Mysterious

Rotating

Circles

Centers

C_1 = \;?
C_2 = \;?
\vdots
C_n = \;?
C_4 = \;?
C_3
= \;?
\vdots

Radii

r_1 =\;?
r_2 =\; ?
r_3 = \;?
r_4 =\; ?
r_n =\; ?

\(\overline{C_1P_1}\)

\(= r_1\)

\(\overline{C_1P_1}= r_1\)

\(\overline{C_2P_2}= r_2\)

\(= r_1 s\)

\(\overline{C_1P_1}= r_1\)

\(\overline{C_2P_2}= r_2\)

\(\overline{C_3P_3}= r_3\)

\(= r_1 s\)

\(=r_1 s^2 \)

\(= r_2 s \)

\(=\left( r_1 s \right) s\)

\(\overline{C_1P_1}= r_1\)

\(\overline{C_2P_2}= r_2\)

\(\overline{C_3P_3}= r_3\)

Here \(\,0<s<1\)

\(\vdots \)

\(\overline{C_nP_n}= r_n = r_1 s^{n-1} \)

\(= r_1 s\)

\(=r_1 s^2 \)

\(= r_2 s \)

\(=\left( r_1 s \right) s\)

\(\,r_1 = \overline{C_1C_2} + \overline{C_2P_1},\)

\(\Rightarrow \overline{C_1C_2} = r_1-\overline{C_2P_1}\)

\(= r_1-r_1s\)

\(= r_1\left(1-s\right)\)

\(\,r_1 = \overline{C_1C_2} + \overline{C_2P_1},\)

\(\Rightarrow \overline{C_1C_2} = r_1-\overline{C_2P_1}\)

\(\,r_2 = \overline{C_2C_3} + \overline{C_3P_2},\)

\(\Rightarrow \overline{C_2C_3} = r_2-\overline{C_3P_2}\)

\(= r_1-r_1s\)

\(= r_1\left(1-s\right)\)

\(= r_1s-r_1 s^2\)

\(= r_1s\left(1- s\right)\)

\(\,r_1 = \overline{C_1C_2} + \overline{C_2P_1},\)

\(\Rightarrow \overline{C_1C_2} = r_1-\overline{C_2P_1}\)

\(\,r_2 = \overline{C_2C_3} + \overline{C_3P_2},\)

\(\Rightarrow \overline{C_2C_3} = r_2-\overline{C_3P_2}\)

\(= r_1-r_1s\)

\(= r_1\left(1-s\right)\)

\(= r_1s-r_1 s^2\)

\(= r_1s\left(1- s\right)\)

\(\overline{C_3C_4}= r_1 s^{2}\left(1-s\right) \)

\(\,r_1 = \overline{C_1C_2} + \overline{C_2P_1},\)

\(\Rightarrow \overline{C_1C_2} = r_1-\overline{C_2P_1}\)

\(\vdots \)

\(\overline{C_nC_{n+1}}= r_1 s^{n-1}\left(1-s\right) \)

\(\,r_2 = \overline{C_2C_3} + \overline{C_3P_2},\)

\(\Rightarrow \overline{C_2C_3} = r_2-\overline{C_3P_2}\)

\(= r_1-r_1s\)

\(= r_1\left(1-s\right)\)

\(= r_1s-r_1 s^2\)

\(= r_1s\left(1- s\right)\)

\(\overline{C_3C_4}= r_1 s^{2}\left(1-s\right) \)

Again \(\,0<s<1!\)

\(\overline{C_1C_2} =r_1(1-s)\)

\(\vdots \)

\(\overline{C_nC_{n+1}}= r_1 s^{n-1}\left(1-s\right) \)

\(\overline{C_2C_3} = r_1s\left(1-s\right)\)

\(\overline{C_3C_4}= r_1 s^{2}\left(1-s\right) \)

Lenght of segments 

Now let's find the coordinates of each point

\(C_n =\left(?, ?\right)\)

C_2 = \bigg( r_1(1-s) \cos(\theta), r_1 (1-s)\sin(\theta) \bigg)
\theta
0\leq \theta\leq 2 \pi
\overline{C_1C_2} =r_1(1-s)
C_1 = (0, 0)

Set

C_2 = \bigg( r_1(1-s) \cos(\theta), r_1 (1-s)\sin(\theta) \bigg)
= \bigg( r_1(1-s) ; \theta \bigg)
\theta
0\leq \theta\leq 2 \pi
C_1 = (0, 0)

Set

\(\leftarrow\)Polar form

C_1 = (0, 0)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)

Set

C_3
= \bigg( r_1(1-s) ;\theta \bigg)
\overline{C_1C_2} =r_1(1-s)
\overline{C_2C_3} =r_1s(1-s)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C_1 = (0, 0)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)

Set

C_3
= \bigg( r_1(1-s) ;\theta \bigg)
\overline{C_1C_2} =r_1(1-s)
\overline{C_2C_3} =r_1s(1-s)
\theta
+ \bigg( r_1s(1-s); 2\theta \bigg)
C_1 = (0, 0)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)

Set

C_3
= \bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
\overline{C_1C_2} =r_1(1-s)
\overline{C_2C_3} =r_1s(1-s)
2\theta
\bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C_1 = (0, 0)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)

Set

C_3
= \bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C_4=
\overline{C_1C_2} =r_1(1-s)
\overline{C_2C_3} =r_1s(1-s)
\overline{C_3C_4} =r_1s^2(1-s)
\bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C_1 = (0, 0)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)

Set

C_3
= \bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
\bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C_4=
+\bigg(rs^2(1-s);3\theta \bigg)
\overline{C_1C_2} =r_1(1-s)
\overline{C_2C_3} =r_1s(1-s)
\overline{C_3C_4} =r_1s^2(1-s)
C_1 = (0, 0)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)

Set

C_3
= \bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
\overline{C_1C_2} =r_1(1-s)
C_4=
\overline{C_2C_3} =r_1s(1-s)
\overline{C_3C_4} =r_1s^2(1-s)
3\theta
\bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
+\bigg(r_1s^2(1-s);3\theta \bigg)
C_1 = (0, 0)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)

Set

C_3
= \bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
\vdots
C_n = \sum_{k=0}^{n-2}\bigg( r_1s^k(1-s) ;(k+1)\theta \bigg)
C_4=
\bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
+\bigg(r_1s^2(1-s);3\theta \bigg)
0\leq \theta \leq 2\pi
0\lt s\lt1

Radii

Centers

r_1 = r_1 s^0
r_2 = r_1 s^1
r_3 = r_1 s^2
r_4 = r_1 s^3
\vdots
r_n = r_1 s^{n-1}
C_2 = \bigg( r_1(1-s) ;\theta \bigg)
C_3
= \bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
\vdots
C_n = \sum_{k=0}^{n-2}\bigg( r_1s^k(1-s) ;(k+1)\theta \bigg)
C_4=
\bigg( r_1(1-s) ;\theta \bigg)
+ \bigg( r_1s(1-s); 2\theta \bigg)
+\bigg(r_1s^2(1-s);3\theta \bigg)
C_1 = (0, 0)
0\leq \theta \leq 2\pi
0\lt s\lt1

Radii

Centers

r_1 = s^0
r_2 = s^1
r_3 = s^2
r_4 = s^3
\vdots
r_n = s^{n-1}
r_1=1
C_2 = \bigg( (1-s) ;\theta \bigg)
C_3
= \bigg( (1-s) ;\theta \bigg)
+ \bigg( s(1-s); 2\theta \bigg)
\vdots
C_n = \sum_{k=0}^{n-2}\bigg( s^k(1-s) ;(k+1)\theta \bigg)
C_4=
\bigg( (1-s) ;\theta \bigg)
+ \bigg( s(1-s); 2\theta \bigg)
+\bigg(s^2(1-s);3\theta \bigg)
C_1 = (0, 0)
s = Slider(0.1, 1.5, 0.01)
t = Slider(0, 2pi, 0.01)
n = Slider(0, 50, 1)

Ln = 0...n

LR = Zip(s^k, k, Ln)

LP = Join({(0, 0)}, Zip((s^k * (1 - s); (k+1) * t), k, Ln))

LS = Zip(Sum(LP, k), k, Ln+1)

LC = Zip(Circle(P, r), P, LS, r, LR)

GeoGebra Script

Check Ben Sparks' construction

using the Spreadsheet in GeoGebra

Link in the description

Mathematical topics

  • Geometry:
    • Rotations, Dilations, Tangency of circles
  • Analytic geometry:
    • Cartesian and polar coordinates
  • Recursive Sequences and Series

There is also a conexion with

Complex Numbers!

Plot all the centers

and join them

\sum_{k=0}^{n}z^k

for \(z=x+iy\)

Geometrical representation

of the geometric series

=\left(re^{i\theta}\right)^k
z^k
=r^ke^{i (k \theta)}
\sum_{k=0}^{n}z^k
=\sum_{k=0}^{n}r^k e^{i(k\theta)}
z=x+iy
=re^{i\theta}

Geometrical representation of the

Geometric series

Geometric series

C_n = \sum_{k=0}^{n-2}\bigg( s^k(1-s) ;(k+1)\theta \bigg)
\sum_{k=0}^{n}z^k
=\sum_{k=0}^{n}r^k e^{i(k\theta)}

Relationship between the Geometric series and the points \(C_n\)

\sum_{k=0}^{n}z^k
C_n = \sum_{k=0}^{n-2}\bigg( s^k(1-s) ;(k+1)\theta \bigg)
=\sum_{k=0}^{n}r^k e^{i(k\theta)}

Relationship between the Geometric series and the points \(C_n\)

\sum_{k=0}^{n}z^k
C_n
= \sum_{k=0}^{n}\bigg(r^k\cos(k\theta), r^k\sin(k\theta) \bigg)
= \sum_{k=0}^{n-2}\bigg( s^k(1-s) \cos((k+1)\theta), s^k(1-s) \sin((k+1)\theta) \bigg)

Re-write both in cartesian form!

Relationship between the Geometric series and the points \(C_n\)

\sum_{k=0}^{n}z^k
= \sum_{k=0}^{n}\bigg(r^k\cos(k\theta), r^k\sin(k\theta) \bigg)
C_n = (1-s) \sum_{k=0}^{n-2}\bigg( s^k \cos((k+1)\theta), s^k \sin((k+1)\theta) \bigg)

Factorize \((1-s)\)

Relationship between the Geometric series and the points \(C_n\)

\sum_{k=0}^{n}z^k
= \bigg( r^0\cos(0), r^0\sin (0) \bigg) + \bigg( r^1\cos(\theta), r^1\sin (\theta) \bigg)+\cdots + \bigg( r^n\cos(n\theta), r^n\sin (n\theta) \bigg)
= \sum_{k=0}^{n}\bigg(r^k\cos(k\theta), r^k\sin(k\theta) \bigg)
C_n = (1-s) \sum_{k=0}^{n-2}\bigg( s^k \cos((k+1)\theta), s^k \sin((k+1)\theta) \bigg)

Relationship between the Geometric series and the points \(C_n\)

\sum_{k=0}^{n}z^k
= \bigg( r^0\cos(0), r^0\sin (0) \bigg) + \bigg( r^1\cos(\theta), r^1\sin (\theta) \bigg)+\cdots + \bigg( r^n\cos(n\theta), r^n\sin (n\theta) \bigg)
= (1-s)\bigg[\big( s^0\cos(\theta),s^0 \sin(\theta) \big) + \big(s^1 \cos(2\theta), s^1\sin(2\theta) \big)+\bigg.
= \sum_{k=0}^{n}\bigg(r^k\cos(k\theta), r^k\sin(k\theta) \bigg)
C_n = (1-s) \sum_{k=0}^{n-2}\bigg( s^k \cos((k+1)\theta), s^k \sin((k+1)\theta) \bigg)
\bigg. +\cdots + \big( s^{n-2} \cos((n-2)\theta), s^{n-2}\sin((n-2)\theta) \big) \bigg]

Relationship between the Geometric series and the points \(C_n\)

\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
\bigg .\bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
C_n =
(1-s) \bigg [ \bigg.
\sum_{k=0}^{n}z^k=
\bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.

Relationship between the Geometric series and the points \(C_n\)

\sum_{k=0}^{n}z^k=
\bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
\bigg .\bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
C_n =
(1-s) \bigg [ \bigg.

Relationship between the Geometric series and the points \(C_n\)

\sum_{k=0}^{n}z^k=
\bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
\bigg .\bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
C_n =
(1-s) \bigg [ \bigg.

Relationship between the Geometric series and the points \(C_n\)

\sum_{k=0}^{n}z^k
C_n
= \bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
=(1-s) \bigg [ \bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
\sum_{k=0}^{n}\mathcal R^k \cos(k \theta),\quad \sum_{k=0}^{n}\mathcal R^k \sin(k \theta)

Relationship between the Geometric series and the points \(C_n\)

\sum_{k=0}^{n}z^k
C_n
= \bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
=(1-s) \bigg [ \bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
\sum_{k=0}^{n}\mathcal R^k \cos(k \theta),\quad \sum_{k=0}^{n}\mathcal R^k \sin(k \theta)

Relationship between the Geometric series and the points \(C_n\)

\sum_{k=0}^{n}z^k
C_n
= \bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
=(1-s) \bigg [ \bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
\sum_{k=0}^{n}\mathcal R^k \cos(k \theta),\quad \sum_{k=0}^{n}\mathcal R^k \sin(k \theta)
\sum_{k=0}^{n}z^k,

we just need to adjust the expression

To obtain the same result

\left\{ \begin{array}{l} z= re^{i\theta},\\ 0\lt r\lt 1\\ 0\leq \theta\leq 2\pi \end{array} \right.

with

\left(\frac{1}{r}-1\right)
\left\{ \begin{array}{l} z= re^{i\theta},\\ 0\lt r\lt 1\\ 0\leq \theta\leq 2\pi \end{array} \right.

with

\sum_{k=0}^{n}z^k,
\left(1-\frac{1}{r}\right) +

Multiply by

Add this

\left\{ \begin{array}{l} z= re^{i\theta},\\ 0\lt r\lt 1\\ 0\leq \theta\leq 2\pi \end{array} \right.

with

Challenge: Use this approach to built it in GeoGebra

\left(\frac{1}{r}-1\right)
\sum_{k=0}^{n}z^k,
\left(1-\frac{1}{r}\right) +

GeoGebra

Desmos

p5.js

Links in the description

Thanks for

watching!

Patreons:

Christopher-Alexander Hermans, Maciej Lasota, Miguel Díaz, bleh, Dennis Watson, Doug Kuhlmann, Newnome Beauton, Adam Parrott, Sophia Wood (Fractal Kitty), pmben, Abei, Edward Huff.

Thanks for

watching!

Patreons:

Christopher-Alexander Hermans, Maciej Lasota, Miguel Díaz, bleh, Dennis Watson, Doug Kuhlmann, Newnome Beauton, Adam Parrott, Sophia Wood (Fractal Kitty), pmben, Abei, Edward Huff.

Rotating circles

By Juan Carlos Ponce Campuzano

Rotating circles

Construction of rotating circles.

  • 63