# The mathematical beauty of epicycles

Juan Carlos Ponce Campuzano

# The mathematical beauty of epicycles

A mysterious curve

## 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?

🤔🔍

RAWK QT8A

Code:

## What are epicycles?

🤔🔍

Work only with

• Simple rotation
• Double rotation

RAWK QT8A

Code:

\left\{ \begin{array}{l} R \cos( \omega t) \\ R\, \sin(\omega t) \end{array} \right. 0\leq t \leq 2\pi

## 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.

## 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 =6,
\omega_3 = -14

👉

\left\{ \begin{array}{l} \displaystyle R_1\cos(\omega_1 t + \phi_1) + \cdots +R_N\cos(\omega_N t + \phi_N) \\ \displaystyle R_1\sin(\omega_1 t + \phi_1) + \cdots +R_N\sin(\omega_N t + \phi_N) \end{array} \right.

In general we have:

0\leq t\leq 2\pi

## What are epicycles?

\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

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.

If we do not specify a phase offset, we can only describe epicycles where the circles start aligned.

## What are epicycles?

\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

This new symbol $$\phi_k$$ indicates how much the disk $$k$$ is initially rotated at time $$t = 0$$, and we called a phase offset.

If we do not specify a phase offset, we can only describe epicycles where the circles start aligned.

## Question

It seems that the geometric shapes described

by epicycles are smooth closed curves.

But, do all epicycles have to be curvy or closed?

🤔

## Line segment

\left\{ \begin{array}{l} R \cos(\omega t) + R \cos(-\omega t)\\ R \sin(\omega t ) + R \sin(-\omega t ) \end{array} \right.
\begin{array}{l} = 2R \cos(\omega t) \\ = 0 \end{array}

Q $$=(2R\cos(\omega t),0)$$

\left\{ \begin{array}{l} R \cos(\omega t) + R \cos(\omega t)\\ R \sin(\omega t ) - R \sin(\omega t ) \end{array} \right.

🤔

## How can we determine the epicycles for our mysterious curve?

let's explore an interesting mathematical property.

## Rotational symmetry exploration

\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 =6, \omega_3 = -14

{1, 6, -14}

{1, 6, -14}

## Rotational symmetry exploration

What is the relationship between the frequencies $$\{1, 6, -14\}$$ and the rotational symmetry of order $$5$$?

🤔

## Rotational symmetry exploration

What is the relationship between the frequencies $$\{1, 6, -14\}$$ and the rotational symmetry of order $$5$$?

🤔

ZT7S CNXU

Code:

## Rotational symmetry exploration

$$1-6=-5$$

$$6-(-14)=20$$

$$-14-1=-15$$

The greatest common divisor (GCD) of $$-5, 20,$$ and $$-15$$ is:

5

## Rotational symmetry exploration

What is the order of symmetry for the values

-2, 5, 19?

## Rotational symmetry exploration

What is the order of symmetry for the values

2, 8, -10?

Note that 2, 8, and -10 have a common factor.

## How can we determine the epicycles for our 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 and $$\omega_1, \omega_2, \omega_3$$ are also real numbers.

## It is going to get complex! 🙃

(\cos t, \sin t)
=\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}
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

## It is going to get complex! 🙃

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

## 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:

$$i\cdot i = -1$$

## 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}

## More mystery curves

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!

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}

🤯

c(t)= \displaystyle \sum_{k=1}^{N} C_ke^{ \omega_k i t }

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!

🤔

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)

👈 DFT

The inverse 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 }
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 (DFT)

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.

😃

Discrete Fourier Transform (DFT)

A nice methaphor 😃

• What does the Discrete Fourier Transform do?

Given a smoothie, it finds the recipe.

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

A nice methaphor 🧐

• What does the Discrete Fourier Transform do?

Given a smoothie, it finds the recipe.

• How?

Run the smoothie through filters to extract each ingredient.

• Why?

Recipes are easier to analyze, compare, and modify than the smoothie itself.

• How do we get the smoothie back?

Blend the ingredients.

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 }

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 }
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Modern digital media

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 }
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Video

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 }
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Images

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 }
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Sound

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 }
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Magnetic Resonance Imaging

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