identical. So when physicists were looking at this, they discovered in the early 1990s and they they brought it to the mathematics community and the mathematicians were initially skeptical because according to pure geometry, these shapes had no business being related but to prove the power of mere symmetry.

[00:27:31]Physicists used it to solve famous problem in enumerative geometry that had stumped mathematicians for century.

[00:27:38]Mathematicians wanted to count the number of rational curves or spheres of a certain degree that could fit inside a specific global manifold known as a quintic three-fold. For degree 1, the answer was known, 2,875.

[00:27:54]For degree 2, it took until the 1980s to find the answer, 69,250.

[00:28:01]Now for degree three, mathematicians have spent years working on a massive computer program to find it and tenatively came up with an answer of around 2.6 billion. Physicists use mirror symmetry to bypass the impossible math of the first manifold. They swapped the problem to the mirror manifold which is a brutally difficult problem of kinding curves transformed into a much simpler problem of calculating integrals. And almost instantly the physicists announced the answer for degree 3 was 307,26,375 and they provided a single formula that calculated the answer for all degrees out to infinity. So initially the mathematicians thought the physicist's code had a bug because it disagreed with their project. But the mathematicians went and they checked and they found a mistake in their own code. So once they fixed it, the math aligned perfectly with the physicist's mere prediction. So there you have it. So the elephant in the room here is the following. In the 1910s, Vesto Slifer observed a lot of stars and galaxies and that they were all moving away from one another. In the 1920, Edwin Hubble refined his observation and quantified how fast the stars and galaxies are moving away from.

[00:29:25]Now they aren't really moving. It's space itself is expanding and the expansion is due to something that would came to be called dark energy. So that's one thing, dark energy. In the 1930s, Fritz Vicki at Caltech did a calculation for a spiral galaxy. Here's a spiral galaxy over here. And you can see there's a bunch of stars and they all appear to be rotating around the center.

[00:29:52]And so he took an average mass per star.

[00:29:55]He had each star tracked all the other stars by gravity. And he showed that this galaxy here was unstable, that it would fly apart. But when he postulated there was additional matter in the galaxy, matter he couldn't see. So he called it dark matter. The gravitational force was strong enough for this to be stable. And in the 1960s and up to 1970, Vera Rubin confirmed all this with incredibly detailed measurements. And so today, physics believe we've been studying 5% of the mass and energy in the universe in the three dimensions that we see around us. And we have 95% of it dark. 27% is dark matter, and 68% is dark energy. This is not a minor screw-up.

[00:30:44]So what do we have? If we assume that 5% of the matter or energy is in three spatial dimensions, then 100% of the matter energy is in three divided by 0.05 or 60 dimensions. Or if we want to get a little trickier, if we assume that 32% of the matter, regular and dark, and the energy is in three spatial dimensions, then 100% of the matter or energy is in three over 0.32 or 9 spatial dimensions. So, as always, experimental observations from theory, these are just back of the envelope things that argue that we probably have more dimensions than we realize here.

[00:31:26]and Ryan Cowan actually observed that we have three complex spatial dimensions and an additional nine compactifified complex spatial dimensions.

[00:31:37]So what did Ron see? Ron saw this.

[00:31:42]We we have our our realm here of matter and there's four dimensions here. Three for space and one for time. time we we the units aren't correct because uh they're they're not they're in seconds rather than meters. So we multiply it by the speed of light. Speed of light is meters/s. So now we have the units of meters and then we translate this back uh into making it perpendicular by multiplying by the square root of minus1. So these are the natural coordinates here for space time. So similarly for dark matter we just put it into an adjacent subspace over here and the three uh the four coordinates we have here again what we did to make this the subspace perpendicular to this one we multiply by the square root of minus1 and we end up with a new set of coordinates here. Then we have the 18 real or nine complex compactified dimensions here. So six of them are tied up with spacetime. So at every point uh x y and z that we have here we we have a kabia manifold the the ho the hose example we saw before and the they all have to have three holes and then there the curvature parameters here that's summarized here. Similarly for dark matter there's four holes and they there's a curvature parameter here. So that takes care of of uh 10 of these dimensions uh for each of these 10 here for matter 10 for dark matter but now they're linked they're linked through another uh clavia manifold that has eight holes in it. Remember three holes here have to connect to the eight holes. four holes here have to connect to the eight holes.

[00:33:33]Then there's one more hole that connects between these two six dimensions, two six real dimensions or three complex dimensions and it has its own curvature.

[00:33:43]So now what's tricky here is how in the world do we get these bridging dimensions that connect matter and dark matter together through this? Um, and that's left as an exercise. It's not obvious. There's a lot of different ways that we could tie these things up. Over here, for this one, we simply had for each of the three dimensions, X, Y, and Z, we had two a a an associated uh uh two real dimensions for X, two for Y, and two for Z. And down here, we have to tie these to those same things. And that's that's a little tricky to figure out.

[00:34:23]And this takes us back to string theory where Ron observed that the electron in fact is 10 closed strings here. They're vibrating like crazy hundreds of millions of times per second. And the end result of all this is that the uh that the electrons here has energy inside it that's like a jelly and that makes it appear to be like a particle but it's actually energetic but it it actually congeals into it uh as a donut there. We can see right through the middle here's the strings here looking right through and we end up with this.

[00:35:02]So the key thing to tie back to Wolram is that the strings are coded with blocks of information and in fact that's what's controlling how all of this works. All of the algorithms that Wolram is talking about are buried here in these information blocks. So there's both uh algorithms here and storage of of the these things.

[00:35:29]And with that we're done.

[00:35:32]Have a