I finally finished ** Our Mathematical Universe** (2014) by

**Max Tegmark**. It took me a while — only two days left on the 21-day library loan. I often had to put it down to clear my mind and give my neck a rest. (The book invoked a lot of head-shaking. It gave me a very bad case of the

*Yeah, buts*.)

I debated whether to post this for Sci-Fi Saturday or for more metaphysical Sabbath Sunday. I tend to think either would be appropriate to the subject matter. Given how many science fiction references Tegmark makes in the book, I’m going with Saturday.

The hard part is going to be keeping this post a reasonable length.

Back in the 1990s I bought *The Nitpicker’s Guide for Next Generation Trekkers*, by Phil Farrand. As the title suggests, it’s a somewhat tongue-in-cheek analysis of little inconsistencies Farrand spotted in *TNG*.

Part of the deal for me was that Farrand was a fellow computer programmer, so I expected he would approach it with more of an understanding of the *possibilities* of the future rather today’s limits.

By the time I finished the book, it was chock full of tiny Post-it note flags on the *many dozens* of pages where I had nits with his nits. The book literally bulged from all the inserted *Yeah, buts*.

No paper and no bulging here, but I have dozens of bookmarks, highlighted bits, and notes where, again,… I have nits.

As I said, the trick is distilling them into a post, and I’m already wasting words.

**§ §**

Tegmark’s Mathematical Universe Hypothesis (MUH) is that all of reality isn’t just *described* by mathematics, it ** is** mathematics. There is no matter, there is no energy, there is no time, there is no change, there is just a mathematical structure.

Which raises the question: *Where is this structure? How is it embodied?*

Tegmark recognizes the conundrum of actually *implementing* a mathematical structure versus that structure just existing in some abstract sense. He asserts the Platonic abstraction is sufficient; implementation is not required.

So the answer is: *Nowhere. It isn’t.*

This is part of my objection to MWI (another theory I feel confuses the description for reality). Under MWI, reality ** is** the Schrödinger wave-function, a mathematical structure. But unless MWI proponents can explain

*where*or

*how*the wave-function exists, MWI devolves into Tegmark’s belief — reality is mathematical.

In fact, Tegmark takes MWI and the block universe (BU) as being correct descriptions and instrumental to his hypothesis. I’ve argued against both the MWI and the BU views previously.

He believes our universe is infinite, and that eternal inflation is true, which are also important parts of his argument. I’ve recently argued against the latter. The former raises some questions: time required to become infinite size; energy required to… etc.)

He embraces String Theory, another debatable physics quest, which he feels supports eternal inflation (and, hence, his Type **II** multiverse).

To be honest, he comes off a bit credulous to me. (His take on quantum computing seemed another example.)

There is a saying that one should have an open mind but not so open one’s brains fall out. Tegmark’s willingness to embrace these ideas (many of which Jim Baggott calls “Fairy Tale physics”) without skepticism felt vaguely evangelical. Or at least wide-eyed. He reminds me a bit of another wide-eyed evangelist, Brian Greene — someone Tegmark cites fairly often.

To his credit, Tegmark does characterize his idea as extreme. He even says he’s about the only physicist who believes this. (If his arguments haven’t convinced his fellow physicists, perhaps there’s a reason? Ironically, those who believe in MWI are definitely leaning his way.)

**§**

The other multiverse types feature *separate* universes. A distinct feature of multiverses is that separation. But our slice of the mathematical multiverse contains all other mathematical structures (at least in the Platonic sense). And in Tegmark’s view, Platonic is good enough.

Tegmark worked on a computer program to list mathematical structures. Essentially it assigns a Gödel number to them. This requires they be *countable*, which is questionable in light of Gödel’s results (that these are *uncountable*).

Tegmark acknowledges this, but skates right past it into a discussion about computational complexity. He wants to generate some of the structure by computation. The problem is he’s already denied time and change, which any computation requires (and Gödel still applies).

He seems to get hung on dilemma horns by needing many different types of infinity to support his arguments, but recognizing the inherent problems of that, he suggests maybe reality is rational. (A point I think might actually be true.)

But if it is, many of those infinities go away, and so does support for much of his argument. It feels like his own points are self-contradictory, which isn’t surprising since he may be arguing something nonsensical.

**§**

One thing I’ve noticed in these fairy tale physics stories is how they often contain what seems a show-stopper that proponents side-step.

MWI raises questions about energy and probability that, unless answered, seem to invalidate the theory. The BU raises questions about the flow of time and how that structure was ever generated in the first place.

On the flip side, both theories cling to tiny bits of flotsam to keep them floating. MWI clings to the Schrödinger equation being all that matters. The BU clings to a questionable (arguably *contrary*) interpretation of Special Relativity (to the point of claiming it’s *proof*).

Tegmark’s MUH floats largely on the observation that particles are characterized by numerical properties: mass, charge, spin, etc. From this he derives the idea that particles are therefore *nothing but* those numerical properties.

But what causes those properties to manifest? Why are they there? To me, reducing particles to just their numerical properties confuses the description — the measurements — for the thing.

It confuses the map for the territory, so it strikes me as a weak premise.

(As an aside, the MUH and BU, because both are static, are incompatible with quantum mechanics which *appears* random and dynamic. This is why Tegmark needs MWI — to account for apparent randomness in a static system. But MWI makes a hash of probability.)

**§**

The show-stopper I see here involves time and our sense of time flowing.

The MUH has the same time flow problem as the BU. Both say time and change don’t exist. The BU posits a static four-dimensional block that contains all of spacetime. The MUH posits a static mathematical structure describing what amounts to a BU.

So where does our sense of time flowing come from? Why do we all seem to share the «now» as well as a mutual past?

Those denying the fundamental nature of time often point to Einstein’s Relativity as demonstrating that time isn’t real. I think this misses a key point: *proper time* — our personal time — *never changes*. Neither does anyone else’s. It always and ever moves at one second per second.

Tegmark’s answer is that our perception of «now» is based on our sensory inputs and memories, and, at any given point along the timeline there is a corresponding sense of «now».

Fine, but why does it occur serially at a fixed rate? Why do we *share* that rate (when we’re in the same frame of reference)?

I think time is fundamental, and, if so,it falsifies the MUH (and the BU).

As an aside, Tegmark speaks freely of “change” and “process” just as those who believe in strict determinism often still speak of “deciding” or “choosing.” It’s almost as if their subconscious was trying to tell them something. 😉

As another aside, despite advocating the MUH, Tegmark does *not* believe in the VR Hypothesis. For one thing, the VRH evolves in time as it’s calculated *Matrix*-style, which is contrary to the static MUH view.

**§**

Speaking of asides, Tegmarks refers to his four flavors of multiverse as *Levels*, but I have trouble seeing them as levels (unless we mean like in video games).

I can see a Type **II** multiverse (eternal inflation; universes with other laws) as being a higher level than a Type **I** multiverse (this infinite universe), since **II** clearly includes **I**.

But a Type **III** (MWI) doesn’t require either of the first types. It could be true in a single finite universe. Type **IV**, likewise, isn’t an extension of the other types; they are not required first steps.

So I’ll call’m *Types*.

**§**

Another characteristic of fairy tale physics is its tendency to resort to what amount to philosophical arguments and to then take them as givens.

Two huge offenders in a lot of hand-waving books are the Anthropic argument (I refuse to call it a *principle*) and the Doomsday argument. They make an appearance here, of course, and are taken as reasonable.

The Anthropic argument is seen as in tension with the Copernican principle (which also doesn’t deserve the term *principle*). The resolution to that tension is some kind of multiverse — the Copernican principle is seen as winning.

And yet Tegmark himself asserts a belief that, not only are we the only intelligence in the galaxy (something I believe), we’re the only intelligence in the visible universe.

Which, if so, makes us Ptolemaic. This universe literally does revolve around us.

Tegmark uses the infinity of multiverse Types **I**, **II**, and **III**, to put Copernicus back in the win again.

But it is a *fact* that [1] there might be just one finite universe and [2] it just turned out the way it did and [3] since it did, here we are, deal with it.

**§**

In the last section of the book, Tegmark turns to social argument, making a plea for a better, more educated, world. He refers to our one spaceship Earth and how we’re all in this together.

Regardless of all that infinity, we ultimately do have just this one very small, very brief, reality. Tegmark isn’t too impressed with what we’re doing with it, and I have to agree with him on that one.

He’s very concerned about the dangers of AI. He’s definitely on the alarmist side. Given his mathematical views, it’s no surprise he asserts that consciousness is what information processing “feels like” (an idea not only with no supporting evidence, but with all known evidence falsifying it).

As such, he believes in the possibility of a superior AI that could, in turn, build even more superior AI. The speed of computers compounds the risk considerably.

He said something else I agree with: This century is unique in that, for the first time, we have the power to wipe ourselves out. If we survive it, we’ll probably have mastered the impulse. But right now, right here, our million year history is at stake.

**§**

Tegmarks suggests his theory does have a possible falsification: If he’s wrong, scientists should discover something they can’t explain with math.

I couldn’t help but think of wave-function collapse. The big complaint theorists have there is the lack of mathematics to describe it.

I also wonder if the results of Turing and Gödel might apply. Both demonstrated limits to mathematics — things it can’t explain.

*Stay narrow-eyed and skeptical, my friends!*

∇

May 9th, 2020 at 10:45 am

Despite my best efforts, this ran long, sorry. I could easily do a couple more posts getting into the things I bookmarked or highlighted, but I’m not sure what the point would be. (And I’m not sure I’m moved to bother.)

Ultimately this is metaphysics, and one has to just decide for oneself. I think the issue of time is a show-stopper.

Also that a static universe, whether timespace block or mathematical structure, raises too many questions about what process generated it. That requires vast computational resources — literally a universe full.

May 9th, 2020 at 12:01 pm

At one point Tegmark is talking about Kolmogorov complexity and he compares two 128×128 pixel bitmaps, one generated randomly, one showing the bits of the square root of two (an irrational number with “random” digits).

His point is that, by Kolmogorov’s measure, the only way to generate the first bitmap is by storing all 16K random digits, creating a long program. But the sqrt(2) bitmap can be produced by a much shorter program. Tegmark suggests 100 bits.

I

thinkhe’s right that a sqrt(2) function would take less than 2,048 bytes, but it might be close. It’s not a trivial algorithm. I do not see it happening in 100 bits, no way, no how.Then he compares a random 32×32 pixel section. In the random bitmap, this requires specifying the full 1,024 bits. Tegmark feels the sqrt(2) example only requires 114 bits — the 100 bits plus 14 more to specify where the square is.

But since I think a sqrt(2) function would be more than 128 bytes, so the random square is actually the smaller one to specify.

In any event:

[A] I’d like to see what’s really required to implement sqrt(2). I’ve never wanted to get into programming square root functions because, as I said, they’re decidedly non-trivial (despite being ancient), but I could use an implementation for a project, so my interest is piqued.

[B] Who knows, maybe Max is right about the 100 bits (but I doubt it). It would be interesting to see how small it could be. One thing that can be missed here is the complexity of the full implementation. The bits of the algorithm are one thing, but the engine that executes them also has bits.

Doing this right requires finding the sqrt(2) Turing Machine. No easy trick.

May 9th, 2020 at 12:05 pm

Here’s a one-line summary: I think Tegmark confuses how we

viewsomething with what itis.May 9th, 2020 at 6:36 pm

Summarizing that book in one post is tough. It took me four, and I felt like I left a lot on the table.

To varying degrees, I find Levels 1-3, in a speculative fashion, more plausible than you do. Level 1 is the consequence of space being flat and continuous. If our measurements keep finding it flat, then we have to add additional assumptions to avoid infinity with all its implications.

I find Level 2 more difficult. It depends on inflation being true, and then a particular model of inflation. To me, at least at present, it seems to involve more assumptions.

Level 3, the MWI, we’ve discussed. I don’t think it requires that reality is the Schrodinger equation, only that the equation is a full description of the relevant dynamics. (The distinction, I admit, is nuanced, but the latter leaves open the possibility that there is still more to learn.)

Any of these could conceivably be falsified by some scientific discovery, one that indicates a minute curvature in space, or that rules out the MWI for some reason.

But I agree that the idea that reality

ismathematics is dubious. Interestingly, as I recall, Tegmark hedges his bets a bit, stipulating that he’s not referring to the nomenclature and techniques of mathematics, but the underlying reality. The question is whether he’s still actually referring to mathematics, or the underlying reality mathematics describes. In a sense, the position between him and other mathematical platonists might be terminological.But I’m not a platonist, mathematical or otherwise, so the MUH doesn’t really work for me. The issue is that I can go about my life by ignoring platonism. There’s nothing in reality I’m going to trip over if it’s true and I’m wrong about it. At least the others are extrapolations of current science. This one seems like pure metaphysics.

May 9th, 2020 at 8:00 pm

“It took me four, and I felt like I left a lot on the table.”Likewise. I do have two earlier posts about specific aspects of his book, and I’ve already gone after the MWI and the BU, so, in a way, I’ve already written more posts.

Now I’ve returned the book and can’t reference it anymore. I might explore that sqrt(2) thing, but I think I’ve said what I have to say about most of it.

“Level 1 is the consequence of space being flat and continuous.”There are geometry options where it’s flat (or indistinguishable from) and yet bounded. The surface of a torus, for instance, is flat to its inhabitants — triangles have 180° sums of angles — but it’s not infinite.

We can agree Type II is less likely than Type I, although, yeah, I still rate the likelihood lower than you do. And I’m pretty sure MWI is false for reasons I’ve elaborated. 😉

“In a sense, the position between him and other mathematical platonists might be terminological.”Yeah, I’m not sure I see much daylight between them. About the only difference is that the Platonic realm is usually thought of as like Heaven — someplace else, not here. MUH doesn’t have that separation.

But MUH is a fundamentally Platonic view.

“This one seems like pure metaphysics.”I think so too. Tegmark does offer a falsification point, and I mentioned wave-function collapse in the post, but is that a case of our just not understanding or is it a case of actually finding something math doesn’t describe?

What about love or justice? I suppose Tegmark would take a reductionist position that the math is in the particles that comprise the beings perceiving love and justice. But that might be a place where reductionism fails. Maybe the laws of QM can’t describe some things. Currently we can barely extract basic chemistry from QM, so it might be our ignorance.

May 9th, 2020 at 9:43 pm

I don’t really understand the topology of a flat torus. Do you know of any source that provides a good explanation? Even Brian Greene fails at this, referencing an old video game metaphor to get the idea across, but punting on the actual topology.

My issue with Tegmark’s falsification criteria is it seems to be the same as the criteria for the less edgy hypothesis that math describes reality. He needs criteria that uniquely falsifies the MUH. Since we’re likely to expand mathematics if necessary (as Newton did to model gravity), the MUH seems safely unfalsifiable.

May 10th, 2020 at 11:19 am

“Do you know of any source that provides a good explanation?”I’ll give it a go. Keep in mind this is an analogy showing how a 2D surface can be flat yet bounded and infinite (in the sense of having no edges). As with most geometry, the principles extend to other dimensions. (For instance, the same basic principles take us from point, to line, to square, to cube, to tesseract, and beyond.)

Start with a (flat, 2D) sheet of graph paper. The horizontal and vertical lines meet at 90° angles. Parallel lines remain parallel. Triangle angles sum to 180°. OTOH, the sheet has edges, so it is in no sense infinite.

Now roll the paper into a tube that joins, say, the top and bottom edges. Nothing has changed in the geometry of the paper. The grid is still 90° everywhere, parallel lines remain parallel, etc. The only change is that now it’s possible to keep moving in the vertical direction forever. There is an infinite direction to the space and no edge.

Finally, bend the tube into a circle and join the ends. This removes the left-right edge, and now all directions in the space are infinite. Pick any direction, you can go that way forever. (That’s the video game part. Some old games wrapped the screen edges. Doing that creates a virtual toroid space. I do that in my Conway Life implementations.)

Crucially, on the paper, the grid is still 90°, parallel lines remain parallel, triangle angles sum to 180° The space, as far as any inhabitant can tell, is flat and infinite.

Note that there is some compression on the inside of the torus compared to the top and bottom, and there is expansion on the outside, again compared to the top and bottom. This comes from bending the tube into a circle. But that is just a metric change not apparent to those who live on the sheet.

Geometrically, the 2D graph paper hasn’t changed. Only its edges have been removed by joining them through a higher dimension. (This does require that higher dimension, but extra dimensions are almost old hat in physics. String theory was up to 26 at one point, IIRC.)

“He needs criteria that uniquely falsifies the MUH.”To specifically invalidate that theory, yes, but I’m not sure “killing two birds with one stone” is necessarily a problem. It’s certainly a problem if you have

supportingevidence for two theories — then you’re kinda stuck looking for a tie-breaker.But regardless, yeah, I think it’s pretty safely in the metaphysics category, so he’s never likely to be faced with actual falsification.

May 10th, 2020 at 1:31 pm

Thanks for the torus explanation. I can see the flat relations being preserved on the initial role into a cylinder. But I’m not clear why the compression and expansion when it’s drawn into a circle don’t alter the metrics.

Why would they be preserved in this relationship but not in a sphere? If we were on the inside, shouldn’t we expect the metrics to resemble the saddle shape? And if on the outside, should they resemble the sphere’s? In both cases, it seems like they would vary depending on which direction we measured?

On falsifiability, the point I was trying to make is if that falsification never happens, it doesn’t really strengthen his theory over the more modest one. (I should have said that explicitly.)

May 10th, 2020 at 2:07 pm

“I’m not clear why the compression and expansion when it’s drawn into a circle don’t alter the metrics.”Because any apparatus for measuring also compresses or expands. There’s no way to detect it.

I think part of the mental leap required here might involve intuitions of “flatness” — which we tend to think of like a table top. The first step, rolling the paper into a tube, illustrates that nothing changes about the geometry of the paper. But the tube isn’t like a table top. It’s meant as an intuitive bridge to a more mathematical intuition of “flatness” — which has to do with angles and parallel lines. Geometry on the tube is clearly the same as geometry on the flat paper.

“Why would they be preserved in this relationship but not in a sphere?”If you draw two parallel lines on a torus, they remain parallel no matter how far you extend them. (Which, as I mentioned, is part of the definition of mathematical flatness.) If you draw two parallel lines on a sphere, they always converge if you extend them.

“If we were on the inside, shouldn’t we expect the metrics to resemble the saddle shape? And if on the outside, should they resemble the sphere’s?”There is no inside or outside. This is a Flatland situation; you have to think of existing entirely

withinthe 2D sheet.Mathematically, a “saddle shape” is a hyperbolic surface (not toroidal). On such a surface, parallel lines always

diverge. Spheres also have a different curvature than toroids. (The torus is a fundamental geometric shape, along with spheres, cubes, and others. Its flatness is part of its distinction. So is having a hole in it. 🙂 )“In both cases, it seems like they would vary depending on which direction we measured?”It does, but there is no measuring “up” or “down” — those directions don’t exist. It would be like us trying to measure in the W (fourth spacial) dimension. The distinctive flatness of a torus means measurements

withinthe 2D space are isotropic (just as they are in our 3D space).“…if that falsification never happens,…”Ah, yes, totally. Lack of evidence isn’t evidence of lack. Quite agree!

May 10th, 2020 at 3:23 pm

I appreciate the explanations. Just to be clear, I don’t doubt that you’re right. But I want to understand why.

On parallel lines, can’t we maintain parallel lines on the surface of a sphere? For example, although longitudes lines converge, that’s only so they can stay perpendicular to latitudes, which remain parallel, all around the globe.

Usually when the shape of space is discussed, it’s triangles that get measured. On a flat surface, a triangle’s angles add up to 180. But on a curved surface, they either add up to more or less than 180, indicating the type of curvature.

What I don’t understand is why a triangle’s angles would add up to 180 on the surface of a torus. Now that I think about it, even the infinite cylinder seems problematic for this.

I could understand all the measuring apparatus warping so that their measurements indicate flatness. But if so, then it seems like that would also happen on a sphere or saddle. What would cause it to happen on one but not the other?

My reference to directions measured were meant in two dimensions. We can measure along the toroidal or poloidal direction, or some diagonal between them. We wouldn’t know which was which unless it altered the measurement.

But it seems like where we are on the torus surface would alter the measurement. On the surface inside the hole, it seems like we’d get saddle dynamics. On the surface outside the whole, a mixture of positive curvatures depending on the orientation of the triangle.

Unless I’m utterly confused, which is probably the case.

May 10th, 2020 at 3:26 pm

BTW, I keep staring at this, but it’s just hitting my clear button.

https://en.wikipedia.org/wiki/Torus#Flat_torus

May 10th, 2020 at 5:01 pm

Heh. At some point this does get seriously mathematical. 😮

The wrapping of the grid into a tube and the tube into a torus is exactly what I was talking about, though. You can see the stretching, but you can also see the grid lines meet at 90° angles.

That’sthe part that matters.May 10th, 2020 at 4:59 pm

“But I want to understand why.”One of my most favoritest of thought modes! 😀

“On parallel lines, can’t we maintain parallel lines on the surface of a sphere?”Not

straightlines, no. Remember that on a sphere, the geodesic, the shortest possible distance, therefore by definition a “straight” line, is an arc on a great circle. With lines of latitude, this is only true of the equator. All non-zero lines of latitude are slices through the Earth that do not intersect the center.Imagine you’re up on the 60th “parallel” and you want to get to a point many hundreds of miles along the parallel to the west. If you follow the parallel you take a longer path than if you follow a great circle arc that takes you north and then back south to intersect the 60th parallel at your destination.

In reality, following the parallel curves away from the geodesic and then curves back.

The lines of longitude, on the other hand,

areall great circles and are therefore “straight” lines (shortest possible paths).“Now that I think about it, even the infinite cylinder seems problematic for this.”You’re not confining your thinking to the paper. Imagine the graph paper is flat and you draw a triangle on it. Let’s make it a simple right triangle with sides of 1, 1, & sqrt(2), units. It lines up on the grid nicely. I’m sure we agree the sum of its angles is 180°.

Now roll the graph paper into a tube.

On the papernothing has changed. It’s easy to think the curve of the tube adds or subtracts to the angle, but it doesn’t. If you lived in the Flatland of the paper, you couldn’t tell the difference. It’s only from our higher dimensional view we can see it.If we join the tube ends, the graph gets stretched or compressed in spots, but that triangle still sits exactly on the grid. Any geometry done reflects a flat surface.

“But if so, then it seems like that would also happen on a sphere or saddle.”As far as any local

sizemetric, absolutely. On a sphere, near the poles, the distance between the lines of longitude would seem not to change. Such a reality would experiencefor realthe same stretching we see on polar projections of the Earth (huge Greenland, for example).And they wouldn’t be able to notice a size difference. What they would see is a singularity where all lines of longitude converge. Things would get very weird as they approached. To people watching from a distance, travelers would seem to shrink as parallel lines converged.

Triangles of any notable size have angles that sum to greater than 180°. Imagine a giant triangle from any two points on the equator to the North Pole. The two angles from the equator already sum to 180° (90° each). Whatever angle they meet at the pole makes the total greater.

In a hyperbolic space, it’s the other way around. Triangles have less than 180° and parallel lines diverge. (Hyperbolic spaces are kinda weird and I haven’t gotten much into them. Distances

expandas you approach and edge or singularity. Hence the diverging lines.)“We can measure along the toroidal or poloidal direction,”Yes, both of which are geodesics. They appear as straight lines to the Flatland inhabitants. As you say, they are the “X” and “Y” axes of the space. (There is no “Z” axis here.)

“On the surface inside the hole, it seems like we’d get saddle dynamics.”You need to think

insidethe surface. The problem with the analogy is how hard it is to shed the 3D view.All you have is the grid made up of toroidal and poloidal lines, and those lines are parallel

and geodesiceverywhere. The X-Y nature of the grid is maintained. That’s simply not true on a sphere or hyperbolic space.May 10th, 2020 at 6:04 pm

Ok, thanks. I’m still not seeing it, but it might just be me. Although that Wikipedia section implies you have to go to 4D to make it work. That’s the point where my mind blanks. And I also see in the Shape of the Universe article that there are other higher dimensional shapes that would accomplish the same thing.

But they all seem to depend on higher dimensions, which I think represents additional assumptions. They might be accurate ones, but until there is something that makes them essential, it seems more conjectural than straight flat space. And it also leaves open what the topology of that outer realm is. Is it itself a 5D or 6D torus, or some other shape, and what is it in? Is there a final outer containing shape, or is it containing dimensions all the way up?

Multiverses seem inevitable in such an arrangement. Of course, infinite space comes with its own craziness. Reality is absurd.

May 10th, 2020 at 7:51 pm

Yeah. On some level, something intuitively absurd is axiomatic. (It makes existing so interesting to contemplate.)

I’ll try one more round. I’m sort of thinking that, if someone with your background and mind is struggling with this, maybe I should have a go at a full post with diagrams. I could use a POV-Ray 3D model to illustrate how parallel lines and triangles work. (That said, I bet someone’s done a good YouTube video. I even have certain someones in mind.)

The other thing is, this is an example of what I was saying about informed intuition. What you’re struggling with is pretty clear to me, but I have the advantage of years of messing around with this stuff. If you really want the idea to lock, you may have to put in some work to get there.

“But they all seem to depend on higher dimensions, which I think represents additional assumptions.”It does, but that one is not uncommon in physics theories. String theory has six extra dimensions.

Even QED assumes a virtual circular dimension. You’ve probably seen the illustrations of a 2D grid (meant to represent 3D space) and little loops at each grid point (and by extension, every point). That’s kind of what QED is. (IIRC, that extra dimension is the spin axis, which in a photon is polarization.)

That said, requiring a higher dimensional

descriptiondoesn’t necessarily require an actual higher dimension.The complex numbers are one example. They are fundamentally two-dimensional, but there is no actual sqrt(-1) dimension. Another example is an improper rotation of a cube. It can’t be physically accomplished due to electromagnetic forces, but it’s a mathematical description that makes perfect sense in 4D but the parts of the cube don’t really need a fourth dimension. The rotation can be accomplished in three dimensions, modulo those EMF issues.

But yes, it’s true that higher dimensions, virtual or real, would be axiomatic.

BTW, I agree: flat space is a lot simpler.

Anyway, here’s my last go (unless you have further questions):

The grid formed by the toroidal and poloidal lines matches the graph paper. As far as any measurement on the surface, those lines always meet at right angles. All the toroidal lines are parallel to each other; all the poloidal lines are parallel to each other.

Consider drawing a line from any point in any direction. Unless it’s a poloidal line, it will cross one at some point with some angle relative to the poloidal line. Since all poloidal lines are parallel, any others our line crosses it must meet at the same angle.

The same argument applies to crossing the toroidal lines. The line we draw (unless itself toroidal) crosses all toroidal lines at the same angle. (It might be helpful to first visualize this on the cylinder. It remains true on the torus.)

A second line parallel to our first line will cross the toroidal and poloidal lines at the same angle. It remains always parallel to the first line. (Just as would be true on flat graph paper.)

I suspect seeing is believing, so maybe I’ll see what I can do. I already have a post pending about the inside of a tesseract.

“Is there a final outer containing shape, or is it containing dimensions all the way up?”One thing I didn’t get to talk about regarding Tegmark is the argument:

If one, then why not many?It’s another sometimes useful heuristic, but not anywhere close to a law. At some point

somethinghas to be axiomatic, to be “one.” We have three generations of matter (only one of which hangs around). Why three? Why two quarks per generation? Why four dimensions?WTFK! 😀

Although, granted, the holy grail TOE would answer those questions.

“Multiverses seem inevitable in such an arrangement.”They do seem popular. I remain skeptical. 😉

May 10th, 2020 at 9:41 pm

Thanks again. I actually don’t think it works in 3D. I’m basing that not off my own understanding, which is admittedly too green to matter, but on the content of the Wikipedia articles.

In the case of toroidal lines, consider such a line running on the surface facing away from the hole, then another one running on the surface on the hole side. The hole side version is going to be shorter. If we then run poloidal lines between the toroidal ones, evenly spaced, it doesn’t seem like they could be parallel.

You can see it in the diagrams. The intersections don’t look perfectly perpendicular.

So it makes sense to me that you have to add at least one other dimension, going from a two circle torus to a three circle one. That extra dimension gives you the degree of freedom to keep things metrically flat. (I guess technically it would be a 5D torus since it has to contain a 3D surface.)

Multiverses seem a lot less likely with straight flat space. (Aside from Tegmark’s Level I multiverse.)

May 10th, 2020 at 11:16 pm

“I actually don’t think it works in 3D.”I’m kinda hurt you don’t believe me. 😦

“In the case of toroidal lines, consider such a line running on the surface facing away from the hole, then another one running on the surface on the hole side.”That gives us two concentric circles in the same plane. (They represent the set of closest and furthest points from the center.)

“The hole side version is going to be shorter.”To a 3D God looking at this from the outside, it

looksthat way, but consider that none of the poloidal lines meet. Every poloidal line you draw has a unique point on both circles. Therefore both circles have the same number of points.If you have two concentric circles on the 2D plane, it’s easy to demonstrate that, despite the radius difference, both circles have the same number of points. Any line drawn from the center outwards encounters a unique point on both circles. There is no point on either circle that does not have a match in such a line.

Likewise on the torus, all points on the inner circle have matching points on the outer circle, because there is always a poloidal circle that connects them. For that matter, the 2D circle proof works in this case, too.

So the inner circle is in no real sense shorter, despite appearances. All toroidal circles have the same length (as far as the torus surface is concerned). For that matter, all poloidal circles are the same length, but that’s obvious.

“If we then run poloidal lines between the toroidal ones, evenly spaced, it doesn’t seem like they could be parallel.”I’m sorry, I’m not clear on your phrasing here. By “between” do you mean from one to the other (as I just described above)? I assume you do, but I’m not sure which set you think wouldn’t be parallel.

The poloidal circles are all parallel to each other because none of them can ever intersect. Lines that never intersect are, by definition, parallel to each other. All the toroidal circles are also parallel to each other because none of them can ever intersect.

So we have two sets of parallel lines, and they’re certainly not on the same angle. Whatever angle they do cross at, if all lines are parallel, it has to be the same angle in all cases.

Given that the toroidal and poloidal lines map directly to the graph paper lines —

where they are definitely parallel and orthogonal— what’s the basis for there being any change to that? There isn’t; the toroid surface maps isometrically to the flat graph paper.“You can see it in the diagrams. The intersections don’t look perfectly perpendicular.”That’s a 2D rendering of a 3D object seen from an angle. The rendering probably even uses perspective. All that distorts angles. It’s not a reliable guide.

“So it makes sense to me that you have to add at least one other dimension, going from a two circle torus to a three circle one.”The 2D torus surface has a flat metric. Maybe this will help (from about halfway down the page, “Dimension 2” section):

It goes on to get very mathematical, but it does contain the phrase,

“but these are not easily visualizable.”May 11th, 2020 at 8:47 am

Don’t be hurt. I take no one’s assertions as golden. (As you recently pointed out, not even Einstein’s. 🙂 We’re all human.)

I think the point about the toroidal and poloidal lines is the the intersections won’t be 90 degrees. The inner circle will be shorter than the outer one. This is made more obvious if the hole is smaller and seen from above.

I will concede that the metric does look preserved on the initial roll into a cylinder. It’s only when it’s twisted into the torus that the distortions come in. Which is probably why adding an extra dimension helps.

I do see some stuff where adding corrugations can help. Essentially wrinkling the inside surface. Doing that can allow the inside one to supposedly be as long as the outside one, although I wonder how the wrinkles themselves don’t add distortions. Eventually you end up with something with corrugations with the corrugations having corrugations!

The assumptions seem to be piling up.

May 11th, 2020 at 11:09 am

The only assumptions we need are ones involving geometry. (Admittedly, somewhat advanced geometry, but it’s not like we’re talking Reimann surfaces or tensors.) I’m jumping to a new thread for some elbow room and a fresh start. See below…

May 11th, 2020 at 11:33 am

When I say I

amhurt, don’t take that too seriously. I’ll get over it. 😉May 11th, 2020 at 2:09 pm

Just to focus on the cylinder, which we agree has the same metric as the flat graph paper: What will become the toroidal circles are straight lines on the cylinder, and what will become the poloidal circles are circles around the cylinder. We agree the circles intersect the straight lines at 90°.

When we bend the cylinder into a torus, the only thing that changes is stretching and/or compression of distances (as seen from a 3D perspective; it’s not apparent on the 2D sheet). The relationship of the poloidal circles to the toroidal lines doesn’t change.

Imagine two travelers who start on the inner latitude, but 1° apart. Both head exactly north (90° from the latitude line). From their perspective they remain 1° apart throughout the journey until they return to the inner latitude line again. Watching from the 3D outside, we see their paths appear to diverge to a max on the outside and then converge again on the inside.

Crucially, their paths never

cross. Which means, by definition, those paths are parallel. And whatotherangle could those paths cross the other lines of latitude than 90°?May 11th, 2020 at 3:22 pm

BTW: You mentioned corrugations. I think maybe you’re referencing John Nash’s work on embedding spaces while preserving distances? It’s work that applies to manifolds in general, but can require up to 17 Euclidean dimensions to apply.

But when applied to a torus, it does preserve distances relative to the flat space. It makes the torus not exactly a torus any more (a toroid shape), but the latitude and longitude circles will all have the same length they do on the flat sheet. The mapping is then

isometricrather than just isomorphic.(In both cases angles and geodesics are preserved.)

May 11th, 2020 at 3:52 pm

That is the one I was referring to. Thanks.

May 11th, 2020 at 11:20 am

“Don’t be hurt. I take no one’s assertions as golden.”I

amhurt because you are asserting your uninformed intuitions over my knowledge and experience, which devalues it. I’ve explained why the apparent distortion does not change the flatness metric, and I’ve cited Wikipedia sources supporting it. You seem to feel your intuition and 2D renderings of 3D objects trump this. I feel like Copernicus. 😮“I will concede that the metric does look preserved on the initial roll into a cylinder.”That’s progress. (I wish you didn’t see it as “conceding” but as understanding and agreeing. This isn’t a fight. I’m trying to teach you something.)

“It’s only when it’s twisted into the torus that the distortions come in.”Okay, let’s consider that distortion. You’re arguing it changes the 90° angles. I’m saying it doesn’t.

Vital Question: Do you understand and agree that, in a 2D plane with concentric circles, the inner circle has as many points on it as the outer circle? The proof of this is that we can draw a line from the common center outwards such that it intersects both circles. For any point on either circle there is such a line and therefore a matching point on the other circle.This is an important point, and if you disagree, we’re stuck. But if so, it’s disagreeing with more than just Einstein’s opinion about intuition, it’s disagreeing about geometry and math.

On the premise we agree, I’ll keep going.

Proven: All the points in the inner circle map to points on the outer circle, and vice versa.We’ve visualized the inner and outer longitudes (toroidal circles). Now visualize the one running along the “top” of the torus and the one on the bottom. We can call these lines of latitude (we could also call them the “X” axis — our “X” coordinate would specify our location around the torus; it would specify which poloidal circle we were on).

We have four longitude circles now, we can fill in a bunch more to form a toroidal tube of circles. Just the toroidal circles but no poloidal ones just yet.

Now let’s start on that inner line of latitude at what we’ll call X=0.0 — an arbitrary point on the toroidal line we’ll label as zero. The point on the opposite side of the torus (across the hole) is the 180° point, and let’s let clockwise (as seen from above) be positive degrees and counter-clockwise be negative degrees.

We can specify our position along the lines of latitude just like we do on Earth.

Now start at latitude 0.0 on the inner circle and draw a line 90° upwards (towards the top of the torus — “north”).

That line intersects the other lines of latitude. It eventually reaches the far side of the tube, crosses that line, and continues downwards (“south”) crossing the lower lines of latitude, until it finally returns to the point were it started, 0.0 on the inner circle of latitude.

Effectively we’ve drawn a poloidal circle, which we can call a line of longitude (or “Y” axis). Now all pairs of latitudes and longitudes are points on the torus.

Big Questions: At what angle do you think our first circle of longitudecrossesthe lines of latitude. Do you disagree the line we drew returns to the starting point or that it is a poloidal line?In particular, if we return to the same point on the inner latitude, at what angle does our path hit that line? We left at 90° — is there any reason to assert we return at a different angle? Isn’t it the case our line of longitude has to meet all the lines of latitude at 90°?

Now move one degree “west” (positive direction; clockwise from above) on the inner circle of latitude and draw another line at a 90° angle northwards. This forms another poloidal circle of longitude. All the previous questions apply; if not 90°, then what?

If the lines of longitude do

notcross the lines of latitude at 90°, what angle do you think it is, and why?Plus More Questions: If we start at +1.0 degrees on the inner circle of latitude, what is the coordinate of the other lines of latitude we cross? In particular, what is the latitude of the outer line? Is it not +1.0 in all cases?If we go halfway around the torus and do this at the 180° point, don’t we hit the matching 180° point on all the toroidal lines?

For any given latitude, a circle of longitude crossed other lines of latitude at the same coordinate. We can imagine an infinite number of poloidal lines, each one crossing all lines of latitude at the same “X” coordinate.

Thus all the points map isomorphically to a flat surface, and so do the angles. The rectangularity of the grid is maintained.

“I do see some stuff where adding corrugations can help.”😀 If we did that, it wouldn’t be a torus anymore. It would generally toriod-shaped, but a torus is a specific mathematical object (which, as you know, has its own Wikipedia page).

May 11th, 2020 at 3:49 pm

On the intersections, I suppose it’s equivalent to the intersection angles of latitude and longitude on a globe. I wouldn’t think those are precisely 90 degrees, but even if they are, it doesn’t mean they’re parallel since the length of each lattitude gets shorter as we approach the poles.

The poloidal lines on the torus never intersect because of the hole, but they do converge toward each other as we approach the innermost surface, and diverge as we move away from it.

Maybe we’re running into why astrophysicists use triangles to determine flatness.

On the corrugations making it no longer a torus, that’s what I would have thought too. But apparently it’s accepted as one.

http://www.science4all.org/article/flat-torus/

May 11th, 2020 at 4:24 pm

“I suppose it’s equivalent to the intersection angles of latitude and longitude on a globe.”As far as the

intersections, yes, exactly.“…it doesn’t mean they’re parallel since the length of each lattitude gets shorter as we approach the poles.”That’s not the determining factor, since (as you keep pointing out) the longitude lines (poloidal circles) also get closer together on the inside of the torus compared to the outside.

On a sphere, 10° east or west along the 10° latitude has more miles than the same 10° along the 80° latitude. Likewise, on an actual physical torus, 10° east or west along the inner latitude has fewer miles than the same 10° along the outer latitude.

What makes them different geometrical spaces is that sphere’s lines of longitude all cross at the poles. On a torus, the lines of longitude never cross or touch. That’s the significant thing.

“The poloidal lines on the torus never intersect because of the hole, but they do converge toward each other as we approach the innermost surface, and diverge as we move away from it.”Right. The hole is the reason. Without the hole, it’s not a torus. (Not the kind we mean here.)

Imagine following two nearby poloidal lines round and round. Effectively, we have two infinite lines that never cross. They are therefore parallel

by definition. That they get closer and further doesn’t really matter geometrically. It’s that they never cross.“Maybe we’re running into why astrophysicists use triangles to determine flatness.”It’s two cases of the same thing. There are three fundamental spaces: spherical, flat, hyperbolic. Parallel lines and triangles act differently in all three for the same basic reason, the metrics of the space.

Parallel lines are hard to test over the distances we need to measure the cosmos. With triangles we use the CMB as two points for the base (and us as the apex). That’s a big triangle.

“On the corrugations making it no longer a torus,”What I mean is that you can no longer locate a point on the surface with the standard toroidal formula. Or that the standard formulas don’t produce the corrugated shape. (Nash’s discovery is fairly recent compared to how long we’ve been messing with toroids.)

It’s not necessary for the point astrophysicists are trying to make about unbounded finite flat space. A 3-torus would look like flat 3D space but be finite yet unbounded.

May 11th, 2020 at 12:25 pm

I don’t know if this helps, it’s a little mathy, but it demonstrates the mapping from the square grid to the torus surface. (It’s just under seven minutes.)

May 11th, 2020 at 6:22 pm

I started a model in case I write a post on this. Not sure if this adds anything at this point, but FWIW:

We’re looking directly (zero angle) at the poloidal circles pointed to by the arrows. It should be apparent the crossings on this line are 90° since we’re head-on.

The concern you have is that it seems the nearby circles, which diverge and converge, may cross at a different angle. But if we rotate the torus or our view to look at those head-on, they’ll be 90°, too. That is the case for all the poloidal circles.

May 12th, 2020 at 5:58 pm

Speaking of hyperbolic spaces, just saw this on YouTube: