Week 214 Finds by John Baez - Golden Ratio, Fibonacci & Quantum Logicians

[Kea]

Regarding This Week's Finds 214
http://math.ucr.edu/home/baez/week214.html

...I tried to post something to the SPR thread but I guess there were too many equations. Anyway, here's a PF thread. After reading TWF, I thought of the following quote:

Continuous geometries...are a generalization of complex projective geometry somewhat in the way that Hilbert space is a generalization of finite dimensional Euclidean space Halperin (1960)

This quote appears in the book Orthomodular Lattices by G. Kalmbach, on page 191. Just below this quote is the example of the Fano plane and its associated complete modular ortholattice on 16 points.

These are the sorts of lattices that quantum logicians like.

Tony Smith, on the page linked by John Baez, mentions the Golden Ratio in connection with the Fano plane. John has mentioned Fibonacci numbers. I thought it would therefore be interesting to bring up the following observations.

In the paper

Geometrical measurements in three dimensional quantum gravity,
J.W. Barrett, http://xxx.lanl.gov/abs/gr-qc/0203018

John Barrett discusses the following remarkable Fourier transform for 6j symbols

\frac{1}{N} \sum_{j_{1} \cdots j_{6}} [ j_{4}, j_{5}, j_{6};<br /> j_{1}, j_{2}, j_{3} ]^{2} H(j_{1},i_{1}) \cdots H(j_{6},i_{6}) = [<br /> i_{1}, i_{2}, i_{3}; i_{4}, i_{5}, i_{6} ]^{2}

N is a normalisation constant. Choosing q =<br /> e^{\frac{i \pi}{5}} as in

A modular functor which is universal for quantum computation,
M. Freedman M. Larsen Z. Wang,
http://arxiv.org/abs/quant-ph/0001108

gives allowable spin values j \in 0, \frac{1}{2}, 1, \frac{3}{2}. The kernel function is given by the Hopf link invariant

H(j,i) = (-1)^{2i + 2j} \cdot \frac{\sin \frac{\pi}{5}(2j +<br /> 1)(2i + 1)}{\sin \frac{\pi}{5}}

Observe that H only takes values \pm \phi , \pm 1 where \phi = 1.61803399 \cdots is the golden ratio

\phi = \frac{\sin \frac{2 \pi}{5}}{\sin \frac{\pi}{5}}

The full kernel therefore takes values in powers of \phi, or in other words the truncated Fibonacci sequence

1 , \phi , \phi + 1 , 2 \phi<br /> + 1 , 3 \phi + 2 , 5 \phi + 3 , 8 \phi + 5

of 7 terms.

Cheers
Kea

 

[Kea]

Now over on SPR a Kiwi mentioned that 3-holed surfaces are a bit like tubular tetrahedra. Actually, think of a tetrahedron made of untwisted ribbons joined together at the vertices. This is exactly the same as a four punctured sphere, with a little stretching and squeezing. Now a ribbon triangle is dual to a vertex of 3 lines, which becomes a trouser diagram, ie. 3 tubes meeting at a vertex. This is a three punctured sphere. Putting 4 of these trousers (qubits) together turns the tetrahedron into the 3-holed surface.

 

[Kea]

Three leafed trees with a root (associativity trees), if thickened, become four punctured spheres too. And associativity arrows as tetrahedra naturally appear as maps joining two 4-holed spheres.

 

[nightcleaner]

Now over on SPR a Kiwi mentioned that 3-holed surfaces are a bit like tubular tetrahedra. Actually, think of a tetrahedron made of untwisted ribbons joined together at the vertices. This is exactly the same as a four punctured sphere, with a little stretching and squeezing. Now a ribbon triangle is dual to a vertex of 3 lines, which becomes a trouser diagram, ie. 3 tubes meeting at a vertex. This is a three punctured sphere. Putting 4 of these trousers (qubits) together turns the tetrahedron into the 3-holed surface.

This is interesting, Kea. The tetrahedron, or sphere with four holes, has four qubits, or three hole spheres, as vertices. One might wonder if the three hole sphere would have two hole spheres as vertices, but I have been dissapointed trying to visualize that. Instead I see a three hole sphere as made up of three ribbons joined at two vertices, each vertice being another three sphere.

I do wonder about the idea that a ribbon is equivalent to a tube (as in the rather sudden appearance of trousers above). In fact a ribbon is not exactly equivalent to a line either. Is it justifiable to morph a one dimensional line, as in a tetrahedron, into a two dimensional ribbon, into a three dimensional qubit, as you have shown here? Can we just assume more dimensions at will?

In the other direction, one might wonder if a five hole sphere should have four hole spheres as vertices. Or, one might wonder, how many holes must a sphere have before the next new hole requires the appearance of at least one four hole sphere as a vertice? I am thinking from the perspective of the cubeoctahedral isomatrix, which would have fourteen holes, and twelve vertices, all twelve vertices being equivalent to four hole spheres.

What branch of mathematics is this anyway? It seems like some kind of topology.

Thanks,

Richard

 

[selfAdjoint]

do wonder about the idea that a ribbon is equivalent to a tube (as in the rather sudden appearance of trousers above). In fact a ribbon is not exactly equivalent to a line either. Is it justifiable to morph a one dimensional line, as in a tetrahedron, into a two dimensional ribbon, into a three dimensional qubit, as you have shown here? Can we just assume more dimensions at will?

Not equivalent. Dual. Maybe Kea can explain that some more - I don't feel up to it.

You are right; this is topology.

 

[marcus]

NC there was one nice elementary fact in what Kea said that would not be hard for you to understand by drawing a quick picture.

a charming thing with Kea though is that if you understand an inch she will immediately scamper off a mile or more into the blue horizon. So it is a bit risky to say that you understand the initially offered inch.

I believe if you simply draw a tetrahedron frame, not the solid block but just the airy frame as if made of aluminum tubing.

I think if you will simply draw this tubular tet, maybe with rather fat tubing,
that you will see that it can be mooshed into a

1. sphere with 3 handles or a
2. surface of a donut with 3 holes

you may see this so quickly that it would be foolish of me to say more at this point

 

[nightcleaner]

Thank you selfAdjoint. I noticed the use of the word Dual, when Kea said that the triangle formed by the ribbons is dual to the three ribbons converging on the vertex. I guess I don't really know what this use really means. I was thinking when I first read it that in the tetrahedron model, the mathematics of the triangular face is related to the mathematics of the three-legged vertice. You can either describe the tet by describing a face and its relationship to the other faces, or by describing the vertex and its relation to the other vertices. So the tet, or other form more generally, has these two descriptions, dual descriptions. One describes the surfaces, the other describes the vertices and edges. Since both maths are describing the same object, the descriptions, however different they may appear in formula, are then dual.

Did I get this right?

I hope you are feeling better soon, selfAdjoint. Your contributions have always been most valuable to me.

Thanks

nc

 

[marcus]

it is in fact foolish to say more for NC benefit, but in case some other reader looks in and is puzzled maybe you can think of those irritating people who sculpt cute dogs out of sausage-shape balloons at children parties.

imagine you had something better than one of those cute red balloon dogs, namely you had an inflatable balloon forming a tubular tet.

then because it is stretchy you could set the thing down on the table, like a 3-side pyramid frame with open sides and open bottom, and you could squash the apex down into the middle of the open bottom

so then it would be like a flat tubey triangle with another point in the middle, and that centerpoint being joined by tubes to the triangles vertices.

and if you think what that looks like it is a surface of 3-hole donut. draw a picture to be sure.

 

[nightcleaner]

AHA!

Thank you Marcus. We will all wear tube-tet hats at the 8piG roast.

nc

 

[Kea]

I am sure looking forward to that 8piG roast. The idea of dual in 2D is that faces go to points, edges to edges, as in the picture.

I must confess, I mentioned ribbons in the hope we could start discussing tortile tensor categories...

 

[nightcleaner]

Twisted tensors, huh? I do want to know more, but I keep running off in my own odd directions. One of those old Chinese guys says you can't poor more tea into a full cup.

So I must empty my cup and resolve to listen more and babble less.

Please tell me more about tortile tensor catagoriess, Kea. I have never heard of them before.

Thank you,

Richard

 

[Chronos]

Hi Richard! Kea is telling one of those 'get it' jokes that aussies are so fond of terrifying us with! Good one Kea!

 

[nightcleaner]

Hi Richard! Kea is telling one of those 'get it' jokes that aussies are so fond of terrifying us with! Good one Kea!

Hi Chronos

I have been googling. I don't usually think of macreme as an Australian art, but I suppose it could be useful if one wanted to try to catch a mountain parrot. Who would have thought a thesis on macreme would garner so many citations? I am very impressed by the stringyness. Still, many birds are interested in string. Why not a parrot after all?

I wonder if I asked the author for a copy of the thesis if I would "get it?"

nc

 

[Kea]

I don't usually think of macreme as an Australian art, but I suppose it could be useful if one wanted to try to catch a mountain parrot.

Mountain parrots are quite adept at cutting ropes, given the opportunity. An old explorer tried to catch one with a wire snare hidden under leaves. The Kea dug under the leaves and flicked the wire out of the way, and took off with the bounty unharmed.

Think about a knot (a closed loop) embedded in ordinary 3-dimensional space. If one wishes to compute a knot invariant using physics (Chern-Simons path integrals) then one needs to thicken the knot into a tube. The tube may be twisted an integral number of times, but this winding number is more clearly represented by the twisting of a ribbon running along a diameter of the tube.

 

[nightcleaner]

Hi Kea

When does one wish to compute a knot invariant using physics (Chern-Simons path integrals)? I will try to read up on this.

Do I assume the ribbon wraps edge to edge, covering the surface of the tube along which it makes its transit? Some number of Pi per length of tube, I guess. Ribbon tiles the tube at some angle dependent on width of ribbon...as width of ribbon approaches zero, angle gets steeper to 90? Maximum width of ribbon is equal to circumfrence of tube. After that the plane of the ribbon width could be rolled up on the tube like plastic wrap on one of those megaphones I sometimes use to drive Tucker into frantic fits of confusion. Very entertaining. Sorry.

So angle of wrap could be expressed as a width of the ribbon in ratio to the circumfrence of the tube. If you know the width of the ribbon and the angle, you know the circumfrence. If you know the width and the number of revolutions, you know the length along the tube. All this assuming a staight tube. If the tube is bent, the ribbon has to be thinner in places, wider in others, and if you unwrapped it, you would see its wave form. If the tube was bent sufficiently, the width of the ribbon in places would be forced to zero and the ribbon would become a set of spindle shapes.

What fun.

It is interesting how when you twist a rope (maybe you are winding it up in a coil after being frustrated in the great Kea Trap debacle) the free end likes to roll back on itself and form spirol twists with a loop. I have often thought that could be a way to get particles from strings.

So back to the topic. You take a dog annoying megaphone (cardboard tube) and you wrap a pretty red christmas ribbon around it, starting at one end and going edge to edge to the other end. You then carefully grasp both ends of the ribbon and remove the tube, then stretch the ribbon to some rather not too tight length, several many times longer than the original tube. Now the ribbon is a spirol. You look at the spirol in a sharp single source light and it looks like a string of beads. At each twist, the ribbon goes through angles to the light, at regular phases going from zero width to the full width of the ribbon. A string of beads, rather like the strobe image of a ball moving through space when the velocity of the ball is twice the diameter of the ball per each flash interval. If you pull the ribbon tight, it turns into a tube again, a much narrower much longer tube. There is a relatonship between long tube and the original shorter one...they are both made of the same material. And there would be some math to describe the ends of the ribbon tube also, I should think.

Well I am sure you have much more compact and precise language to talk about these interesting things. Surely these kinds of images are what we need for geometry in higher dimensions. Now there is an explore I can get into.

An old explorer, actually a refugee but with a hint of pride, from a culture that was rapidly outgrowing its old clothes, once got tangled up in his nets while trying to mist-trap migrating birds. He hung upside down from the gently swaying branches for some time, his face now a few inches, now a few feet from the litter of the forest floor.

In the third story canopy of the forest above him, a bright eye watched, amused, but also intrigued by the unusual sight. Now there was a bit of rope that could be interesting. Bright feathers and the rush of wings from there to here.

So. I was going to tell you about the sound of raven wings on a cold night in midwinter when the silence in the deep woods was a bell, and the whoosh whoosh whoosh of air through feather was the beginning and the end of the universe, terrible tingling fear and the bright cold stars dying behind wide dark wings. Whoosh. A raven. Instead, the above.

It is spring in Minnesota, the first buds pushing out tips of green on the red dosier dogwood brush that hugs the ditches. So it is Autumn in New Zealand. I hope your harvest is the beauty in the bounty.

Blessed be,

Richard

 

[Kea]

Do I assume the ribbon wraps edge to edge, covering the surface of the tube along which it makes its transit?

Hi nightcleaner

Actually, the ribbon runs along the center of the tube. In other words, for each disc slice of the tube, choose a diameter so that they all meet up smoothly as you run around the tube.

I was disappointed with the movie. Were you?

Kea

 

[nightcleaner]

Hi Kea
Do the two edges of the ribbon make a double helix?

I havn't seen the movie, but found the trailer by google a few days ago. The trailer is mysterious and provocative, gives me shivers. Someone from a far universe reporting the end of the world on a failing radio...what is she saying anyway?

nc

 

[selfAdjoint]

Hi Kea
Do the two edges of the ribbon make a double helix?

I havn't seen the movie, but found the trailer by google a few days ago. The trailer is mysterious and provocative, gives me shivers. Someone from a far universe reporting the end of the world on a failing radio...what is she saying anyway?

nc

Yes if you twist a ribbon as Kea describes, the edges will make a double helix.

What movie is this you two are discussing?

 

[nightcleaner]

Hi selfAdjoint

That would be telling.

Interesting things happen to a twisted ribbon if you pull one edge of the double helix into a straight line, forcing the other edge to wind around it.

Richard

 

[Kea]

Do I assume the ribbon wraps edge to edge, covering the surface of the tube along which it makes its transit?

As usual, nightcleaner has his own wonderful and illuminating perspective on things. If one traces each end of the diameter ribbon around the tube one gets two curves, just like the boundary of a wrapped ribbon. But the number of windings will be different. Nightcleaner, what is the relationship?

Actually, I've only seen the first movie - it just came out here. The whooshing story is indeed creepier. Sounds more enjoyable! I'll try to view it from a distant planet.

And yes, the desiduous trees in the city here are blazing autumn in the low morning sun. Personally I prefer the native evergreens, lush under a weight of snow.

Cheers
Kea

 

[Kea]

What movie is this you two are discussing?

Nightcleaner, we can't be so mean to selfAdjoint! Here's a clue...a quote...

For a long period of time there was much speculation and controversy about where the so-called 'missing matter' of the Universe had got to. All over the Galaxy the science departments of all the major universities were acquiring more and more elaborate equipment to probe and search the hearts of distant galaxies, and then the very centre and the very edges of the whole Universe, but when eventually it was tracked down it turned out in fact to be all the stuff which the equipment had been packed in. (1992)

 

[marcus]

Nightcleaner, we can't be so mean to selfAdjoint! Here's a clue...a quote...

For a long period of time there was much speculation and controversy about where the so-called 'missing matter' of the Universe had got to. All over the Galaxy the science departments of all the major universities were acquiring more and more elaborate equipment to probe and search the hearts of distant galaxies, and then the very centre and the very edges of the whole Universe, but when eventually it was tracked down it turned out in fact to be all the stuff which the equipment had been packed in. (1992)

sounds harmless enough, mostly harmless anyway

 

[nightcleaner]

I guess pi and two pi. I can almost visualize it. The diameter has to rotate twice to return to its original configuration, while the wrapping ribbon only goes around once before returning to any chosen zero.

Fill the tube with a helix ribbon and slice it in half like slicing a sausage, not the usual way, but from end to end. What happens to the helix ribbon, Kea? What shapes will it fall into?

Sorry about the raven wing thing, just a personal image that comes back to me at odd moments, and it occurred to me strongly as I was writing then. Winter is my favorite season. Sometimes I dream of moving to Antarctica, but it is already too late. I am skipping work to write this letter.

Yodhi, anyone?

Richard

 

[Kea]

I guess pi and two pi.

One to two. Exactly.

Ahhh. In this thread I feel like I'm in a sea of sanity amidst a world gone mad. Discussions of North American politics I can safely ignore, but politics in Physics is like a hot sun soaking up the sea's surface. Maybe they're hoping they'll find the keys by burning up the ocean. I just hope they still have 99 cent cans of beans at the supermarket.

Cut the tube like a sausage? Well, let's see. When the axis of the cut lines up with the diameter it will cut right through the ribbon. That will happen once a twist. Otherwise the cut hits the centre of the ribbon. So we'll get lots of little half-width half-twisted ribbons.

All the best everybody
Kea

 

[nightcleaner]

Hi Kea

You are lucky to be able to ignore the insanity. I have made efforts to isolate myself from the culture, but still have to admit my part in it. They said lead, follow, or get out of the way, and I wasn't interested in going the direction they were going in, so I tried to get out of the way.

My place in the woods is a joy to me but sometimes I worry my conscience about if I am guilty of the mayhem by abstemption. Would that one, and so many other, horrible thing have happened if only I had been there to tell them not to do it?

I said no a long time ago and it was spitting into the tempest. You can't go up a stairway when everyone is going down. At the first and every opportunity I made myself small. I have been relegated to the night. I lived for a long time in places no one else wanted. I found friends among the other outcasts and refugees. We wanted to live on the land, and then we learned to live in the land, and now we are becoming part of the land. We still feel like aliens. Maybe when we are the land we will feel at home.


I nibble the new leaves that the deer live upon. Tiny buds of aspen and ash are sweet and then bitter. This spring is slow coming, the weather cold and damp. Rain, fog, ice crystals, wind, broken clouds and a mocking wink of sun, then rain again, it changes every time you go out the door. I got off work early and came home under a starry sky. Maybe tomorrow will be clear and warm. I would like to work in the garden. And this week the horse-logger from Hovland has promised to come with two of his friends to finish putting up a roof on the cottage. Maybe I will have a place to live soon!

Yes, I see the little pieces of ribbon, half as wide, one half twist long. But my visualization keeps wanting to put in curves. Why isn't there a diagonal slice through the half turn? Can't quite get used to that. Probably an artifact of imagining something with thickness being cut with a knife...if the ribbon were a true two dimensional surface would it have any thickness?

Let's look closely at the cross cut, that is the place where the ribbon is cut across. Let's imagine that the ribbon is composed of a two dimensional array of beads, and that the knife works by cutting the imaginary string between two adjacent beads. The beads of course are glued to the string so they don't go flying all over the place! Then at the site of the cut, every string from one side to the other of the ribbon is cut, and the angle is ninety.

Now in the next row of beads, the knife edge must find the exact center of the ribbon, and the exact center of every row of beads after that until it comes to the next cross cut.

Well if the beads are really points and have no dimension I suppose we can do that. The tiniest tilt will move every point on the ribbon out of the way of the knife edge, exept the center most point, in all cases. So we do cut straight down the center axis of the tube which coincides with the center line of the ribbon, and at regular intervals we also cut a line at ninety across the ribbon.

In the real world all our models must have some thickness so a perfect knife edge will not simply cut the center point of every row but will cut a varying bevel. That bevel preserves the information of the spirol along one edge of the half twist.

Of course, there is another difference between the spirol wrap on the surface of a tube and the spirol twisted ribbon inside the tube. The edge of the spirol wrap can be completely defined with one spirol line, since the wrap places opposite edges of the surface ribbon together. But the spirol twist in side the tube has two edges which are nowhere placed side by side, so defining the position of the edges of the spirol twist as they touch the tube from the inside requires two spirol lines on the tube. As a result, if you wanted the spirol wrap outside the tube to match up edge to edge with the spirol ribbon edges inside the tube, you would need two outside ribbons.

But what has all this to do with macreme? And where is my autographed copy of that doctoral dissertation?


nc

 

[selfAdjoint]

sounds harmless enough, mostly harmless anyway

Hitchhiker's Guide to the Universe. Is it OUT? Or does somebody have a time machine?

 

[marcus]

Hitchhiker's Guide to the Universe. Is it OUT? Or does somebody have a time machine?

the movie is out

 

[marcus]

Alan Rickman provided the voice for the morose robot Marvin

http://www.imdb.com/title/tt0371724/

This site gives the other cast, with photographs, filmographies, bios, and some other details about the film version. Comment appeared favorable but did not inspire me with enthusiasm to see the film. the book is such a good read I would be cautious about overlaying it with someone else's visuals.

 

[Kea]

Hitchhiker's Guide to the Universe. Is it OUT? Or does somebody have a time machine?

No, it's not. But Hitchhiker's Guide to the Galaxy is. And Marcus is right about the film.

 

[Kea]

In the real world all our models must have some thickness so a perfect knife edge will not simply cut the center point of every row but will cut a varying bevel. That bevel preserves the information of the spirol along one edge of the half twist.

Great thoughts, nightcleaner. You want to replicate double helices? The maths does promise such wonderful things, but the details live in the (not too distant) future.

A tortile tensor category is a braided monoidal category for which every object has a twist map A \rightarrow A and also a dual object A*.

as discussed in M.C. Shum's paper (J. Pure. Appl. Alg. 93 (1994) 57-110)
- sorry, no web version.

Now given any category whatsoever, one can create a tortile tensor category out of it by labelling 2-stranded knotty diagrams with bits of the category. For example, as above, one might be considering quantum group like categories with 4 allowable spin values A, B, C, D ... or maybe A, C, G, T ...

Kea


Let's try a little picture, courtesy of
http://biology.clc.uc.edu/courses/bio104/dna.htm

 

[Kea]

And as I've just discovered in

Topological Quantum Computation in TGD Universe
M. Pitkanen
http://www.physics.helsinki.fi/~matpitka/articles/tqc.pdf

the angle shift for nucleotides in DNA happens to be \frac{2 \pi}{10}.

 

[Kea]

For those interested in homotopy theory:

Recall the well-known connection between Hopf fibrations (linking numbers) and the group of classes of maps from the 3-sphere to the 2-sphere

\pi_{3} (S^{2}) = \mathbb{Z}

where one can think of the 3-sphere as the unit quaternions. Similarly, considering the 7-sphere instead, one finds

\pi_{7} (S^{4}) = \mathbb{Z} \oplus \mathbb{Z}_{4}

so octonionic analogues of Hopf links give rise to a four element torsion group. Actually, it is better to think of this as a quaternionic analogue of complex geometry, and the 7-sphere as lying in \mathbb{H} \times \mathbb{H}.

 

[Kea]

And the reason I'm mentioning this is, of course, that it appears in the basic twistor correspondence, which relies on the fourfold cover of the conformal group by the twistor group SU(2,2), and where its elements do in fact correspond to spins in the massless free field equations.

 

[nightcleaner]

Hi Kea

Can't say Marcus didn't warn me, can I?

I'll google Homotopy theory and see if I can make anything of it.

nc

 

[selfAdjoint]

nightcleaner, here is a neat visual of the Hopf Fibration. http://www.cs.ubc.ca/nest/imager/contributions/scharein/hopf/hopf.html

It maps the three-sphere (all of our 3-D space plus a point at infinity) into the two-sphere (like a basketball) with the fibers (pre-images of two-sphere points under the mapping) being one-spheres (circles). Well, it's clearer in the video.

 

[marcus]

More of Robert Scharein's topological graphics

nightcleaner, here is a neat visual of the Hopf Fibration. http://www.cs.ubc.ca/nest/imager/contributions/scharein/hopf/hopf.html

It maps the three-sphere (all of our 3-D space plus a point at infinity) into the two-sphere (like a basketball) with the fibers (pre-images of two-sphere points under the mapping) being one-spheres (circles). Well, it's clearer in the video.

beautiful
by truncating I found this

http://www.cs.ubc.ca/nest/imager/contributions/scharein/

Robert Scharein is in the U. British Columbia dept of computer science and
he likes visualizing knots and making online animations of topological things

he calls his page of images "knot plot" and he has links to other people's knot pages too

 

[nightcleaner]

nightcleaner, here is a neat visual of the Hopf Fibration. http://www.cs.ubc.ca/nest/imager/contributions/scharein/hopf/hopf.html

It maps the three-sphere (all of our 3-D space plus a point at infinity) into the two-sphere (like a basketball) with the fibers (pre-images of two-sphere points under the mapping) being one-spheres (circles). Well, it's clearer in the video.

Hi selfAdjoint, and Marcus

Pretty pictures.

It loads slow on the old stretch string I have for a land line, but I had the patience to watch it do its thing one time through. My impression was of a view from the inside of a translucent torus, pan back and spin the torus, exit torus wall and watch torus shrink in thickness and expand in circumfrence until it is a tube joined in a circle.

I probably missed the point.

I would like to know about the evolution of the terminology, one-sphere two-sphere and so on. One-sphere is a circle, two-sphere is a basketball, three-sphere is all of our three dimensional space mapped to a point at infinity...I should have thought a three-sphere to be a sphere in three dimensions, in otherwords a basketball, but apparently the conventional usage followed a different logic. Why are those numbers connected to those images instead of the ones that seem reasonable to me?

Is a two-sphere a simple circle or is it structurally a tube?

Thanks

nc

 

[selfAdjoint]

Hi selfAdjoint, and Marcus


I would like to know about the evolution of the terminology, one-sphere two-sphere and so on. One-sphere is a circle, two-sphere is a basketball, three-sphere is all of our three dimensional space mapped to a point at infinity...I should have thought a three-sphere to be a sphere in three dimensions, in otherwords a basketball, but apparently the conventional usage followed a different logic. Why are those numbers connected to those images instead of the ones that seem reasonable to me?

Is a two-sphere a simple circle or is it structurally a tube?

Thanks

nc

The terminology actually goes down to zero. The zero-dimensional sphere is a pair of points. The terminology describes the dimension of the sphere ITSELF, not of the space it lies in, because a given dimensional sphere could lie in any larger dimensionl space. So the dimensionality of the embedding space is not well-defined, but the dimension of the sphere itself, is.

A basketball (you have to think of the skin of it as infinitely thin) is two-dimensional. That is you can set up a coordinate system like latitude and longitude - two coordinates - to describe it. And yes that simple coordinate system is singular at the poles, so you have to cover the basketball with at least two open sets, for example one of them can be the northern hemisphere, plus a tad below the equator, and the other would be the southern hemisphere, plus a tad above the equator. The tads are to make sure the two sets overlap and cover the whole sphere. And you can set up separate non-singular two-dimensional coordinate systems in each of these open sets and derive simple rules for converting from one to the other on the tads where they overlap. This is exactly what we mean by "a two-dimensional manifold." So two-sphere is the right name for the basketball.

And similarly the three dimensional sphere is locally made of three dimensional space. The idea of "all of three space plus a point at infinity" is based on stereographic projection; you map the whole space into the sphere minus the one point you project from, and then insert that one point. This concept of the three sphere (there are others), is the one Hopf invoked in defining his fiber space.

 

[kneemo]

... three-sphere is all of our three dimensional space mapped to a point at infinity...I should have thought a three-sphere to be a sphere in three dimensions, in otherwords a basketball, but apparently the conventional usage followed a different logic.

Is a two-sphere a simple circle or is it structurally a tube?

selfAdjoint's logic is consistent for the 2-sphere and the 3-sphere. The description of the 3-sphere being our space mapped to a point at infinity follows from the 2-dimensional analogy.

Consider a 2-dimensional plane (like the x,y plane), extending out to infinity. Following tradition, let this 2D plane be the complex plane \mathbb{C}. Now imagine a basketball-like sphere floating above this plane and identify a north pole. Start drawing lines from this pole down to the plane and you'll notice the line passes through the a point p_0 on the sphere down till it hits a point z_0 on the plane. This sets up a correspondence between points on the sphere p and points on the plane z.

The next question to ask is: doesn't the correspondence break down for the point at infinity on the plane? The answer is yes, sorta. We have to consider the 'point at infinity' on the complex plane as different than all the other points p being mapped to the sphere. The correspondence, or 'stereographic projection' maps the complex plane plus the 'point at infinity' to the sphere S^2, which in our complex case is the Riemann sphere \mathbb{CP}^1.

What selfAdjoint is doing is considering our 3-dimensional space as a space analogous to the plane in the 2D example. We imagine there is the analog of a 'point at infinity' for our 3-dimensional space and map it to a sphere in 4-dimensions. So the Riemann 2-sphere construction is good for Pac-Man in flatland, but will not do for us 3-dimensional beings. We up everything one dimension, and struggle to visualize this higher sphere, just as Pac-Man struggles to imagine a basketball-like sphere.

Regards,

Mike

 

[nightcleaner]

Thanks, selfAdjoint and Kneemo

I presume this topology is well-developed and mathematics is not going to tremble because I cough about it. Oh well.

Still, stubborn individualist that I am, I have to examine the ground carefully before I can commit myself to walk upon it, no matter how many others have already danced lightly across the gap.

The essense of my complaint is that the mathematics of topology, beautiful as it is, starts with the assumption of a one-to-one correspondence between points, hence the mapping from the two-sphere pole to the plane. Points are perfectly small, so there is no problem mapping the two-sphere to the plane, until you come to the final, polar point, which then has to have lines that map out to the infinite edges of the plane, and therefore has to have a one-to-infinity correspondance. The lines from the pole to infinity are all parallel to the plane, but of course they can't be because they have to meet the plane at infinity...but hey, it's only the one point that is the exception to prove the rule, and all goes well as long as you don't get too close to the singularity. In mathematics. Where points are perfectly small.

However, if the plane and the sphere are quantized, so that the points are not perfectly small, we immediately run into trouble with the topology. It is no longer possible to draw a line from the pole to the plane which passes though only one point on the sphere and one point on the plane. That works pretty well for the southern hemisphere, but the the closer one comes to the north pole position, the more points the line has to intersect, if the points have measurable size. This reflects the fact that, intuitively, the plane, which is open to infinity, should be larger in surface area and so have more (quantized) points than the sphere can have.

If we assume a quantized surface on the sphere, there is a different problem. How do you build a perfect sphere of finite size from other, smaller spheres? My conjecture is that small spheres of uniform size cannot be tiled perfectly on the surface of a larger sphere without leaving irregular gaps on the surface.

This leads me to wonder about the geometry of the gaps. If we define the radius of the large sphere R and the radius of the tiling spheres r then there must be a relationship between R and r that would minimize the amount of gap. If there are minima there must also be maxima. Perhaps the maxima occurs when the gap becomes large enough to admit one more sphere on the surface.

But I have not had much luck finding a mathematics of the gap. (I bow here to Garth, who put forward the idea of a "God of the Gaps", altho I can't say that what I am describing has anything to do with what he was talking about.)

Leaving math and physics for a moment, as long as I am thinking of it, Garth's God of the Gaps seems to me to be the God of the leftover places, widely worshipped in Europe and America to this day, by means of ancient custom and ritual, who was also associated with the commons, the green spaces, the unfenced land. Probably a descendent of Pan, the great god of wilderness, no longer contiguous.

People harvesting fields in Europe used to leave a small patch uncut, to honor the god of the leftovers. The idea was that the harvestors drove the god back swath by swath, until at the end, the leftover god was confined to a single sheaf of grain. This sheaf was cut and tied with special honors, brought into the house and hung on the wall all winter, as a way to ensure a good harvest for the following year.

Today many people like to decorate their walls each autumn with wreaths and sprays of wheat or other grain, or even the boughs of trees and weeds picked from the roadsides. Ok, they just do it because it is pretty, not to honor the Great God Pan.

I suppose that the fencing of the commons, the death of Pan, autumnal decoration, and god of the gaps has little to do with that single point at the pole that has to stand in for everything. It is a rainy day in May in Minnesota, and this new moisture should bring the leaves out. They are tardy this year, having suffered a late frost again. My world is closing on the singularity, where everything is nothing. I hold the fragrant spring soil in my fingers, knowing that someday even this will be dust scattered in darkness.

But for this little time we have light and warmth and fragrance.

I bid you all enjoyments, mapping this infinity onto one grain of sand.

Be well,


Richard T. Harbaugh

 

[Kea]

Goodness! Everyone has been so busy.

You're quite right, nightcleaner, ordinary spheres are just boring classical point set topology. There is such a thing as a fuzzy sphere in the LQG literature but I don't think that's what you're after either.

The thing is, even though knots are just made out of bits of ordinary curves, endowing them with an orientation and an interpretation in terms of some nice algebraic structure turns them into a representation of something vastly more abstract. Then one doesn't care about the ordinary points on the strings because physical points become something else entirely.

Unfortunately, the literature is full of papers that don't see it this way and that give the misleading impression that points in a physical geometry really are just ordinary points.

Regards to all
Kea

 

[Kea]

A rainy day in May

...as Jupiter
On Juno smiles, when he impregns the Clouds
That shed May Flowers...
Paradise Lost, Milton

No longer poetry month, but what the heck.

 

[Kea]

more on lines

Consider the category Set of sets with functions as arrows. In this category there is the set of 3 bananas in Marcus's bowl, and also the set of 3 oranges in selfAdjoint's bowl. These are different sets. There are functions between these sets, such as the function that takes the top banana to the top orange, the middle banana to the middle orange, and the bottom banana to the bottom orange. This particular function happens to be one-to-one. Anyway, functions are represented by lines (actually arrows).

This use of 1-dimensionality represents the fact that the sets are different or separated. The sets themselves are represented by points, but hopefully it is clear that these points aren't points in the usual sense. So for someone living with the logic of Set, many sets are distinguishable.

Now if one lumps finite sets of the same cardinality into a single point n these might be objects in some other category, such as the category of n \times m matrices (the matrices are the arrows).

Sorry everybody, if this has gotten a little off-topic.