## Dante and the 3-sphere

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### Dante and the 3-sphere

The 3-sphere is the 3D surface of a 4D ball. It can exist on its own without the ball. It is the higher dimensional analog of the 2D surface of a 3D ball. Dante visualized the universe as a 3sphere. Here is chapter 2 of a book called Art Meets Mathematics in the Fourth Dimension
￼￼Lipscomb,S.

Chapter 2 describes Dante's vision. It has good illustrations. http://www.springer.com/cda/content/doc ... 532-c1.pdf

The 3-sphere, denoted S3 is sometimes called the hypersphere (like an ordinary sphere surface but one dimension higher. In cosmology today one possible version of the 3D space we live in is S3, it is boundaryless and has finite 3D volume, finite circumference.

Here is the google cache HTML version, which is missing illustrations.

Marshall
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### Re: Dante and the 3-sphere

Lipscomb, the author of "Art meets mathematics in the 4th dimension" cites the 1979 article in the American Journal of Physics, by Mark Peterson:
http://scitation.aip.org/content/aapt/j ... 19/1.11968

Dante and the 3‐sphere
Mark A. Peterson
Am. J. Phys. 47, 1031 (1979); http://dx.doi.org/10.1119/1.11968

We describe three different methods of visualizing the ’’closed universe’’ S3, and point out language in Dante’s Divine Comedy which suggests that he visualized his universe in the same ways, making his universe topologically S3

Peterson's 1979 article is behind the paywall---I'd have to hike over to the university's physics library and find it in the stacks.

ROVELLI cites Peterson in a recent article

http://arxiv.org/abs/1508.00001
Michelangelo's Stone: an Argument against Platonism in Mathematics

Carlo Rovelli
(Submitted on 2 Aug 2015)
If there is a "platonic world" M of mathematical facts, what does M contain precisely? I observe that if M is too large, it is uninteresting, because the value is in the selection, not in the totality; if it is smaller and interesting, it is not independent from us. Both alternatives challenge mathematical platonism. I suggest that the universality of our mathematics may be a prejudice hiding its contingency, and illustrate contingent aspects of classical geometry, arithmetics and linear algebra.
7 pages, 3 figures

Marshall
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### Re: Dante and the 3-sphere

BTW Lipscomb, in his wide-audience book, cites not only Mark Peterson's scholarly paper but also quotes from another wide-audience book, by someone named Stillwell. Here's a sample passage:
==quote==
We continue with our quote from Stillwell’s book:

With this sophisticated model of a finite universe, the Church was able to hold out against infinite space for a few centuries. But eventually infinite flat space came to be generally accepted for its greater simplicity, despite some uneasiness about infinity . . .
In the twentieth century, cosmology returned to the idea of a finite universe, and physicists now look back in admiration to Dante’s Paradiso, seeing in it a good description of the simplest finite universe, which we now call the 3-sphere

==endquote==

I find this a charming thought, if only it were true. SOME physicists seem to prefer the idea of a spatially finite universe. Maybe you could say that cosmology has RETURNED to seriously considering it as possible. But the issue is still unresolved. We don't know. Lots of people use the infinite flat space model because it's convenient and consistent with observations made so far. A very large 3-sphere would fit observations equally well. So far the observations made can not distinguish.

Marshall
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### Re: Dante and the 3-sphere

Amazing. It's a bit like learning Dante made the Poincare conjecture or figured out 1 point compactification of hyperspheres.

I've always liked, aesthetically, the universe as S3. Taken to the limit, the S3 boundary becomes a hyperplane. It takes big balls to make a universe like that. :-o

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### Re: Dante and the 3-sphere

Yes, I also like the Hypersphere. If you tell some one they could live on the 3D surface of a Hypersphere and get to move in 3 directions.. like North/South or East/West.. no problem. But introduce the 3rd Direction (Up/Down) and tell them it doesn't take them closer or further from the center of the Hypersphere and their brains go Blooey.

Some people just can't wrap their minds around a Hypersphere. True, there is a 4th Direction that will take them towards or away from the center but it's hard to visualize for most people.

Bestest from
Dave :^)

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### Re: Insider's View

Marshall wrote:The 3-sphere is the 3D surface of a 4D ball. It can exist on its own without the ball. It is the higher dimensional analog of the 2D surface of a 3D ball.

As much as I find utility in 3-spheres, the above statements seem inconsistent. The 2-sphere does have an inside and so does not seem analogous to whatever 3-sphere the authors describe.

This is where Jorrie (and Wikipedia) often disagree with my understanding. I assert the balloon analogy requires an inside, they don't. Presumably Dante et al. don't either. If an inside-less 3-sphere is somehow topologically possible, the geometry would seem to be more complicated and thus, less attractive.

A "3-sphere" is a legitimate conceptual simplification. Why not go one step further? We draw a circle, assign three dimensions to it (I call that a "3-ring" but "3-circle" is fine) and ask, does it have an inside?

If it doesn't, its going to be tough maintaining that it's a circle. If it does, we either have to add a new 4th dimension for the ring to exist in (adding complexity) or we can use the one we have, time. (As I chose to do.) Owing to unidirectionality, time requires a radial orientation from a central origin in that model. What a coincidence!

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### Re: Dante and the 3-sphere

A ring is a good example! Personally I don't think a ring has an "inside" unless the ring is EMBEDDED in a higher dimensional space, so something else exists which the ring lives in.

But imagine that all existence is concentrated in that ring. There are 1D galaxies (short line segments) subsets of which are 1D stars maybe even 1D planets passing back and forth thru them. Nothing exists except for that ring.

How could an "inside" or an "outside" exist? Nothing exists except the ring. There are no external "points" you can "point to".

As a topological construction, take a line segment [0, 1] and identify the endpoints. Where is the inside, where is the outside? Nowhere. The unit interval exists on its own, not embedded in any larger space.

In math one can define a topological space without it being embedded as a subspace of anything. Nothing simpler!

Marshall
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### Re: Dante and the 3-sphere

It is odd to live in a boundary layer, in the S3 hypothesis, and say "the bulk" just doesn't exist. If there is a 4-sphere, one has to ask why nothing exists within. Yes, as Marshall says, one can define a topological space, but when does nature listen to our definitions? Like Farad, I wonder how our S3 would stay spherical if the 4th d wasn't real in some radial sense. This sounds naive, but either you have 4 dimensions or you don't.

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### Re: Access

Marshall wrote: I don't think a ring has an "inside" unless the ring is EMBEDDED in a higher dimensional space, so something else exists which the ring lives in. ...one can define a topological space without it being embedded as a subspace of anything.

OK. I might define such a "ring" as a line segment where, moving from the center to one end, upon passing through it, I appear immediately at the other end, moving toward the center. That works ... sort of.

Two problems:

1. It's the continuous endpoint transition that identifies the structure as a ring rather than merely a line segment. That requires endpoints (albeit undetectable ones), which would seem to preclude any infinite universe of this type.

2. Radius of curvature. A sphere of any dimension (including a 1-sphere or circle) is defined as "the set of points that are all at the same distance r from a given point ... This distance r is the radius of the ball, and the given point is the center of the mathematical ball." The rules required to get around this will, in my view, add more complexity than allowing the additional dimension. Within the universe C = 2πr, but that somehow doesn't apply to the ring itself - two different geometries.

Remember that time must still be accommodated by any model, regardless. A radial field seems as good a place as any for it. That access to the radius is limited, compared to the ring (objects move ever outward to an expanding space), is given by the unidirectional nature of time.

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### Re: Dante and the 3-sphere

How swiftly we all departed the Art forum. Hehehe!

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### Re: Dante and the 3-sphere

I may be very naive, but why do you need any inside or outside?

Just think of a curved segment, in such a way that the two ends coincide. Don't care about what is on the convex or concave side: there is no convex or convex side, abstractly: the segment simply is curved and folds on itself. We do not care about what is outside the segment.

Now do the same with a surface. We are out of the Cartesian plane and do not care which side the surface bends (which side is concave and which is convex): simply, the surface folds on itself. There is nothing outside the surface. Possibly difficult to imagine for us, not so much for Flatland inhabitants... we are just telling them their world is not flat but curved, but they do not even know what flat - or convex or concave surface - means!

Now do the same with a three-dimensional space. It becomes as difficult for us as the previous point was for Flatland people. But why should we wonder about something existing outside such a curved three-dimensional space? we don't now what space flatness - or curved or concave spaces - means!

I apologize if this is too trivial...
(and for forgetting about art myself, but Marshall himself was a bit tricky in putting this in Art, merely because he referred to Dante)

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### Re: Dante and the 3-sphere

Neuro that's a good clear discussion. Thanks! We will probably get back to Dante and the medieval vision of the Hypersphere. It's attractive by its beauty. And Dante's universe had no inside or outside BTW.

But anyway, to respond: FARADAY somebody, maybe you, mentioned TOPOLOGY. In topology a sphere has no shape, a ring has no shape. It is just a limp loop of string. Topology is about the basic connectivity and layout.

So we are getting into semantics. A topological 3-sphere is a set with a certain topology. It has no inside or outside, unless it is embedded in some other space. It has no curvature because the curvature is not defined on a topological space. that is just how it is.

To define curvature you need some kind of differential manifold structure ON TOP OF the topological space. Then you have a GEOMETRY.

You still do not need an inside or an outside. the manifold structure can be defined without an embedding, purely internally.

BUT if you want to study the TIME EVOLUTION OF THE GEOMETRY, say using the GR equation, then you need another dimension.

An analogy might be imagined, with our ring, the topological one-sphere S1. One can take the cartesian product of that with the real line R (to allow for time) and get a cylinder S1 x R.
Still no "inside or outside". No embedding. All existence (now including historical evolution) concentrated on the cylinder. But we can think of geometry being put on the cylinder in the manner of GR "spacetime" geometry. So it can have ripples and bumps, detected internally if at all since there is no outside.

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### Re: Dante and the 3-sphere

To get back to Dante, one way to make a topological 2-sphere is to take two disks and identify points on their boundary.

Like cut out two similar disks of tissue paper and put one on top of the other, and glue them along their edges.

If the two disks are not embedded in any higher dimensional space (all existence concentrated on the two disks) then what you get by gluing is a 2-sphere with no inside or outside.

If you step that up one dimension the construction becomes two topological 3-balls identified along their 2d surfaces---forming a 3-sphere with no inside or outside.

This was Dante's vision of the Universe.

One 3-ball was terrestrial, the other was angelic. One had subterranean layers of hell with Evil at the center. The other had concentric heavenly spheres with Good at the center.

Apparently human nature was part of the adhesive that glued the two 3-balls together to form the 3-sphere.

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### Re: The Hole Ball of wax.

Braininvat » September 6th, 2015, 11:30 am wrote:How swiftly we all departed the Art forum. Hehehe!

Hey, I brought art!

neuro wrote:Just think of a curved segment, in such a way that the two ends coincide.

A sphere is a collection of points equidistant from a center. We care by definition.

Marshall wrote:In topology a sphere has no shape, a ring has no shape. It is just a limp loop of string. Topology is about the basic connectivity and layout.

They always use the example of a coffee mug (handle) being topologically equivalent to a donut. This is because they each surround a hole. A hole is a region which separates the surround. Dimensions provide the potential for separation.

Anyway, I didn't realize we were in Art. Didn't mean to distract.

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### Re: The Hole Ball of wax.

do you mean beacause you have used colors in your print? :°)

neuro wrote:Just think of a curved segment, in such a way that the two ends coincide.

A sphere is a collection of points equidistant from a center. We care by definition.

(the surface only of the sphere...)

I don't thing that Dante needed the spheres to be spheres rather than any other solid, except for their smoothness and an esthetic aspect of harmony and "perfection".

I imagine that modern views of the universe as a three-sphere would not care much about the distance of the points of such three-sphere from any "center" that would possibly not even exist (because if the universe is a three-sphere then it is finite, and if you represent its shape then there will be points that do not belong to it and therefore "do not exist")

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### Re: Named It & Framed It

neuro wrote:do you mean beacause you have used colors in your print?

As my first piece didn't make much of an impression...

Portrait of an Expanding 3-Ring Circus

Like flower, bursting forth in Spring,
the future never fails to bring,
expansion of its spatial ring.

It's not exactly Dante but, I think it has universal appeal. Enjoy!
Last edited by Faradave on September 6th, 2015, 4:34 pm, edited 2 times in total.

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### Re: Dante and the 3-sphere

So the universe has a fancy gilt frame! Who knew? I got a big bang out of your art, I mean Art, FD.

OK, I accept that the bulk may not exist. Though if they ever find a 4D rotation of the 3-sphere, then I will be absolutely POLE-axed.

TheVat