A Rose is a rose

Posted by Tino Cogin of Samos January 18, 2025

A Rose is a rose

Specification

A rose is the set of points in polar coordinates specified by the polar equation^[2]

r=a\cos(k\theta )

or in Cartesian coordinates using the parametric equations

{\begin{aligned}x&=r\cos(\theta )=a\cos(k\theta )\cos(\theta )\\y&=r\sin(\theta )=a\cos(k\theta )\sin(\theta )\end{aligned}}

Roses can also be specified using the sine function.^[3] Since

\sin(k\theta )=\cos \left(k\theta -{\frac {\pi }{2}}\right)=\cos \left(k\left(\theta -{\frac {\pi }{2k}}\right)\right)

Thus, the rose specified by $r = a sin(kθ)$ is identical to that specified by $r = a cos(kθ)$ rotated counter-clockwise by $⁠ π / 2 k ⁠$ radians, which is one-quarter the period of either sinusoid.

Since they are specified using the cosine or sine function, roses are usually expressed as polar coordinate (rather than Cartesian coordinate) graphs of sinusoids that have angular frequency of $k$ and an amplitude of $a$ that determine the radial coordinate $r$ given the polar angle $θ$ (though when $k$ is a rational number, a rose curve can be expressed in Cartesian coordinates since those can be specified as algebraic curves^[4]).

General properties

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Artistic depiction of roses with different parameter settings

Roses are directly related to the properties of the sinusoids that specify them.

Petals

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Graphs of roses are composed of petals. A petal is the shape formed by the graph of a half-cycle of the sinusoid that specifies the rose. (A cycle is a portion of a sinusoid that is one period T = ⁠2π/k⁠ long and consists of a positive half-cycle, the continuous set of points where r ≥ 0 and is ⁠T/2⁠ = ⁠π/k⁠ long, and a negative half-cycle is the other half where r ≤ 0.)
- The shape of each petal is same because the graphs of half-cycles have the same shape. The shape is given by the positive half-cycle with crest at $(a,0)$ specified by $r = a cos(kθ)$ (that is bounded by the angle interval $- ⁠ T / 4 ⁠ \leq θ \leq ⁠ T / 4 ⁠$ ). The petal is symmetric about the polar axis. All other petals are rotations of this petal about the pole, including those for roses specified by the sine function with same values for $a$ and $k$ .^[5]
- Consistent with the rules for plotting points in polar coordinates, a point in a negative half-cycle cannot be plotted at its polar angle because its radial coordinate $r$ is negative. The point is plotted by adding $π$ radians to the polar angle with a radial coordinate $| r |$ . Thus, positive and negative half-cycles can be coincident in the graph of a rose. In addition, roses are inscribed in the circle $r = a$ .
- When the period $T$ of the sinusoid is less than or equal to $4 π$ , the petal's shape is a single closed loop. A single loop is formed because the angle interval for a polar plot is $2 π$ and the angular width of the half-cycle is less than or equal to $2 π$ . When $T > 4 π$ (or $| k | < ⁠ 1 / 2 ⁠$ ) the plot of a half-cycle can be seen as spiraling out from the pole in more than one circuit around the pole until plotting reaches the inscribed circle where it spirals back to the pole, intersecting itself and forming one or more loops along the way. Consequently, each petal forms two loops when $4 π < T \leq 8 π$ (or $⁠ 1 / 4 ⁠ \leq | k | < ⁠ 1 / 2 ⁠$ ), three loops when $8 π < T \leq 12 π$ (or $⁠ 1 / 6 ⁠ \leq | k | < ⁠ 1 / 4 ⁠$ ), etc. Roses with only one petal with multiple loops are observed for $k = ⁠ 1 / 3 ⁠, ⁠ 1 / 5 ⁠, ⁠ 1 / 7 ⁠$ , etc. (See the figure in the introduction section.)
- A rose's petals will not intersect each other when the angular frequency $k$ is a non-zero integer; otherwise, petals intersect one another.

Definition

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In terms of other physical constants, $α$ may be defined as:^[4] $\alpha ={\frac {e^{2}}{2\varepsilon _{0}hc}}={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c}},$ where

$e$ is the elementary charge (1.602176634×10⁻¹⁹ C^[5]);
$h$ is the Planck constant (6.62607015×10⁻³⁴ J⋅Hz⁻¹^[6]);
$ħ$ is the reduced Planck constant, $ħ = h /2 π$ (1.054571817...×10⁻³⁴ J⋅s^[7])
$c$ is the speed of light (299792458 m⋅s⁻¹^[8]);
$ε$ ₀ is the electric constant (8.8541878188(14)×10⁻¹² F⋅m⁻¹^[9]).

the quantity in this list that does not have an exact value in SI units is

the electric constant aka vacuum permittivity

Vacuum permittivity, commonly denoted $ε 0$ (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric constant, or the distributed capacitance of the vacuum. It is an ideal (baseline) physical constant. Its CODATA value is:

ε₀ = 8.8541878188(14)×10⁻¹² F⋅m⁻¹.^[1]

It is a measure of how dense of an electric field is "permitted" to form in response to electric charges and relates the units for electric charge to mechanical quantities such as length and force.^[2]

The value of ε₀ is defined by the formula^[3]

$\varepsilon _{0}={\frac {1}{\mu _{0}c^{2}}}$

where c is the defined value for the speed of light in classical vacuum in SI units,^[4]^: 127 and μ₀ is the parameter that international standards organizations refer to as the magnetic constant (also called vacuum permeability or the permeability of free space). Since μ₀ has an approximate value 4π × 10⁻⁷ H/m,^[5] and c has the defined value 299792458 m/s, it follows that ε₀ can be expressed numerically as^[6]

$\varepsilon _{0}\approx {\frac {1}{\left(4\pi \times {10}^{-7}\,\mathrm {N/A^{2}} \right){\left(299\,792\,458\,\mathrm {m/s} \right)}^{2}}}\approx {8.854\,187\,8176}\times {10}^{-12}\,\mathrm {F{\cdot }m^{-1}} .$