Type any function and see its shape straight away. Plot up to three at once in three colours to compare them, drag the trace point to read a value off a curve, and see where each one crosses zero.
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Fourier Epicycles
About this simulation
This simulation draws a curve with a chain of rotating circles, each turning at a whole-number multiple of the base speed. Students pick a square, sawtooth or triangle wave or a heart, star or infinity shape, then set the number of circles from 1 to 30 and toggle the circles and trail. Readouts show the circles used, the RMS error and the pen position. Teachers can use it to introduce Fourier series, sums of sine waves and approximation, and to show the Gibbs overshoot at the corners of a square wave.
What students can explore
- Pick Square wave and raise Circles from 1 to 30 while watching the RMS error readout fall.
- Switch to Heart and compare 5 circles with 30 circles.
- Turn off Show circles to see only the pen path.
- Try Sawtooth wave and Triangle wave and compare how fast each one converges.
- Turn Trail on and off to follow or clear the drawn path.
Learning objectives
- 1Explain that a repeating curve can be built from rotating circles or sine waves.
- 2Observe how adding terms improves an approximation.
- 3Interpret an error measure as the circles increase.
- 4Compare how different waves need different numbers of terms.
Questions teachers ask
Is the Fourier epicycles simulation free to use?
Yes. It is completely free and runs in any modern web browser, including on interactive whiteboards, tablets, Chromebooks and phones. There is nothing to install and no sign-up or account is needed.
What is a Fourier series?
It is a way to write a repeating function as a sum of sine and cosine waves of different frequencies. Each term in this simulation is one rotating circle.
Why do square wave corners look uneven?
Near a jump, a Fourier sum overshoots the true value by about 9 percent, called the Gibbs phenomenon. Adding more circles narrows the overshoot but does not remove it.
How do I use the Fourier epicycles simulation?
Choose a shape, start with a few circles and slowly add more. Watch the RMS error readout and the pen position to see how close the drawing is to the target.
Embed this simulation
Drop it into your LMS, blog, or class site — free.
<iframe src="https://kiwibee.io/en/simulations/fourier-epicycles-3d-simulation/embed" title="Fourier Epicycles | KiwiBee" loading="lazy" allowfullscreen style="width:100%;height:min(70vh,620px);min-height:420px;border:0;border-radius:16px"></iframe>Continue exploring
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