Math Rose Patterns

Loading engraving table...

Presets

Curve

Pen, surface and look

Show pen arm (flat table only) Animate the pen drawing Glass dome (dome surface only) Auto-orbit camera

Live readouts

Graphs

Polar radius r = sin(n theta) over one turn, marker = pen angle

Distinct Maurer vertices 360/gcd(d,360) for every step d (current d highlighted)

Vertex angle i*d mod 360 deg for the first 90 vertices

Pen arm: carriage radius r (orange) and arm angle / 360 (blue) while drawing

Numerical checks

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How to use

Choose the petal number n and the Maurer step d (degrees). The brass arm turns by d degrees for every vertex and slides its carriage to radius sin(n x angle); straight chords between vertices make the Maurer rose, the smooth orange line is the rose curve itself. Switch the Surface to lift the same pattern onto a glass dome, a cone, a saddle or a ripple. Drag to orbit, wheel to zoom.

The science

The rose curve in polar form is r = sin(n theta). For integer n it has n petals when n is odd and 2n petals when n is even. One petal spans theta from 0 to pi/n and encloses A = pi/(4n), so the total enclosed area is pi/4 (n odd) or pi/2 (n even). Its length is the integral of sqrt(r^2 + (dr/dtheta)^2) dtheta over one turn.

P_i = ( sin(n i d) cos(i d), sin(n i d) sin(i d) ), i = 0..360, angles in degrees

The Maurer rose (Peter Maurer, 1987) joins these points with straight chords. Because the angle i d only takes integer degree values, the sequence is periodic with period 360/gcd(d,360): it closes after that many distinct vertices (d = 71 is coprime to 360, so all 360 vertices are distinct; d = 60 gives 6 vertices and a hexagram-like star). The number of steps d in 1..180 with gcd(d,360) = 1 equals phi(360)/2 = 48.

The 3D table is a mechanical reading of the same maths: the arm angle is alpha = i d, the carriage radius is r = sin(n alpha) (negative r means the carriage slides through the pivot to the other side of the double-ended beam), and the pen position on the table is (r cos alpha, r sin alpha). Lifting onto a surface adds a height z(x,y): dome z = sqrt(1 - rho^2) (unit sphere), cone z = 0.9 (1 - rho), saddle z = 0.35 (x^2 - y^2), ripple z = 0.12 cos(8 rho), with rho = sqrt(x^2 + y^2).

Simplifications: the pen follows the Maurer chords at a constant parameter speed (no inertia, no friction); the physical beam is a rigid line, so the real mechanism ignores carriage mass; 1 table unit is shown as 100 mm only to give the readouts a scale. The marble, brass and glass are rendered from baked Blender textures; the curves themselves are generated live.

What the checks mean

Distinct vertices counts unique rendered vertex positions and compares with 360/gcd. Petal area is a shoelace sum over one lobe of the rendered rose, expected pi/(4n). Rose length compares the rendered polyline with a 400,000-point Simpson integral of sqrt(r^2 + r'^2). Pen kinematics reads the world position of the pen tip from the 3D scene graph after the arm moved and compares it with the chord point. Closure verifies vertex number 360/gcd equals vertex 0. d scan repeats the vertex count for every d from 1 to 180 and checks that exactly 48 steps give a full 360-vertex figure.

Keyboard and accessibility

Space pauses, R resets, C re-runs the checks (while the page has focus).
All controls are native inputs with labels.