In my presentations I sometimes talk about my four years in engineering school as being something of a disaster. The way I present this fact is through this image, and the math/science types get the pun – Four Year / Fourier Transformation!
Now despite the joke, I have to be honest: I had just the vaguest understanding of the Fourier transformation. I remember learning it many years ago and finding it quite beautiful as an idea. I also knew it was foundational to all digitization of analog signals. But that was pretty much it. So the other day, I decided to understand it better by engaging in a conversation with Claude.
Along the way, Claude told me about an interesting visual consequence of the Fourier Transformation: that any image could be recreated by moving circles. That seemed intriguing so I spent a few hours with Claude to create a simulation to demonstrate this process.
All I have to say is Wow!! And just in case it wasn’t clear, I did not write a single line of code.
Note: The first draft of the prose below was written by Claude. I inserted the somewhat silly story at the beginning, and edited the rest of the piece, to be sure that it contained no errors.
It’s Circles All the Way Down
There’s an old story about a woman who attended a lecture on astronomy. After the astronomer explained how the Earth orbits the sun, she approached him with confidence. “That’s all very well,” she said, “but everyone knows the world is really resting on the back of a giant turtle.”
The astronomer, trying to be polite, asked, “But what is the turtle standing on?”
Without missing a beat, she replied, “You’re very clever, young man, but it’s turtles all the way down.”
While her cosmology might have been questionable, her intuition about infinite recursion was surprisingly profound. In the mathematical world of Fourier transforms and drawing, we discover something equally remarkable: when it comes to creating any shape or curve you can imagine, it really is circles all the way down.
In our complex world, we’re constantly surrounded by patterns, sounds, and shapes that seem impossibly complicated. A symphony orchestra creates rich, layered music from dozens of instruments. Digital images capture detailed scenes with millions of pixels. Complex engineering signals carry vast amounts of information. The question arises: how do we make sense of this complexity?
The answer lies in one of mathematics’ most powerful ideas: that complex things can often be understood by breaking them down into simpler, fundamental components. Just as a master chef can taste a complex sauce and identify its individual ingredients, mathematicians have developed tools to decompose complicated signals and patterns into their basic building blocks. And as we’ll discover, when it comes to drawing, those building blocks are circles – spinning, nested, interconnected circles that go on indefinitely, and through their synchronized dance, create any shape of image you can imagine.
Imagine you’re tracing the outline of a drawing – perhaps a cat, a face, or any shape you can think of. As you trace with your finger, you’re creating two signals: your x-coordinate over time and your y-coordinate over time. These position signals can be fed into a mathematical process called the Fourier transform.
The result is extraordinary: each frequency component from the Fourier analysis corresponds to a circle rotating at a specific speed. The transform tells you exactly how big each circle should be, how fast it should spin, and where it should start. When you connect all these spinning circles together in a chain – with each circle’s center attached to the edge of the previous circle – the final endpoint traces out your original drawing perfectly.
Picture this as a mechanical system: a large circle spinning slowly, with a smaller circle attached to its edge spinning faster, with an even smaller circle attached to that one’s edge spinning faster still, and so on. As this chain of circles rotates, each at its own predetermined speed, the final point draws your shape with mathematical precision. And as you can imagine, the more circles you add the more accurate the image is to the original.
What you’re about to explore is not just a clever mathematical trick, but a window into one of the most fundamental and beautiful concepts in mathematics and physics. Every curve you draw will be reborn as a dance of circles, each one contributing its own frequency to recreate your artistic vision with perfect mathematical harmony.
The beauty of this system is its flexibility and precision. With just a few circles, you get a rough approximation of your drawing (or one of the images you pre-select). Add more circles, and the reproduction becomes increasingly detailed and accurate. With enough circles, you can recreate incredibly intricate images, from simple geometric shapes to complex portraits.
While closed shapes create the most visually appealing continuous animations, the mathematics works for any path you can imagine. Open curves, your signature, branching patterns, or any arbitrary squiggly line – all can be decomposed into rotating circles that will faithfully trace your original path.
Click this link or the image below to explore the
Fourier Transform Explorer
While watching circles trace out drawings is mesmerizing, the Fourier transform’s most important applications actually lie in the world of sound and signal processing. This mathematical tool that creates such beautiful visual art is the same one that powers much of our modern world.
Consider listening to your favorite song, what reaches your ears is a single, complex sound wave – a continuous stream of air pressure changes. Yet somehow, your brain effortlessly separates this into distinct instruments, voices, and melodies. How is this possible?
The secret lies in frequency analysis. Every sound, no matter how complex, is actually made up of simple sine waves of different frequencies mixed together. A guitar chord that sounds rich and full is really just several pure tones (frequencies) playing simultaneously. Your ear naturally performs this decomposition, which is why you can distinguish between a guitar, drums, and vocals all playing at once.
The Fourier transform is the mathematical tool that makes this decomposition explicit. It takes any complex signal and reveals exactly which frequencies are present and how strong each one is. Think of it like a prism breaking white light into a rainbow – the Fourier transform breaks complex signals into their frequency spectrum.
If you recorded someone playing three piano keys simultaneously, the Fourier transform would tell you exactly which three notes were being played, even though you started with just one mixed-up audio signal. It works by systematically comparing your signal to sine waves of every possible frequency, asking “how much of this particular frequency is present in my signal?”
This frequency analysis powers everything from noise-canceling headphones to audio compression, from medical imaging to telecommunications. The same mathematical principle that creates beautiful circle animations is quietly working behind the scenes in countless technologies we use every day.
This connection between Fourier transforms and circular motion reveals something profound: there’s a deep mathematical relationship between frequency analysis and rotation. The same tool that helps us understand sound, process digital signals, enhance medical images, and compress data can also create mesmerizing visual art from spinning circles.
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