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

[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.