In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the concept of a Maurer rose was introduced by Peter M. Maurer in his article titled A Rose is a Rose...http://www.toodoc.com/Peter-M.-Maurer-ebook.html. A Maurer rose consists of some lines that connect some points on a rose curve.

Definition

Let r = sin(nθ) be a rose in the polar coordinate system

Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....

, 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

Degree (angle)

A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

, not radians

Radian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

.

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

Polygonal chain

A polygonal chain, polygonal curve, polygonal path, or piecewise linear curve, is a connected series of line segments. More formally, a polygonal chain P is a curve specified by a sequence of points \scriptstyle called its vertices so that the curve consists of the line segments connecting the...

 with vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

 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

Maple (software)

Maple is a general-purpose commercial computer algebra system. It was first developed in 1980 by the Symbolic Computation Group at the University of Waterloo in Waterloo, Ontario, Canada....

 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);