Have you ever followed your curiosity and found yourself in the wonder of exploration, where the thrill of the unknown pulls you in every direction? That's the magic of following your curiosity, and it's what makes life a never-ending adventure.

In this beautiful world of ours, we're blessed to have mathematics - a treasure trove of metaphors and analogies that help us understand the mysteries of our universe. Differential Geometry is one of those gifts, and it teaches us an essential lesson: Follow your curiosity and trust where it leads.

I follow my curiosity and don’t question it wherever it takes me. That is how at any given moment, I can end up in Carbondale visiting Buckminster Fuller Dome, or flying a plane in Wonder Lake.

Think of your curiosity as a landscape with hills and valleys, represented by the principal curvatures - the steepest slope and the shallowest slope. These slopes may not seem interesting individually, but when you multiply them, you get a curvature that unlocks the surface's secrets.

Carl Friedrich Gauss called this concept Theorema egregium[1], and it's mind-boggling. By knowing the product of the two principal curvatures, you can determine the curvature of the surface at any given point. And if you apply this concept to the Earth, you can prove that it's not flat [2].

But why does this matter? Because following your curiosity is like exploring the principal curvatures of your life. You never know where they will lead you. It could be teaching yourself Differential Geometry, creating random projects, or writing an essay like this one. And the beauty of it all is that the more you follow your curiosity, the more you realize that the sum of your experiences is far greater than their individual parts.

In fact, when you multiply your curiosities, you get a greater purpose, a curvature that defines the path you were meant to follow. And the best part is that you can discover this purpose by merely following your curiosity, without questioning it.

So please, follow your curiosity, let it take you places you've never been before. Even if you face obstacles or uncertainty, trust that your curiosity will guide you to the answers you seek. Don't hold back, don't question it, just let it lead you to the intersection of your curiosity and your potential, where the greatest discoveries await.

If Einstein's use of the same math in relativity isn't enough to convince you, I don't know what will.

Embrace the unknown, the unexplored, and the uncharted, and let your curiosity lead the way. The universe is waiting for you to discover its secrets.

Who knows where it might lead you? All I know is that the adventure is worth it.

**Notes**

[1] The Theorema egregium (k1 • k2 = K) by Carl Friedrich Gauss.

The two principal curvatures of a surface can be defined mathematically as follows:

- k1 is the maximum curvature of the surface in a particular direction, and
- k2 is the minimum curvature of the surface in a perpendicular direction.

Mathematically, the Theorema egregium states that k1 • k2 = K, where K is the Gaussian curvature of the surface.

[2] Spheres have a surface of constant curvature. However, there are many other surfaces, like the pseudosphere, that also have a surface of constant curvature. Following this thread further, we can learn about the work of Gauss's student Riemann, who developed the concept of non-Euclidean geometry, and ultimately, we can delve into Einstein's theory of general relativity.

Thanks to Étienne Ghys and Salman Ansari for reading the drafts of this essay.

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