Chapter 1: The Intriguing Nature of Symmetry

Tidiness is indeed a virtue, and symmetry frequently embodies beauty. – Winston Churchill

While it’s widely acknowledged that symmetry is a hallmark of beauty, the question remains: how and why is that the case? I, too, grappled with this question until I stumbled upon a fascinating mathematical puzzle.

Imagine eleven stones arranged in rows of three. What is the greatest number of rows you can create?

At first glance, this puzzle may seem disconnected from the concept of symmetry. Compared to articles that illustrate symmetry's appeal using celebrity features, this might appear somewhat dry and less intuitive. However, just as treasures aren't typically hidden on the earth's surface, the allure of symmetry requires us to delve deeper.

Let’s begin by exploring the surface—the simplest scenario. We know that any two points define a straight line. Thus, to align three points on a straight line, the third point must fall on the line created by the other two.

Now, consider five stones. By drawing two intersecting lines and placing point A1 at their intersection, you can arrange A2, A3, A4, and A5 along the two lines.

From this illustration, it becomes clear that maximizing the number of points at the intersections of these lines allows for the greatest number of rows to be formed.

If we increase the count to six points, we can draw four lines that intersect pairwise and assign the six points to the remaining intersections.

As we add more points, it’s evident that they should be strategically placed at the intersections of the lines, creating a more symmetrical arrangement (including central and axial symmetry).

For the scenario involving eleven stones, we can employ the principle of axisymmetric shapes. Since the number of points on the axis is always odd, three points will lie on the axis of symmetry, while the remaining eight will be distributed on either side, ensuring that each point serves as a common point for at least three lines.

This arrangement showcases the beauty of symmetry—natural and striking. As Plotinus observed, beauty is more akin to a light that dances over the symmetry of objects rather than symmetry itself.

Thank you for joining me on this exploration!

Chapter 2: The Visual Charm of Symmetry

Discover how symmetry enhances our perception of beauty in this engaging video titled "The Beauty of Symmetry."

Explore the reasons behind our attraction to symmetry in the enlightening video "Why is Symmetry Beautiful?"