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, mirror, Newnome Beauton, Adam Parrott, Sophia Wood (Fractal Kitty), pmben, Abei, Edward Huff.