Transcription
If we ever discovered an alien monolith on Mars and it had a single sentence written on it, I believe it would be this equation. To a non-mathematician, this looks like a random collection of letters, but to a physicist, this is the God equation. It is spooky. It connects the five most fundamental numbers in the universe. numbers that were discovered centuries apart in completely different fields of study by people who never met each other.
Zero is the symbol of the void. One is the symbol of unity. Pi is the geometry of the circle. E is the calculus of growth. And I is the algebra of the imaginary. They shouldn't speak the same language. Growth has nothing to do with circles. Imaginary numbers have nothing to do with basic arithmetic. But this equation says they are not only related, they are inseparable.
Most textbooks will show you the algebra. They will show you the trick to prove it. But today, we aren't going to look at the trick. We're going to look at the mechanism. We're going to look at the gears and cogs that make this machine work. Because when you see how the pieces click together, you realize this isn't just math. It is the source code of reality.
To understand the God equation, we have to meet the characters. In the first one is the most misunderstood, E. Most people think E is just a magic constant, like a random serial number. But where does it come from? In 1683, Jacob Berni was looking at interest rates. He asked a simple question. If I have $1 and I get 100% interest, I end up with $2. But what if I cheat the system? What if I calculate the interest every 6 months? I get a little more. What about every day? Every second? Intuitively, you might think you could get infinite money. If you compound continuously, shouldn't your growth explode to infinity? Berni found something strange. No matter how fast you chop up time, even if you calculate interest every nancond, you hit a wall. You can never force growth past a specific number. That number is E. E isn't just a number. It is the universal speed limit of growth. Keep this in mind. E is the number that wants to push forward.
But before we move on, there's a hidden trap, a detail that ruins the math for students all over the world. Degrees. We're used to measuring circles in 360°. But why 360? It's an arbitrary number humans made up because it divides easily by 12. The universe doesn't care about our arbitrary numbers. If you try to do calculus using degrees, the equations break. You get messy, ugly constants everywhere. The math screams at you that you're doing it wrong. But if you measure the angle by taking the radius and wrapping it along the edge, you get the radian. Oiler's identity teaches us a humbling lesson. The universe only speaks to us when we stop using human units and start using natural ones.
Now we meet the villain, the number that broke mathematics. I. Renee Deart, the father of the graph, coined the term imaginary as an insult. He thought these numbers were useless trash. For a 100red years, mathematicians played a game of pretend. They used i to solve equations, but they hid it in the final answer because they were embarrassed. But Oiler didn't care about embarrassment. He looked at the geometry. Think about the number line. If I multiply by -1, I flip 180°. So what number if I multiply it twice equals1? Geometrically that means what rotation done twice equals a 180° flip. The answer is obvious. A 90° turn. I is not a ghost number. I is a left turn. This was the hidden detail that changed everything. We're not dealing with phantom numbers. We're dealing with a two-dimensional map. The real numbers are east and west. The imaginary numbers are north and south. So, we have E, the machine of growth, and I, the machine of rotation. Now, let's force them to collide.
Now, we enter the engine room. How do we combine growth and rotation? We can't just multiply them. We need to look at their DNA. Isaac Newton showed us that every function can be broken down into an infinite sum of numbers. A tailaylor series. This is the infinite machine of e to the x. It adds every single power of x forever. Now look at the DNA of the circle. S and cosine. They look broken. S only has the odd powers. Cosine only has the even powers. And the signs are flip-flopping plus and minus. They look like debris from an explosion. But watch what happens when we drop our left turn agent I into the E machine. We replace X with I * X. Because I is a rotation, every time we multiply by it, we turn 90°. One multiplication up, two multiplications left, which is negative. Three multiplications down negative I. Four multiplications right positive again. This cycle positive positive negative negative forces the infinite series to rearrange itself. It's like a magnet sorting metal filings. The real parts lock together to form cosine. The imaginary parts lock together to form sign. And suddenly the equation appears. E to the ix equ=cossine x + i sinx. By giving growth in imaginary dimension, it transformed into rotation.
If that infinite series felt too abstract, here's the intuitive picture that most people miss. In calculus, the derivative tells you your velocity. The derivative of e to the iix is i times itself. Remember what multiplying by I does? It rotates you 90°. This equation is literally saying your velocity is your position rotated 90°. Think about that. If you're running forward, but something keeps pushing you 90° to the left, you don't go straight, you turn. And if that push always happens, you run in a perfect circle. This equation is the mechanical instruction for how to draw a circle. Always turn left. But here's the most beautiful detail of all, the one that usually gets left out. We usually write this equation on flat paper, but it describes a three-dimensional reality. Imagine x is time. As time moves forward, the value e to the iix traces a corkcrew, a helix spiraling through the complex universe. If you look at this helix from the side, you see a cosine wave. If you look at it from the top, you see a sine wave. This tells us something profound. S and cosine, the waves of sound, light, and music, are just 2D shadows of a higher dimensional spiral. The universe isn't waving up and down. It is spiraling.
You might be thinking, "This is a cool trick, but I is still imaginary. It doesn't exist in the real world." You would be wrong. 150 years after Oiler, we looked at the smallest particles in the universe, electrons. We tried to write an equation for how they move. Irwin Schroinger discovered that you cannot describe fundamental reality using only real numbers. The fundamental equation of quantum mechanics has an I right there in the front. Matter, you, me, this screen behaves like a wave. And that wave is described by Oilers's formula. Nature doesn't just use this equation for poetry. She uses it to build protons.
So, what happens when we rotate halfway around this spiral? We plug in pi. The rotation completes a 180° turn. The imaginary part vanishes. The spiral hits the floor and we land precisely on -1. Move the one to the other side and you get the inscription on the tombstone of mathematics. E to the i pi + 1 equals 0. analysis, geometry, algebra, the void and the unit, all balanced in perfect silence. It suggests that if you look deep enough into the code of the universe, the distinctions we make between things are just illusions. Deep down in the source code, everything is one.
If you are still watching, it means you appreciate the craft. Making animations with code takes a lot of time, but your support keeps this engine running. So please resonate with the algorithm and I will see you in the next dimension.