The Beauty and Obsession with Shapes in Mathematics and Physics

Mathematicians and physicists share an undying fascination with shapes—both simple and complex. These intricate forms appear in their daily endeavors, revealing profound truths and intricate patterns. But why do these professionals find such beauty and obsession in the world of shapes?

Simple vs. Complex Shapes: A Beautiful Observation

The concept of shapes, as observed by 16-year-old Blaise Pascal in his Essay on the Conics, showcases a perfect blend of simplicity and complexity. Pascal, a prodigious mathematical mind, published a theorem on Hexagrammum Mysticum in 1639, demonstrating that the intersections of the opposite sides of a hexagon inscribed in a circle or a conic section lie on a straight line. This remarkable observation is both simple and awe-inspiring.

Consider a circle or another conic section, and select any six points on it: A, B, C, D, E, and F. Connect these points to form a hexagon. No matter their order, the intersections of the opposite sides (AB and DE, BC and EF, and CD and FA) will lie on a straight line. This theorem, now known as the Hexagrammum Mysticum Theorem, exemplifies the elegance and beauty inherent in the world of shapes. It is a testament to the inherent simplicity and complexity that coexist in mathematical observations.

Topological Spaces and Shape

Topological spaces underpin the notion of closeness and continuity, extending far beyond the confines of traditional mathematical and physical applications. For example, if a behavioral scientist quantifies the psychological proximity of individuals, the topological space can represent the range of relationships from strangers to very close friends.

Strangers: 0 to 1 (semi-open interval) Exists some empathy: 0 to 1 (open interval) No empathy: Circle Independent sexual and platonic attractions: Cylinder Both directions: Torus Polyamorous relationships: Complicated space

These spaces are rich with variation, demonstrating how certain shapes naturally emerge in the study of human behavior. Once the open sets are defined, a topological space is formed, capturing the essence of space with a defined shape. Examples like the Klein bottle and Riemann surfaces illustrate the complexity of these shapes, which often cannot be visualized in our three-dimensional world.

Comparing Art and Mathematical Shapes

Mathematical shapes transcend art in terms of their abstractness, dimensionality, and embeddability. For instance, the Klein bottle and complex functions like z^(1/2) require higher-dimensional spaces to avoid self-intersection. Additionally, advanced topological concepts like Zariski and Grothendieck topologies present even more abstract and complex spaces that are difficult to visualize directly.

Further Reading and Resources

Exploring the realm of shapes further, readers can delve into the following resources:

The Shape of Space by Jeffrey Weeks: A nontechnical introduction to topology, perfect for those with a basic understanding of the subject. Sketches of Topology: A blog that visualizes complicated topological spaces, appealing to those interested in the geometric aspects of topology. Counterexamples in Topology: A deeper dive for those interested in the intricacies and counterintuitive aspects of topological spaces. Mathematics, Music, and Meaning by Dmitri Tymoczko: A talk exploring how interesting shapes can arise in the context of musical chords, bridging the gap between mathematics and music.

Mathematics and physics offer a boundless and fascinating exploration into the world of shapes, revealing their simple yet complex beauty in endless variations.