Topological Wonders: Unleashing Imagination in Mathematics

Dedicated to Bella L, her insightful studies, and her delightful article on Tacos.

Recently, I came across a charming article that highlights the enchantment of several unique topological ideas, particularly focusing on quotient maps and the resulting quotient spaces.

Introduction: Dividing a Taco Wrap into Eight Equal Pieces

I don’t intend to overshadow Bella L's work, as her explanations are clearer than mine. Therefore, I encourage you to familiarize yourself with her problem and its solution as detailed in her article.

How To Divide A Taco Wrap into 8 Equal Pieces with 1 Cut?

This is a FANTASTIC PROBLEM that sets the stage for discussing various topological concepts, beginning with a fundamental issue like Bella's.

These concepts include quotient spaces and identification maps, as well as covering spaces and fiber bundles. We will utilize this scenario to elucidate these ideas.

I find it amusing how the video concludes resembling a Granny Smith apple: delightful! My interests span delicious food, mathematics, affectionate interactions, and engaging discussions with cherished friends.

Quotient Maps, Projections, and Identifications

A quotient map identifies sets of points within a topological space as equivalent, effectively "gluing" them together. This formal process helps in visualizing how edges unite in a topological setting. For instance, by identifying two edges of a rectangle, we can transform it into a cylinder. Observing videos that demonstrate how this identification occurs in a two-dimensional manifold significantly aids in understanding. For example, a quotient map that identifies all points on the edge of Bella's disk at the start of her video results in a sphere, which we can visualize by bending and suturing a disk, as illustrated below.

Suturing a disk into a sphere through a quotient map

To delve deeper into these concepts, I recommend Chapter 7, "Surfaces By Identification," from Bert Mendelson's "Introduction To Topology." While texts like James Munkres's "Topology" provide more rigorous explanations, they can be overwhelming for beginners. I believe one should gain some experience before tackling Munkres's text.

There are also excellent online resources, including videos by mathematics educators. However, if you, like me, have challenges with auditory comprehension, the traditional approach of reading beautifully crafted texts at your own pace is still preferable.

Let’s start with a straightforward example: take a strip of paper and bend it into a ring. A quotient map identifies the edges brought together, yielding a cylinder—the simplest instance of a surface defined through identification. This surface is also developable; rolling it does not distort the geometry of figures drawn on its surface. Angles remain consistent, meaning triangles on a flat sheet correspond to triangles made of geodesics on the cylinder and vice versa. The sum of angles in triangles remains 180º, Pythagorean Theorem applies, and the geometry is entirely Euclidean.

This flatness manifests in the paper's ability to bend without creasing, given an ideal infinitely thin strip. In contrast, a thicker material would experience bending moments due to layers at varying heights, leading to different radii of curvature. However, in the ideal case, the only distinction between our cylinder and a flat paper strip is that, if you were a tiny creature on the surface, you could walk in a specific direction and eventually return to your starting point.

A related concept is projection, which collapses multiple points in various instances of a base space to a single point in that base. Projection is commonly associated with covering spaces, Cartesian products, or fiber bundles. A fiber bundle generalizes the Cartesian product, having a local resemblance to it while allowing for intriguing global topological properties. Projection is also a concept used in vector spaces to break down a vector into lower-dimensional components within a "base" space and its orthogonal complement.

All of these ideas—quotient spaces, Cartesian products, covering spaces, fiber bundles, and vector spaces—are distinct yet lead to varied theoretical outcomes. What I admire about Bella's problem is its ability to intersect these concepts, encouraging readers to explore different paths from this common point.

The Taco Division Problem

Before we explore more generalized ideas, let’s analyze a broader version of Bella’s problem using recursive quotient maps to identify equal sectors of a paper disk.

Let’s define notation for Bella’s problem: let a?,? represent the nth vertex of the disk folded m times as shown below. Each fold doubles the number of sectors, resulting in 2? after m folds. After three folds, as in Bella’s problem, we have eight vertices labeled a?,?.

Initially, Bella folds the disk in half. Let f: ??????? be the quotient map that folds a sector ?? along its axis of symmetry (refer to the sector on the right side of my diagram) and identifies the surfaces that come into contact. For clarity, we define each sector with one edge removed to prevent overlap with its neighbor. The first fold along the horizontal diameter unites two semi-disks and their circumferences, leading to the identifications a? and ??, a? and ??, and so forth. The second fold along the vertical diameter creates two equivalence classes. Ultimately, the last fold identifies all octant sectors.

We can view the recursive folding as arranging the circle into a preimage of the final identified sector:

Next, we consider what happens when we identify all copies in the cover and bisect the sector. To address this, we must determine the set of points in the original disk mapped to the line down the center of the sector. This creates an asterisk-shaped network that divides the original disk into eight distinct sectors, thereby completing Bella's demonstration.

You can visualize how the eight segments of the cut are identified in the image above. The first fold about the horizontal diameter connects ? with ?, ? with ?, and so forth. The second fold about the vertical diameter identifies ??? with ??? and ??? with ???. Finally, all eight points correspond to the ends of each segment resulting from the cut that divides the disk into eight separate sectors.

Thus, the entire disk can be interpreted as a covering space or a fiber bundle—a trivial fiber bundle equivalent to a Cartesian product between the sector and a set of eight discrete points. The base space is B, and its eightfold cover is ? (in cover terminology, we often denote the cover with a tilde over a symbol, such as X and X-tilde). Here, P represents a fiber of the cover or bundle.

Fascinating Fiber Bundles

I will return to fiber bundles that transcend simple Cartesian products in a future article focusing on the remarkable Hopf fibration, a particular topology of unit quaternions.

The Hopf Fibration, showcasing the topology of the three-sphere or unit quaternions

The Hopf fibration warrants a dedicated article, which I provide here:

Topological Magic: Bizarre Twists in Space

The exquisitely beautiful Hopf fibration serves as a generalization of Cartesian products into fiber bundles.

But before we delve into such wonders, let’s complete our discussion on surfaces formed through gluing!

More Surfaces Through Identification

Two other notable instances of flat (Euclidean) quotient spaces are the Möbius strip, achieved by identifying two opposite sides of a rectangle, and the Klein bottle, which is derived from the Möbius strip by suturing two halves of the Möbius strip’s edge. The Möbius strip, when formed from a strip with an aspect ratio of at least 3:1, is surprisingly flat and Euclidean. Remarkably, the Klein bottle's surface is also Euclidean. Together with the more familiar plane, cylinder, and torus, the Klein bottle and Möbius strip encompass all possibilities when we inquire about the geodesically complete two-dimensional smooth, flat manifolds.

Visualizing the creation of a Möbius Strip through twisting a strip of paper

In fact, the Klein bottle and torus are the only compact flat, geodesically complete two-dimensional manifolds.

These five surfaces formed through the identification of rectangle sides all possess Euler characteristics of 0. I discussed the Möbius strip and the Klein bottle in my article here:

Weird and Divine, O Wondrous Bottle of Klein!

This article explores the Klein bottle, Möbius strip, curled dimensions in the universe, and the fascinating world of topology.

In it, I further elaborate on the suturing process.

Now, let’s examine a few examples in the diagram below, demonstrating how to glue edges of a rectangular strip of paper together.

If we take our cylinder and curve it into a circle, gluing the two remaining edges together without a twist results in a torus. We thus identify A?B and D?C. If we then identify edges AB and CD, as we traverse from A to B on the top green line in the bottom left diagram, we also traverse from C to D on the bottom green line. However, if we attempt the same process to form a cylinder—reaching the stage A?B and D?C but flipping one of the green edges before gluing—while traversing from A to B on the top line, we run from D to C on the bottom line, resulting in a Klein bottle. This transformation necessitates the cylinder intersecting itself in three dimensions, and we must employ the full four dimensions of the Whitney embedding theorem to achieve a true embedding without self-intersections.

The Universe of All Five Possibilities for a Globally Flat Two-Dimensional Topology

In the diagram above, I have sketched each topological object within its fundamental polygon, a notation for constructing surfaces by identifying edges of flat regular polygons, such as equilateral triangles, regular hexagons, or most commonly, squares or rectangles. For instance, if we glue edges BC and AD together in the same direction—meaning our quotient map equates A?B and D?C—we obtain a cylinder. Conversely, if we swap the orientation of one line—so that A?C and D?B, imposing a half twist on the strip before gluing the edges—we create a Möbius strip.

However, with this gluing concept, I must address the remaining possibilities.

The two remaining possibilities are compact and possess non-zero Euler characteristics, making them incapable of maintaining a flat embedding isometric with the plane, according to the Gauss-Bonnet Theorem. The sphere and the real projective plane demonstrate this. Their Euler characteristics of 2 and 1, respectively, preclude them from being flat, regardless of the dimensions in which we embed them. The Gauss-Bonnet theorem states that a compact two-dimensional manifold has an average scalar curvature of 2? times the Euler characteristic, indicating that these two compact surfaces can never achieve a flat embedding since their average scalar curvature across the surface is non-zero.

The sphere is formed by gluing adjacent sides of a rectangle. Interestingly, both the projective plane and Klein bottle can also be created by gluing adjacent rectangle sides, but they ultimately yield the same topology as that resulting from the opposite side identification procedure.

Why is this? Surely, we can define the metric as the distance between points in the original flat rectangle, yielding a Euclidean metric. Amazingly, both the Klein bottle and torus can be assigned flat metrics when embedded in four dimensions. However, this must be done cautiously; when we glue edges in specific ways, we open the possibility of creating a two-sided geodesic polygon with a non-zero corner angle. Essentially, a closed figure with two straight sides enclosing a non-zero area.

Applying a Euclidean metric to this figure leads to contradictions arising from reaching the same point via different "seams." In the image above, adjacent sides of a square are quotiented to form a sphere. If we attempt to assign the resulting figure the square's Euclidean metric, we encounter contradictions. Starting at point U, there are two geodesics leading to point V, intersecting at both U and V at right angles. In Euclidean space, two individuals traveling along geodesics at right angles will move further apart. This principle stems from Pythagoras's theorem. Even in spherical geometry, geodesics can only meet at antipodal points on the sphere.

The identification of adjacent sides excludes a Euclidean metric, and this necessity for distance compatibility across different identification boundaries serves as a foundation for one proof of the renowned Gauss-Bonnet theorem. This insight, alongside Gauss's famous Theorema Egregium, underpins the workings of the Gauss-Bonnet theorem.

An intriguing nuance arises here: not every flat surface is smooth! In mathematical terms, "smooth" implies that every differential operator of any order exists; in simpler terms, the manifold is infinitely differentiable in every direction. A stronger condition than smoothness is "analytic," meaning that the infinite set of derivatives in any direction fully defines the manifold's coordinate functions over a non-zero interval via convergent Taylor-Maclaurin series concerning a path parameter along the defined geodesic line.

We can smoothly embed an n-dimensional smooth manifold into a space of 2n dimensions, as per the Whitney Embedding Theorem. This theorem ensures that smoothness and analyticity remain intact during the embedding process. However, the Whitney Theorem does not consistently preserve isometry (the metric). For the four-dimensional embedding of the torus and Klein bottle, isometry occurs "by coincidence," which is not a prediction of the theorem.

Astonishingly, smoothness is a more stringent logical requirement than flatness. While we need four dimensions for a smooth, isometric (i.e., flat) embedding of a torus, the remarkable Hévéa torus is a three-dimensional torus that is flat (capable of supporting a Euclidean geometry via its metric) but is not smooth—it has a fractal-like structure. Nash embeddings provide embeddings that maintain the metric, but second derivatives lack continuity, resulting in peculiar structures like the three-dimensional torus.

One begins with the familiar "doughnut" surface of the torus embedded in three dimensions. While this is a smooth embedding, it is not isometric, as evidenced by the distortion of the checkerboard pattern on the "rubber" hose that is glued into a torus.

Roughly, the Hévéa torus starts with a smooth, non-flat embedding, followed by an infinite sequence of waves and corrugations with decreasing amplitude and increasing frequency, altering distances anisotropically (with a preferred direction) to “straighten” deviations from isometry. The result is a flat torus with a fractal-like corrugation structure. First derivatives and the metric exist, but second derivatives are undefined. This would yield a world where Euclidean geometry is applicable, yet Newton's second law would hold no significance!