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In geometry, the concept of a Maurer rose was introduced by Peter M. Maurer in his article titled A Rose is a Rose...[1]. A Maurer rose consists of some lines that connect some points on a rose curve.
Contents
Definition
Let r = sin(nθ) be a rose in the polar coordinate system, where n is a positive integer. The rose has n petals if n is odd, and 2n petals if n is even.
We then take 361 points on the rose:
(sin(nk), k) (k = 0, d, 2d, 3d, ..., 360d),where d is a positive integer and the angles are in degrees, not radians.
Explanation
A Maurer rose of the rose r = sin(nθ) consists of the 360 lines successively connecting the above 361 points. Thus a Maurer rose is a polygonal curve with vertices on a rose.
A Maurer rose can be described as a closed route in the polar plane. A walker starts a journey from the origin, (0, 0), and walks along a line to the point (sin(nd), d). Then, in the second leg of the journey, the walker walks along a line to the next point, (sin(n·2d), 2d), and so on. Finally, in the final leg of the journey, the walker walks along a line, from (sin(n·359d), 359d) to the ending point, (sin(n·360d), 360d). The whole route is the Maurer rose of the rose r = sin(nθ). A Maurer rose is a closed curve since the starting point, (0, 0) and the ending point, (sin(n·360d), 360d), coincide.
The following figure shows the evolution of a Maurer rose (n = 2, d = 29° ).
Images
The following are some Maurer roses drawn with some values for n and d:
Maple code
Below is the Maple code for plotting Maurer roses (different values of n and d can be chosen by the user):
with(plots) : n:=7:d:=29: Rose:=plot(sin(n*t),t=0..2*Pi,coords=polar,thickness=2): K:=360: for k from 0 to K do k1:=k*d*Pi/180:k2:=(k+1)*d*Pi/180: Point[k]:=pointplot([sin(n*k1),k1],coords=polar,color=blue): Maurer_rose[k]:=listplot(sin(n*k1),k1, sin(n*k2),k2,coords=polar,color=blue):od: Maurer_rose:=display(seq(Maurer_rose[k],k=0..K)): Point:=display(seq(Point[k],k=0..K)): display(Rose,Point,Maurer_rose);