Polytropy

The last post, “Knottedness,” constructed Alexander’s Horned Sphere and proved, or sketched the proof, that

• the horned sphere itself is topologically a sphere, and in particular is simply connected, meaning

  • it’s path-connected: there’s a path from every point to every other point;

  • loops contract to points—are null-homotopic;

• the space outside of the horned sphere is not simply connected.

This is paradoxical. You would think that if any loop sitting on the horned sphere can be drawn to a point, and any loop outside the horned sphere can be made to sit on the sphere and then drawn to a point, then we ought to be able to get the loop really close to the horned sphere, and let it contract it to a point, just the way it could, if it were actually on the horned sphere.

You would think that, but you would be wrong. Mathematics is pathological that way. Some of us may note the pathologies, then pass to areas where intuition works better. Other persons may become obsessed with the pathologies.

If not by just reading this blog, you might find such persons by doing an image search on the Alexander Horned Sphere. I made such a search, but could not find images like the ones I am going to give here. I am trying to clarify why the horned sphere has the properties it does.

The idea of the proof of those properties is that the horned sphere is the intersection of a descending chain,

of sets X_n that are not simply connected, but are compact.

I did not try to define compactness in the last post. Perhaps one does not normally learn about this concept until one has spent some time with calculus. Calculus is the practical side of what has the theoretical side called analysis. From high school I have Apostol’s Mathematical Analysis (second edition, Reading, Mass.: Addison-Wesley, 1974); we used selections from it with Mr Brown to learn about uniform convergence, a concept needed also for the proof of Dirichlet’s Theorem that I discussed in “LaTeX to HTML.”

Apostol defines compactness for ℝⁿ on page 59; for an arbitrary metric space, page 63. Thus I must have learned about compactness when I started to work my way through Apostol before graduate school. I did some of this work at the farm.

Alexander’s Horned Sphere sits in ℝ³, which is an example of ℝⁿ, which is an example of a metric space, which is an example of a topological space. Such a space was apparently not defined till the 20th century. It is a set of points, and certain subsets are called closed, and these must satisfy certain axioms, namely:

• the union of two closed sets is closed;

• the intersection of any collection of closed sets is closed (here the intersection of the empty collection is understood to be the whole space);

• the empty set is closed.

That’s all.

The friend who gave me the book about Helaman Ferguson that inspired my last post (and now this one) gave me also Siobhan Roberts, King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry (Toronto: Anansi, 2006). There’s a foreward by Douglas R. Hofstadter, whom I discussed in “Writing, Typography, and Nature.” In Roberts’s book, Hofstadter recalls discovering, in adolescence, Kelley’s General Topology. This “did not, in its hundreds of pages, contain a single diagram.” At first, Hofstadter was not dismayed.

Within a few years, however, I discovered that I personally could not survive in such an arid atmosphere. Diagrams … were the oxygen of mathematics to me, and without them I would simply die. And thus, when the air of abstraction for abstraction’s sake became too thin for me to breathe, I wound up with no choice by to bail out of graduate school in mathematics. It was a terrible trauma.

Had he known about Coxeter’s geometry, he might have stayed in mathematics.

There are many kinds of mathematics. Some can be visualized in a two-dimensional diagram; some not. The ellipsis in the quotation of Hofstadter stands for a parenthesis:

Diagrams (or at least mental imagery that could be thought of as personal, inner diagrams) were the oxygen of mathematics to me …

Reading anything gives you mental imagery, unless you are not really reading, but only looking at the pages. However, topology as such is so abstract that drawings on paper may mislead. I was thrilled to learn in graduate school that you could prove a lot of interesting theorems, just from the axioms that I wrote above. I was so thrilled that I wrote about the axioms to a friend in New York who had got a job with a publisher.

The friend was not amused, but he did send me the book of mathematical short stories that his firm published: Rudy Rucker (editor), Mathenauts: Tales of Mathematical Wonder (Arbor House, 1987). A theme of one of the stories (“The New Golden Age,” by the editor) is that, if you figure out a way for people to enjoy mathematics without actually understanding it, then the work they enjoy may be that of cranks.

The friend himself later became a chiropractor.

The notion of a topological space is general enough that the structures of a logic can serve as the points of a topological space. The closed sets are the sets of models of theories. For example, the union of the sets of models of formulas φ and ψ respectively is the set of models of the formula (φ ∨ ψ). The connection between topology and logic was a reason why I decided to specialize in model theory.

In “Boolean Arithmetic,” I worked out the proof that the space of models of propositional logic is compact. The topology of that space is the topology of a Cantor set, discussed in “The Tree of Life.”

As a quotation in the last post noted,

there is a Cantor set of ‘bad’ points on the horned sphere.

We constructed that sphere by, for each finite binary sequence σ, attaching the horns H_σ0 and H_σ1 to H_σ. Here if σ is the empty sequence, we can understand H_σ to be the ball B₀. Each infinite binary sequence τ yields the sequence of horns H_τ|n, where τ|n is the sequence (τ(k): k < n) of the first n entries of τ. The union (over n) of the horns H_τ|n has a unique “tip,” a unique accumulation point that doesn’t belong to it. Such points are the “bad” points mentioned in the quotation.

The same sequence τ determines the model of propositional logic in which the true propositions P_n are those for which τ(n) = 1.

In our present situation, working inside Euclidean space, I suggested in the last post that being compact is equivalent to being closed and bounded.

• Being bounded means being part of a ball. A ball is the inside of a sphere, this being understood in the usual sense of the set of points sharing a common distance from a single point, which is then the center of the sphere and the ball.

• A set F is closed if every point not belonging to F is the center of a ball with no points in common with F. I believe the letter F is commonly used for closed sets because it stands for the French fermé(e).

A subset K of Euclidean space is closed and bounded if and only if, whenever K is covered by a collection of balls, meaning the union of the balls includes K, a finite number of those balls are enough to cover K. This theorem, or the difficult part, the “only if” part, is called the Heine–Borel Theorem. The condition involving coverings is now a standard definition of a compact set (in German, kompakt), and I quoted the definition in “Poetry and Mathematics” as being “so poetic.”

The complement of a ball is a closed set. Then being compact is equivalent to the condition whereby, if the intersection of a family of closed subsets is empty, then some finite number of those closed subsets have empty intersection. This yields that the outside of the horned sphere fails to be simply connected, as in the last paragraph of the previous post.

But why does the proof work? Where does the horned sphere come from?

I used the idea of contorting a lump of clay to capture a loop of string. Freeing the string would mean contracting it to a point. During this contraction, the loop would trace out a surface S. Being closed and bounded, this surface would be compact. Its points would therefore have a minimum distance from the horned sphere. Proof: The horned sphere B being closed, each point of the surface S would be center of a ball that contained no points of B; but finitely many of those balls would cover S, by its compactness. The minimum radius of those finitely numerous balls would be the minimum distance from S to B.

Thus, for any distance, no matter how small, there should be arms of the lump of clay

• that are that close together, but

• that the loop would have to slip through to be free.

An obvious first attempt is to have two arms that grow indefinitely close, as in the figure below. This attempt fails, because the compactness of figure ensures that the two arms will actually touch.

Still, the figure can have, attached to it somehow, infinitely many pairs of arms, with no lower bound on the gap between them, as below.

But how can such arms trap the string? The string cannot initially be made to pass through all of the hoops that are almost formed by the arms; for some of those hoops will be too narrow. (If they are not, then we are back at the first failed attempt.)

However, two hoops, if linked, can form another hoop, which the string would have to break out of to be free.

Thus we let the gap between two arms be almost closed by two more pairs of arms, and so on. This leads to Alexander’s construction, which I have tried to depict below. The original pdf file has size A1, if you want to see more detail.

Each column converges to Alexander’s Horned Sphere