Maurer Roses
The faint gold flower is a rose curve. The web of lines is that same rose visited in jumps: start at the centre, turn d degrees, mark where the rose is at that angle, turn again, 361 points in all, and join each mark to the next with a straight line. The petal number n sets the flower and the step d decides which of its points get joined, so moving d by a single degree can turn a star into a lattice.
| Points on the walk | 361 |
|---|---|
| Petals of r = sin(nθ) | n if odd, 2n if even |
| Walk closes after | 360 / gcd(d, 360) steps |
| Default rose | n 9, d 121 |
What you are looking at
The thin gold curve is the rose r = sin(n theta), a flower with n petals when n is odd and 2n petals when n is even. The coloured lines are the walk: the points at 0, d, 2d and so on up to 360 times d degrees, 361 points in all, each one joined to the next. Every line starts and ends on the rose. Short lines that hug a petal glow gold and long lines that leap across the flower are violet, so the colour tells you how far each jump goes.
The default is n 9, d 121: a nine-petal rose walked in steps of 121 degrees. Three steps make 363 degrees, one turn and 3 degrees more, so the walk draws near-triangles that creep round 3 degrees at a time until they have built a nine-pointed star.
Pick a preset, or choose Your own and move the sliders. The link in your address bar always reproduces the rose on your screen.
The math
A rose curve is r = sin(n theta) in polar coordinates, where theta is the angle and r the distance from the centre. Guido Grandi studied these curves in the early 1700s and called them rhodonea. The petal rule comes from what happens after half a turn. When n is odd, sin(n(theta + 180)) = -sin(n theta), and a negative distance points the opposite way, so the second half turn traces the same n petals again. When n is even the second half turn draws n new petals, 2n in all.
The walk was described by Peter Maurer in the American Mathematical Monthly in 1987. It visits the angles 0, d, 2d and so on, read round the circle, so it comes back to its start after 360 / gcd(d, 360) steps, where gcd is the greatest common divisor. When d shares no factor with 360, as 71, 97 and 121 do not, the walk uses all 360 whole degrees before it closes. When it does share one, the walk closes early and then repeats itself: d = 39 closes after 120 steps, and d = 120 after only 3, which draws a single triangle.
Why one degree matters so much: when d is close to a whole fraction of a turn, the walk nearly closes into a polygon and then drifts. 121 is one degree more than a third of a turn, so it draws triangles. 71 is one degree less than a fifth, so it draws pentagons that drift backwards.
How it is computed
The 361 points come straight from the formula. Each angle is first reduced to a whole number of degrees between 0 and 359, so no rounding error builds up along the walk, then turned into a position on the rose. The chords are drawn in walk order as the picture builds, and each chord is coloured by its length. The caption reports how many steps the walk takes before it first returns to its start. The petal rule, the 361 points and the closing step are all checked by the tests, and everything is computed in your browser.
Questions
What is a Maurer rose?
A rose curve r = sin(n theta) sampled at every d degrees, 361 points from 0 to 360 times d degrees, with each point joined to the next by a straight line. Peter Maurer described it in 1987. Two whole numbers, n and d, are all it takes.
Why does the picture change so much when d moves by one degree?
The chords join points d degrees apart. When d is near a whole fraction of a turn, such as 121 near a third or 71 near a fifth, the walk almost closes into a triangle or a pentagon and then drifts by the difference. A different d means a different near polygon and a different drift, so a different web.
Why do some settings look sparse?
When d shares a factor with 360 the walk comes back to its start early and then retraces itself. With d = 120 it closes after 3 steps and draws one triangle; with d = 39 it closes after 120 steps.