Juan Carlos Ponce Campuzano
Independent Mathematics Educator
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.
Rotating circles
By Juan Carlos Ponce Campuzano
Rotating circles
Construction of rotating circles.
