A little motion,
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Listen to this idea
Imagine a pen on the end of an arm. Now attach that arm to the end of another turning arm. Each movement is
simple. Their combination draws the curve you see.
Show me the arms
When the rhythms meet again
If one arm makes a whole number of turns while the other also makes a whole number, both can return to their
starting positions together. The pen closes its loop.
A tiny change in speed can make that reunion take much longer. You get new loops in between, weaving a much
denser drawing. Try “Almost a circle” and then “Trace it all.”
There’s a musical connection
A rotating point’s horizontal and vertical positions each move like a smooth wave. Add the positions of two
rotating points, and you’re adding waves. Combining waves is also central to sound. This drawing is a visual
relative of that idea, not a simulation of a musical instrument.
Curious about the mathematics?
The pen’s position is the sum of two circular motions:
x = a cos(t) + b cos(kt + φ)
Here, a and b are the arm lengths, k is the inner arm’s rotation speed relative to
the outer arm’s speed, and φ is its starting angle. Both angles are measured against the canvas, not
against the other arm.
A rational speed ratio gives a closed path. An irrational ratio would never close exactly. All the finite
decimal settings in this playground are rational, even when their paths take a long time to close.
Welcome to Wonderlattice.
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A small experiment in enjoying mathematical ideas through play. Change something. Follow what catches your eye.
Make something you like.
Each little world connects a piece of mathematics to something you can play with: shapes, sounds, crowds, games,
patterns you can make. There is no lesson to finish and nothing to get right.
Make it yours
Everything here runs in your browser, even offline. There are no accounts, no tracking, no cookies, and no AI
chat. Sound is off until you turn it on.
Your privacy
Wonderlattice collects nothing. The site is hosted by Cloudflare, which keeps standard access logs (IP address,
time, page) under its own privacy policy. My trail and your language choice are kept in this browser only, and
only when you use them; clearing site data removes them. Narration uses your browser’s voices: some online
voices send the text being read (these explanations, never your notes) to the browser maker’s speech service.
Who makes this
Wonderlattice is a free, noncommercial hobby project by Eyal Weiss. Say hello at
eyal8488@gmail.com or on
GitHub .
The models here are simplified for play and explanation; they are not predictions, measurements, or advice.
Provided as is, without warranty. Links lead to independent sites; no affiliation or endorsement is implied.
Rubik’s Cube® is a registered trademark of Spin Master Toys UK Limited; Wonderlattice is not affiliated with it.
The code and words are free to reuse under the MIT licence. The portraits of historical mathematicians are in
the public domain, except the photo of John Conway by Thane Plambeck (cropped,
CC BY 2.0 ).
The drawn visitors are playful sketches, not likenesses.