Mystery curves

Mystery curves

Mystery curves

Goal:

Explore mystery curves

Mystery curves

  1. Intuitive definition of mystery curves.
  2. How to construct these curves
  3. Explore some of its mathematical properties.
  4. Propose a challenge for you.

Route

1. Definition

{x(t)y(t)a≤t≤b\left\{ \begin{array}{l} x(t) \\ y(t) \end{array} \right. a\leq t \leq b
\left\{ \begin{array}{l} x(t) \\ y(t) \end{array} \right. a\leq t \leq b

Epicycles

(x(t),y(t))\big(x(t) , y(t) \big)
\big(x(t) , y(t) \big) 
Simple rotations
{Rcos⁡(ωt)R sen(ωt)0≤t≤2π\left\{ \begin{array}{l} R \cos( \omega t) \\ R\, \text{sen}(\omega t) \end{array} \right. 0\leq t \leq 2\pi
\left\{ \begin{array}{l} R \cos( \omega t) \\ R\, \text{sen}(\omega t) \end{array} \right. 0\leq t \leq 2\pi

1. Definition

{R1cos⁡(ω1t)+R2cos⁡(ω2t)R1 sen(ω1t)+R2 sen(ω2t)\left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t)\\ R_1 \,\text{sen}(\omega_1 t) + R_2 \,\text{sen}(\omega_2 t) \end{array} \right.
\left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t)\\ R_1 \,\text{sen}(\omega_1 t) + R_2 \,\text{sen}(\omega_2 t) \end{array} \right.

1. Definition

Double rotations
{R1cos⁡(ω1t)+R2cos⁡(ω2t)+R3cos⁡(ω3t)R1 sen(ω1t)+R2 sen(ω2t)+R3 sen(ω3t)\left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\ R_1 \,\text{sen}(\omega_1 t) + R_2 \,\text{sen}(\omega_2 t) + R_3 \,\text{sen}(\omega_3 t) \end{array} \right.
\left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\ R_1 \,\text{sen}(\omega_1 t) + R_2 \,\text{sen}(\omega_2 t) + R_3 \,\text{sen}(\omega_3 t) \end{array} \right.

1. Definition

🤔

{R1cos⁡(ω1t)+R2cos⁡(ω2t)+R3cos⁡(ω3t)R1 sin⁡(ω1t)+R2 sin⁡(ω2t)+R3 sin⁡(ω3t)\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.
\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.

2. Construction in GeoGebra

R1=1,R2=12,R3=13R_1 = 1, R_2 = \frac{1}{2}, R_3 = \frac{1}{3}
R_1 = 1, R_2 = \frac{1}{2}, R_3 = \frac{1}{3}
ω1=1,ω2=6,ω3=−14\omega_1 = 1, \omega_2 =6, \omega_3 = -14
\omega_1 = 1, \omega_2 =6, \omega_3 = -14

Mystery curve = ?

Steps to contruct Epicycles

A = (0, 0)
c = Circle(A, 1)
B = Point(c)
t = Slider(0, 2 pi, 0.001, 0.4)
B' = Rotate(B, t)
v1 = Vector(A, B')
d = Circle(B', 1/2)
C = Point(d)
C' = Rotate(C, 6 * t, B')
e = Circle(C', 1/3)
v2 = Vector(B', C')
D = Point(e)
D' = Rotate(D, -14 * t, C')
v3 = Vector(C', D')
locus = Locus(D', t)

a.k.a. GeoGebra Script

2. Construction in GeoGebra

Steps to construct mystery curve

R1 = 1
R2 = 1/2
R3 = 1/3
w1 = 1
w2 = 6
w3 = -14
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)
a = Curva(fx(t), fy(t), t, 0, 2 pi)

a.k.a. GeoGebra Script

2. Construction in GeoGebra

3. Rotational symmetry exploration

{1, 6, -14}

{1, 6, -14}

3. Rotational symmetry exploration

What is the relationship between the frequencies {1,6,−14}\{1, 6, -14\} and the rotational symmetry of order 55?

🤔

3. Rotational symmetry exploration

1−6=−51-6=-5

6−(−14)=206-(-14)=20

1−(−14)=151-(-14)=15

The greatest common divisor of −5,20,-5, 20, y 1515 is:

5

1,61, 6 y −14-14 congruent 1 modulus 5

1,6,−14=1(mod 5)1,6,-14 =1 \left(\text{mod} \, 5\right)

3. Rotational symmetry exploration

1. Look for rotational symmetries of order 3 and 4.

2. What is the order of symmetry of the numbers 2, 8, -10?

Note: The common factor for the numbers 2, 8 y -10 is 2.

3. Rotational symmetry exploration

Activity

  1. Define the values of R1R_1, R2R_2 and R3R_3 using the command
    • ​​​RandomUniform( <Min>, <Max> )
      • ​Da por resultado un número real aleatorio a partir de una distribución uniforme en el intervalo [Mínimo, Máximo].
  2. Define the values of ω1\omega_1, ω2\omega_2 and ω3\omega_3 using the command
    • ​​RandomBetween( <Min> , <Max> )
      • ​Genera un número entero aleatorio entre el mínimo y el máximo (inclusive).

Example:

R1 = RandomUniform(0.5, 4.5)
w1 =  RandomBetween(0, 9)

3. Create a boton with the command UpdateConstruction()

3. Rotational symmetry exploration

Example:

R1 = RanformUniform(0.5, 4.5)
w1 =  RandomBetween(0, 9)

3. Create a boton with the command UpdateConstruction()

3. Rotational symmetry exploration

Activity

Share your work in GeoGebra Classroom

Join in with the code:

YHEG X7HK

or use the link:

https://www.geogebra.org/classroom/yhegx7hk

In Twitter/Mathstodon

@jcponcemath

Actividad

By email

jcponcemath@gmail.com

3. Rotational symmetry exploration

c(t)={R1cos⁡(ω1t)+R2cos⁡(ω2t)+R3cos⁡(ω3t)R1 sin⁡(ω1t)+R2 sin⁡(ω2t)+R3 sin⁡(ω3t)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) = \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.

4. More mystery curves

0<Ri∈R,  ωi∈Z0\lt R_i\in \mathbb R, \;\omega_i\in \mathbb Z
0\lt R_i\in \mathbb R, \;\omega_i\in \mathbb Z
(cos⁡t,sin⁡ t)(\cos t, \sin\, t)
(\cos t, \sin\, t)
=cos⁡t+i sin⁡ t=\cos t + i \,\sin \,t
=\cos t + i \,\sin \,t
=eit= e^{i t}
= e^{i t}
(R1cos⁡(ω1t),R1sin⁡(ω1t))\big(R_1 \cos (\omega_1 t), R_1 \sin (\omega_1 t)\big)
\big(R_1 \cos (\omega_1 t), R_1 \sin (\omega_1 t)\big)
=R1eiω1t= R_1e^{i \omega_1 t}
= R_1e^{i \omega_1 t}
i=−1i = \sqrt{-1}
i = \sqrt{-1}

4. More mystery curves

c(t)=R1eω1it+R2eω2it+R3eω3itc(t)= R_1e^{ \omega_1 i t}+R_2e^{ \omega_2 i t}+R_3e^{ \omega_3 i t}
c(t)= R_1e^{ \omega_1 i t}+R_2e^{ \omega_2 i t}+R_3e^{ \omega_3 i t}
c(t)={R1cos⁡(ω1t)+R2cos⁡(ω2t)+R3cos⁡(ω3t)R1 sin⁡(ω1t)+R2 sin⁡(ω2t)+R3 sin⁡(ω3t)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) = \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<Ri∈R,  ωi∈Z0\lt R_i\in \mathbb R, \;\omega_i\in \mathbb Z
0\lt R_i\in \mathbb R, \;\omega_i\in \mathbb Z
c(t)=C1eω1it+C2eω2it+C3eω3itc(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}+C_3e^{ \omega_3 i t}
Ci∈C,  ωi∈ZC_i\in \mathbb C, \;\omega_i\in \mathbb Z
C_i\in \mathbb C, \;\omega_i\in \mathbb Z

4. More mystery curves

c(t)=eit+12e6it+i3e−14itc(t) = e^{it} + \dfrac{1}{2}e^{6 i t} + \dfrac{i}{3}e^{-14 i t}
c(t) = e^{it} + \dfrac{1}{2}e^{6 i t} + \dfrac{i}{3}e^{-14 i t}

4. More mystery curves

c(t)={cos⁡(t)+1/2cos⁡(6t)+1/3 sen(14t) sen(t)+1/2 sen(6t)+1/3cos⁡(14t)c(t) = \left\{ \begin{array}{l} \cos( t) +1/2\cos(6 t) +1/3\,\text{sen}(14 t) \\ \,\text{sen}( t) +1/2 \,\text{sen}(6 t) +1/3 \cos(14 t) \end{array} \right.
c(t) = \left\{ \begin{array}{l} \cos( t) +1/2\cos(6 t) +1/3\,\text{sen}(14 t) \\ \,\text{sen}( t) +1/2 \,\text{sen}(6 t) +1/3 \cos(14 t) \end{array} \right.

4. More mystery curves

Challenges in GeoGebra

c(t)=C1eω1it+C2eω2it+C3eω3itc(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}+C_3e^{ \omega_3 i t}

1. Build mystery curves using complex numbers

2. Generalize the construction for nn terms

c(t)=C1eω1it+C2eω2it+⋯+Cneωnitc(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}

4. More mystery curves

Twitter/Mathstodon

@jcponcemath

Email

jcponcemath@gmail.com

Share your work in GeoGebra Classroom: YHEG X7HK

https://www.geogebra.org/classroom/yhegx7hk

References

🔗 Creating symmetry:

The artful mathematics

of wallpaper patterns

 

Frank A. Farris

 

🔗 Infinite mystery curve

Resources

Book of activities: Mystery curves

https://www.geogebra.org/m/pkbhvz4s

Activity: Change of frequencies

https://www.geogebra.org/m/xs7tmvy2

Applets used in this presentation

https://www.geogebra.org/m/btdzzzdt

Video Workshop - Institut GeoGebra Valencia

https://youtu.be/n24wrb7K3Gc

Thanks for your attention!

ABC of Mathematics:

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Complex Analysis:

A visual and interactive introduction

Acceso libre: 🔗 complex-analysis.com

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Mystery curves

By Juan Carlos Ponce Campuzano

Mystery curves

Exploring mystery curves with GeoGebra

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