Rose (mathematics)
Encyclopedia
In mathematics
, a rose or rhodonea curve is a sinusoid
plotted in polar coordinates. Up to similar
ity, these
curves can all be expressed by a polar equation of the form
If k is an integer, the curve will be rose shaped with
When k is even, the entire graph of the rose will be traced out exactly once when the value of θ changes from 0 to 2π. When k is odd, this will happen on the interval between 0 and π. (More generally, this will happen on any interval of length 2π for k even, and π for k odd.)
If k ends in 1/2 (ex: 0.5, 2.5), the curve will be rose shaped with 4k petals.
If k ends in 1/6 or 5/6 and is greater than 1 (ex: 1.16666667, 2.8333333), the curve will be rose shaped with 12k petals.
If k ends in 1/3 and is greater than 1 (ex: 1.333334, 2.333334), the curve will be rose shaped and will have:
If k ends in 2/3 and is greater than 1 (ex: 1.666667, 2.666667), the curve will be rose shaped and will have:
If k is rational
, then the curve is closed and has finite length. If k is irrational
, then it is not closed and has infinite length. Furthermore, the graph of the rose in this case forms a dense set
(i.e., it comes arbitrarily close to every point in the unit disk).
Since
for all
, the curves given by the polar equations
and 
are identical except for a rotation of π/2k radians.
Rhodonea curves were named by the Italian mathematician Guido Grandi
between the year 1723 and 1728.
where k is a positive integer, has area
if k is even, and
if k is odd.
The same applies to roses with polar equations of the form
since the graphs of these are just rigid rotations of the roses defined using the cosine.
When d is a prime number then n/d is a least common form and the petals will stretch around to overlap other petals. The number of petals each one overlaps is equal to the how far through the sequence of primes this prime is +1, i.e. 2 is 2, 3 is 3, 5 is 4, 7 is 5, etc.
In the form k = 1/d when d is even then it will appear as a series of d/2 loops that meet at 2 small loops at the center touching (0, 0) from the vertical and is symmetrical about the x-axis.
If d is odd then it will have d div 2 loops that meet at a small loop at the center from ether the left (when in the form d = 4n − 1) or the right (d = 4n + 1).
If d is not prime and n is not 1, then it will appear as a series of interlocking loops.
If k is an irrational number (e.g.
, 
, etc.) then the curve will have infinitely many petals, and it will be dense
in the unit disc.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a rose or rhodonea curve is a sinusoid
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...
plotted in polar coordinates. Up to similar
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
ity, these
curves can all be expressed by a polar equation of the form
If k is an integer, the curve will be rose shaped with
- 2k petals if k is even, and
- k petals if k is odd.
When k is even, the entire graph of the rose will be traced out exactly once when the value of θ changes from 0 to 2π. When k is odd, this will happen on the interval between 0 and π. (More generally, this will happen on any interval of length 2π for k even, and π for k odd.)
If k ends in 1/2 (ex: 0.5, 2.5), the curve will be rose shaped with 4k petals.
If k ends in 1/6 or 5/6 and is greater than 1 (ex: 1.16666667, 2.8333333), the curve will be rose shaped with 12k petals.
If k ends in 1/3 and is greater than 1 (ex: 1.333334, 2.333334), the curve will be rose shaped and will have:
- 3k petals if k is even, and
- 6k petals if k is odd.
If k ends in 2/3 and is greater than 1 (ex: 1.666667, 2.666667), the curve will be rose shaped and will have:
- 6k petals if k is even, and
- 3k petals if k is odd.
If k is rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
, then the curve is closed and has finite length. If k is irrational
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
, then it is not closed and has infinite length. Furthermore, the graph of the rose in this case forms a dense set
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
(i.e., it comes arbitrarily close to every point in the unit disk).
Since
for all
, the curves given by the polar equations and are identical except for a rotation of π/2k radians.
Rhodonea curves were named by the Italian mathematician Guido Grandi
Guido Grandi
thumb|Guido GrandiDom Guido Grandi, O.S.B. Cam., was an Italian monk, priest, philosopher, mathematician, and engineer.-Life:...
between the year 1723 and 1728.
Area
A rose whose polar equation is of the formwhere k is a positive integer, has area
if k is even, and
if k is odd.
The same applies to roses with polar equations of the form
since the graphs of these are just rigid rotations of the roses defined using the cosine.
How the parameter k affects shapes
In the form k = n, for integer n, the shape will appear similar to a flower. If n is odd half of these will overlap, forming a flower with n petals. However if it is even the petals will not overlap, forming a flower with 2n petals.When d is a prime number then n/d is a least common form and the petals will stretch around to overlap other petals. The number of petals each one overlaps is equal to the how far through the sequence of primes this prime is +1, i.e. 2 is 2, 3 is 3, 5 is 4, 7 is 5, etc.
In the form k = 1/d when d is even then it will appear as a series of d/2 loops that meet at 2 small loops at the center touching (0, 0) from the vertical and is symmetrical about the x-axis.
If d is odd then it will have d div 2 loops that meet at a small loop at the center from ether the left (when in the form d = 4n − 1) or the right (d = 4n + 1).
If d is not prime and n is not 1, then it will appear as a series of interlocking loops.
If k is an irrational number (e.g.
, , etc.) then the curve will have infinitely many petals, and it will be denseDense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in the unit disc.
See also
- Lissajous curveLissajous curveIn mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equationswhich describe complex harmonic motion...
- quadrifoliumQuadrifoliumThe quadrifoliumis a type of rose curve with n=2. It has polar equation:r = \cos, \,with corresponding algebraic equation^3 = ^2. \,Rotated by 45°, this becomesr = \sin \,...
- a rose curve with k = 2. - Maurer roseMaurer roseIn geometry, the concept of a Maurer rose was introduced by Peter M. Maurer in his article titled A Rose is a Rose.... A Maurer rose consists of some lines that connect some points on a rose curve.-Definition:...
- Foucault pendulumFoucault pendulumThe Foucault pendulum , or Foucault's pendulum, named after the French physicist Léon Foucault, is a simple device conceived as an experiment to demonstrate the rotation of the Earth. While it had long been known that the Earth rotated, the introduction of the Foucault pendulum in 1851 was the...