Modular Multiplication Circle: Math Makes Beautiful Curves

How Simple Math Makes a Complex Form

In our Modular Multiplication Circle, the simplest of math is made astonishingly beautiful

Just as the fractals in our fractal tools, we continue the GoRhyme theme of beauty in rhythm by looking at epicycloid curves. From a simple calculation, complexity emerges. Mesmerizing.

Epicycloid curves emerge from straight lines drawn inside a circle. This tool takes a simple circle, divides it into points, and then connects the points by a simple formula. The math is easy; the forms it creates are complex, rhythmic, and artistic.

How the Modular Multiplication Circle Works

The Setup

The tool starts with a circle. What we choose is the number of points the circle is divided equally into. We will call this N. The points start at 0 and continue to N minus 1.

The Rule

Now we can take any point on the circle and call it k. We then multiply k by the multiplier we choose, which usually gives us a large product, divide that number by N, and keep the remainder. We then connect k and the remainder point with a line. Every point gets exactly one line, and when all N are drawn, the picture is complete. Visually, these lines create what appears to be a smooth curve, though every line is perfectly straight; the curve is an illusion created by their overlap.

The Multiplier

Along with N, we also choose a multiplier, M. Interestingly, M dictates the final curve shape that forms. Starting at the lowest, an M=2 makes a cardioid (one lobe), M=3 makes a nephroid (two lobes), and higher multipliers give epicycloids with more lobes. The number of lobes is M minus 1.

Mandala Mode

In our Modular Multiplication Circle tool, we also have mandala mode. The formula for this differs from that of our other tool option. Instead of connecting a point to only one other point per calculation, it connects that point to every point from 1 to N/2, cycling through all possibilities. After that, it moves to the next point and repeats the calculations. The result is a complex web that symmetrically connects all possible points.

Now Play

The best way to see the differences between how curves grow is to experiment. Try M=2 for a classic cardioid, make M higher to see how having more lobes changes the shapes within the circle. Try animating to see how curves move in real time.

How to use this tool

Pick a mode in the Modular Multiplication Circle. Multiply draws the classic times-table circle: each point connects to its multiple, and the lines reveal a hidden curve. Mandala layers every chord length at once into a dense, glowing web.

The controls:

Points (N) sets how many points sit around the circle. Fewer points show crisp individual lines and clear structure; more points blend into smooth curves or a luminous disc. Mandala mode looks best in the lower range, where you can see the petals.

Multiplier (M) is the times-table factor in Multiply mode. M=2 gives a heart-shaped cardioid, M=3 gives two lobes, and higher values fold into denser blooms. Try setting M near a divisor of N for clean symmetry.

Line opacity controls how strongly the lines build up. Low opacity lets overlapping lines glow where they cross.

Speed sets how fast the animation moves.

Palette changes the colors. Spectrum cycles the full rainbow around the figure; the others are single-color themes.

Glow makes overlapping lines add toward light, like long-exposure photography. Turn it off for a flatter, line-drawing look.

Animate sweeps the multiplier in Multiply mode so you can watch the curve morph, or slowly rotates the weave in Mandala mode. Reset returns everything to the start.