Maurer rose
Walk around the rose r = sin nθ in steps of d degrees and join the dots: 361 straight lines weave a rose.
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Formulas and key values
- Polar equation
- Construction rule
Join the points of the rose at these angles in order with straight segments.
- Closes after · steps at mostIn the graph = 360
n is the rose’s petal number and d the step in degrees between joined points.
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Start the video01 · Definition
What is the Maurer rose?
Peter Maurer described this construction in the American Mathematical Monthly in 1987. Take the rose r = sin nθ, mark the 361 points at angles 0°, d°, 2d°, … 360d°, and connect them in order with straight segments. The same rose underlies every picture; the step d decides the weave.
Because the angles are taken in whole degrees, the polygon closes after at most 360 steps. Steps that share a large factor with 360 revisit the same few points and give sparse stars; steps like 29° or 71° visit many points and give dense, lace-like patterns.
02 · Properties
Key properties
- Vertices: (sin(nkd°), kd°) in polar form, for k = 0, 1, …, 360.
- The rose behind it, r = sin nθ, has n petals for odd n and 2n for even n.
- The polygon closes after 360 / gcd(d, 360) steps at most.
- Different d give different weaves on the same rose.
- Described by Peter M. Maurer in 1987.
03 · Questions
Maurer rose: common questions
- What is the equation of the Maurer rose?
- Polar equation: r = sin(nθ). n is the rose’s petal number and d the step in degrees between joined points.
- How is the Maurer rose constructed?
- Construction rule: θ_k = k d^circ, k = 0, 1, dots, 360, Join the points of the rose at these angles in order with straight segments.. n is the rose’s petal number and d the step in degrees between joined points.
04 · Keep exploring