What are the most beautiful ideas you've encountered? By beautiful, I mean ideas that surprised you, were intensely intellectually satisfying, and changed how you see the world.

For instance, I was blown away when I learned about integration. I had been struggling to calculate the distance a ball traveled given its velocity and felt like it was impossible to do exactly. Then I learned about integrals and that there are exact, closed-form solutions to the problems I faced and was simply floored.

Here are the most beautiful ideas I've encountered, in no particular order:

• NP-completeness and P = NP ?
• Universal Turing machines
• Decidability and undecidable problems
• Countably vs. uncountably infinite sets
• Proof by induction (and other methods of proof)
• Calculus (differentiation, integration, and the connection between the two)
• Fourier transforms
• AM/FM radio
• The connection between electricity and magnetism
• Public-key encryption
• General/special relativity
• Evolution via natural selection
• Emergence

These ideas all come from math and the hard sciences. This is likely at least partially the result of a bias, either in the subjects I've studied or the types of ideas I find intellectually stimulating.

It's also possible that many of the incredible ideas from other fields have become so pervasive in our culture that they no longer seem incredible. For instance, the concept of "consciousness" would be startling and beautiful if you learned it as an adult, but I don't know when I learned about it; I feel like I've known it forever. Contrast that with the math behind AM radio, which I can pinpoint exactly when I learned.

The most controversial explanation is that other fields don't have any (or as many) such ideas. Despite being a bit of a math and science bigot, I hope that's not true, because it would mean there are fewer beautiful ideas in the world.

I came up with the above list in about 10 minutes. Take a few minutes and come up with the most beautiful and significant ideas you've learned.