The Most Unique Shapes in Mathematics: Focusing on Plane Subsets

Delving into the intriguing world of plane subsets, we uncover a multitude of fascinating and unique shapes that challenge our conventional geometric understanding. Starting with subsets from the book ldquo;Counterexamples in Analysisrdquo; by Gelbaum and Olmstead, this article explores shapes that are not just visually complex but also deeply significant from a mathematical standpoint.

Unique Shapes in the Plane

While the shapes described below may seem abstract and unvisualizable, each has contributed uniquely to our understanding of mathematics and set theory. Let's explore these intriguing subsets one by one.

1. A Set with Exactly Two Points in Common with Any Line

This particular set, often referred to as a counterexample in analysis, challenges the intuitive belief that a set could have exactly two points in common with any line. While a circle, for instance, might intersect a line at most at two points, it's not immediately obvious that such a set can exist. Its existence is a testament to the vast and unexpected realms of mathematics.

2. Non-Measurable Subsets with At Most Two Points Common with Any Line

Building on the previous concept, this set takes the idea a step further. It is a non-measurable subset of the unit square, meaning it defies the conventional notion of measurable sets. The non-measurability concept, while complex, indicates that this set lacks a well-defined or even approximable area. This peculiar property makes the set a cornerstone in measure theory and set theory.

3. A Connected Set That Becomes Totally Disconnected by Removing a Single Point

This shape, known as Cantor's tent or Cantor's leaky tent, offers a fascinating insight into the world of connected and disconnected sets. It is a connected set that can be visualized (albeit as a dusty line) and demonstrates a transition between being connected and totally disconnected upon the removal of a single point. This set's structure is preserved in the Wikipedia page dedicated to its exploration, showcasing its distinct characteristics.

4. A Simple Curve in the Unit Square with Infinite Length Between Any Two Points

Another intriguing example is a simple curve within the unit square that has an infinite length between any two points on the curve. This curve is a counterexample in analysis and challenges our understanding of length and continuity. A rigorous understanding of real analysis is necessary to comprehend the intricacies of such a curve.

5. Bonus Shape in 3D: More Weird Spheres

In three-dimensional space, we encounter shapes that defy conventional wisdom, such as the wierd spheres mentioned. These shapes are homeomorphic to a sphere's surface, have a nonzero but arbitrarily small surface area, and enclose a finite but arbitrarily large volume. The construction of such shapes is complex, with examples including surfaces such as Gabriel's Horn, which exemplifies the counterintuitive relationship between surface area and volume. While the description of these shapes is intricate, understanding them broadens our appreciation of mathematical concepts and anomalies.