Juan Carlos Ponce Campuzano
Mathematics Educator
The mathematical
beauty of epicycles
Juan Carlos Ponce Campuzano
School of Environment and Science
A mysterious curve
The mathematical beauty of epicycles
What are epicycles?
A circle moving on another circle.
What are epicycles?
A circle moving on another circle, on another circle.
What are epicycles?
A circle moving on another circle, on another circle, on another circle...
What are epicycles?
\left\{
\begin{array}{l}
x(t) \\
y(t)
\end{array}
\right. 0\leq t \leq 2\pi
\big(x(t) , y(t) \big)
How can we represent them mathematically?
A parametric function!
\big(x , y \big)
What are epicycles?
🔍
Code:
ABEV FXP8
What are epicycles?
🔍
Work only with
- Simple rotation
- Double rotation
Code:
ABEV FXP8
\left\{
\begin{array}{l}
R \cos( \omega t) \\
R\, \sin(\omega t)
\end{array}
\right. 0\leq t \leq 2\pi
Simple rotations
\big(R\cos(\omega t),R\sin(\omega t)\big)
\text{with \,}0\leq t\leq 2\pi
Both represent
the same thing!
Simple rotations
\(\sin x\)
Simple rotations
\(\sin x\) and \(\cos x\)
Double rotations
\left\{
\begin{array}{l}
R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t)\\
R_1 \,\sin(\omega_1 t) + R_2 \,\sin(\omega_2 t)
\end{array}
\right. 0\leq t \leq 2\pi
Triple rotations
\left\{
\begin{array}{l}
R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\
R_1 \sin(\omega_1 t) + R_2 \sin(\omega_2 t) + R_3 \sin(\omega_3 t)
\end{array}
\right.
Triple rotations
\left\{
\begin{array}{l}
R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\
R_1 \,\sin(\omega_1 t) + R_2 \,\sin(\omega_2 t) + R_3 \sin(\omega_3 t)
\end{array}
\right.
R_1 = 1, R_2 = \dfrac{1}{2}, R_3 = \dfrac{1}{3}
\omega_1 = 1,
\omega_2 =\;\,?,
\omega_3 = \; ?
👉
Complete last task!
6
-14
Triple rotations
\omega_1 = 1,
\omega_2 =\;\,?,
\omega_3 = \; ?
6
-14
Why there is rotational symmetry?
Triple rotations
\omega_1 = 1,
\omega_2 =\;\,?,
\omega_3 = \; ?
6
-14
Is this symmetry related to these numbers?
\left\{
\begin{array}{l}
R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\
R_1 \,\sin(\omega_1 t) + R_2 \,\sin(\omega_2 t) + R_3 \sin(\omega_3 t)
\end{array}
\right.
A parametric function with sin and cos functions
0\leq t\leq 2\pi
What are epicycles?
\left\{ \begin{array}{l} \displaystyle \sum_{k=1}^{3} R_k\cos(\omega_k t ) \\ \displaystyle \sum_{k=1}^{3}R_k\sin(\omega_k t ) \end{array} \right. \; 0\leq t\leq 2\pi
What are epicycles?
In general we have:
This new symbol \(\phi_k\) indicates how much the disk \(k\) is initially rotated at time \(t = 0\), and we called a phase offset.
\left\{ \begin{array}{l} \displaystyle \sum_{k=1}^{N} R_k\cos(\omega_k t + \phi_k) \\ \displaystyle \sum_{k=1}^{N}R_k\sin(\omega_k t + \phi_k) \end{array} \right. \; 0\leq t\leq 2\pi
What are epicycles?
If we do not specify a phase offset, we can only describe epicycles where the circles start aligned.
\left\{ \begin{array}{l} \displaystyle \sum_{k=1}^{N} R_k\cos(\omega_k t + \phi_k) \\ \displaystyle \sum_{k=1}^{N}R_k\sin(\omega_k t + \phi_k) \end{array} \right. \; 0\leq t\leq 2\pi
\(\phi_k\) is called a phase offset
mysterious curve
Remember our
mysterious curve
How can we determine the epicycles for our mysterious curve?
mysterious curve
c(t) = \left\{
\begin{array}{l}
R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\
R_1 \sin(\omega_1 t) + R_2 \sin(\omega_2 t) + R_3 \sin(\omega_3 t)
\end{array}
\right.
(0\lt R_k\in \mathbb R, \;\omega_k\in \mathbb Z)
- \(R_1, R_2, R_3\) are positive real numbers
- \(\omega_1, \omega_2, \omega_3\) are integers (\(\ldots,-2,-1,0,1,2,\ldots\))
How can we determine the epicycles for our mysterious curve?
It is going to get complex!
=\cos t + i \sin t
= e^{i t}
\big(R_1 \cos (\omega_1 t), R_1 \sin (\omega_1 t)\big)
= R_1e^{i \omega_1 t}
i = \sqrt{-1}
(\cos t, \sin t)
0\lt R_k\in \mathbb R, \;\omega_k\in \mathbb Z
It is going to get complex!
c(t) = \left\{
\begin{array}{l}
R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\
R_1 \sin(\omega_1 t) + R_2 \sin(\omega_2 t) + R_3 \sin(\omega_3 t)
\end{array}
\right.
c(t)= R_1e^{ \omega_1 i t}+R_2e^{ \omega_2 i t}+R_3e^{ \omega_3 i t}
0\lt R_k\in \mathbb R, \;\omega_k\in \mathbb Z
It is going to get complex!
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}
C_k \text{ is a complex number}, \;\omega_k\in \mathbb Z
It is going to get complex!
C_k = a + i b = (a,b)
The mysterious curve
c(t) = e^{it} + \dfrac{1}{2}e^{6 i t} + \dfrac{i}{3}e^{-14 i t}
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}
C_1=1,\,C_2=\dfrac{1}{2},\, C_3= \dfrac{i}{3}
\omega_1=1,\,\omega_2=6,\, \omega_3=-14
1
The mysterious curve
c(t) = \left\{
\begin{array}{l}
\displaystyle \cos( t) +\frac{1}{2}\cos(6 t) +\frac{1}{3}\sin(14 t) \\
\\
\displaystyle \sin( t) +\frac{1}{2} \sin(6 t)\, +\frac{1}{3} \cos(14 t)
\end{array}
\right.
Using properties of complex numbers:
\(i\cdot i = -1\)
c(t) = e^{it} + \dfrac{1}{2}e^{6 i t} + \dfrac{i}{3}e^{-14 i t}
More mystery curves
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}
In fact we can use this equation to create more complex epicycles!
Discrete Fourier Transform
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
In fact we can use this equation to create more complex epicycles!
c(t)= \displaystyle \sum_{k=1}^{N} C_ke^{ \omega_k i t }
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
\(\Bigg\{\Bigg.\)
Discrete Fourier Transform
👈 DFT
The inverse of the DFT
👈
(DFT)
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}
They are very similar to this:
\(\Bigg\{\Bigg.\)
Discrete Fourier Transform
(DFT)
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
They are very similar to this:
\(\Bigg\{\Bigg.\)
Discrete Fourier Transform
c(t) = \left\{
\begin{array}{l}
\text{Sum of a bunch sines and cosines} \\
\text{Sum of a bunch sines and cosines}
\end{array}
\right.
(DFT)
Discrete Fourier Transform
(DFT)
A nice methaphor 😃
What does the Discrete Fourier Transform do?
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
Given a smoothie, it finds the recipe.
A nice methaphor 😃
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
🍓 🍌🥭🍏
🫐🍍🥝🍫
How does it do that?
A nice methaphor 😃
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
How does it do that?
Run the smoothie through filters to extract each ingredient.
🍓 🍌🥭🍏
🫐🍍🥝🍫
A nice methaphor 😃
🍓 🍌🥭🍏
🫐🍍🥝🍫
Why do we want to do this?
Recipes are easy to analyse, compare, than the smoothie itself.
A nice methaphor 😃
🍓 🍌🥭🍏
🫐🍍🥝🍫
How do we get the smoothie back?
Blend the ingredients.
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
A nice methaphor 😃
🍓 🍌🥭🍏
🫐🍍🥝🍫
How do we get the smoothie back?
Blend the ingredients.
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Applications of the DFT
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
Modern digital media
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Applications of the DFT
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
Video
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Applications of the DFT
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
Images
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Applications of the DFT
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
Magnetic Resonance Imaging
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Applications of the DFT
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
Sound
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Try it yourself!
R1 = Slider(0, 5, 0.01, 1, 200)
R2 = Slider(0, 5, 0.01, 1, 200)
R3 = Slider(0, 5, 0.01, 1, 200)
w1 = Slider(-10, 10, 1, 1, 200)
w2 = Slider(-10, 10, 1, 1, 200)
w3 = Slider(-10, 10, 1, 1, 200)
fx(x) = R1 * cos(w1 * x) + R2 * cos(w2 * x) + R3 * cos(w3 * x)
fy(x) = R1 * sin(w1 * x) + R2 * sin(w2 * x) + R3 * sin(w3 * x)
c = Curve(fx(t), fy(t), t, 0, 2pi)
The beauty of mathematics shows itself to patient followers.
- Maryam Mirzakhani
The mathematical beauty of epicycles
By Juan Carlos Ponce Campuzano
The mathematical beauty of epicycles
The mathematical beauty of epicycles
