The Math Behind Maurer Roses

2026-01-02•By MathArt

At first glance, a Maurer Rose looks like chaos. It is a mesh of sharp, angular lines that somehow form a soft, organic shape. But underneath that chaos is a perfect, smooth curve known as a Rose Curve.

A Maurer Rose is essentially a game of "connect the dots" played on top of that smooth flower.

The Foundation: The Rose Curve

Before we can draw the chaotic version, we need the smooth version. A standard rose curve is defined by the polar equation:

r = \sin(n \theta)

Where:

$r$ is the radius (distance from the center).
$n$ is an integer that determines the number of petals.
$\theta$ is the angle (from 0 to 360 degrees).

If $n$ is odd, the rose has $n$ petals. If $n$ is even, it has $2n$ petals.

The Twist: The Maurer Step ( $d$ )

In 1987, Peter M. Maurer introduced the concept of the "Maurer Rose" in an article for The American Mathematical Monthly. His idea was simple but powerful: instead of drawing the curve smoothly (incremementing the angle by a tiny amount like 0.1°), what if we skipped ahead by a large amount, say $d$ degrees, at every step?

The algorithm works like this:

Start at angle 0.
Draw a line to the point on the rose curve at angle $d$ .
Draw a line to the point at angle $2d$ .
Continue until you close the loop (usually 360 steps).

Mathematically, we are connecting the points:

(\sin(n k), k) \quad \text{for } k = 0, d, 2d, 3d, \dots, 360d

Interactive Laboratory

Use the widget below to see this relationship.

Toggle "Show Original Rose Curve" to see the smooth $r = \sin(n\theta)$ foundation.
Adjust $n$ to change the number of petals.
Adjust $d$ to change how we connect the dots.

n=6, d=71

Show Original Rose Curve

Petals (n)6

Determines the shape of the underlying flower.

Angle Step (d)71°

How many degrees to skip before drawing the next line.

Notice the patterns?

When $d$ is small, the lines are short and trace the original curve closely.
When $d$ is large, the lines jump across the center, creating the "web" effect.
When $d = 29$ or $71$ , you get stark geometric beauty because these numbers don't divide evenly into 360, causing the lines to cover the entire space before repeating.

References & Further Reading