What quantity did you change?
Labs / Mathematics
Fourier Epicycles
Each spinning circle is one sine wave — a single harmonic — and stacking them tip-to-tip adds those waves together. Add more circles to sharpen the traced shape, or drag the canvas to hand-crank the whole machine and watch the pen draw the wave.
Labs / Mathematics / Fourier Epicycles
Fourier Epicycles
Each spinning circle is one sine wave — a single harmonic — and stacking them tip-to-tip adds those waves together. Add more circles to sharpen the traced shape, or drag the canvas to hand-crank the whole machine and watch the pen draw the wave.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
terms 6A₁ 1.273RMS err 0.000
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- With one circle you get a pure sine. How many harmonics does a square wave need before its edges look truly flat?
- Why do the wiggly overshoots near each jump never fully disappear, no matter how many terms you add?
- Switch between square, sawtooth, and triangle — why do the circles shrink so much faster for the triangle?
- Watch the RMS error as you drag the harmonics slider. Does doubling the terms halve the error?
- Which wave's reconstruction gets closest to ideal with the fewest circles, and what does that say about how "smooth" it is?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.