Cardioid Generator

About Cardioids

A cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. The name comes from the Greek word "kardia" meaning heart, due to its heart-like shape. Cardioids appear in various natural phenomena, including the caustics seen in coffee cups and the pickup patterns of certain microphones.

Construction Lines Explained

• • Fixed Circle (dashed gray): The base circle around which another circle rolls
• • Rolling Circle (solid gray): A circle of equal radius that rolls around the fixed circle without slipping
• • Trace Point (bright colored dot): A point on the circumference of the rolling circle that traces the cardioid
• • Path History (faded trail): Shows the recent path traced by the point as it moves

Times Table Pattern

Enable "Times Table Pattern" mode to explore a fascinating mathematical phenomenon: when you place points evenly around a circle and draw lines connecting each point n to point (multiplier × n) mod (total points), beautiful patterns emerge.

When the multiplier equals 2, you get a perfect cardioid! Try different multipliers (like 3, 4, 5, or even decimals) to discover other curves like nephroids, ranunculoids, and more. This demonstrates how cardioids can emerge from simple modular arithmetic patterns.

Fibonacci Mode

Enable "Fibonacci Mode" to connect point n to point Fibonacci(n) mod (total points). The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13...) creates unique organic patterns that reflect the mathematical beauty found in nature, from spirals in seashells to the arrangement of sunflower seeds. The resulting patterns exhibit both the regularity of modular arithmetic and the exponential growth characteristics of the golden ratio.