Hans Walser, [20181201]
Ford Circles and an Amusing Function
Link between the Ford circles and the functions described by Steinerberger (2018), Fibonacci numbers, and the Golden ratio.
Just visualizations. Proofs are left to the reader.
Figure 1 gives a collection of Ford circles.
Fig. 1: Ford circles
We use the function:
(1)
Apart from the insignificant factor , these are the functions described by Steinerberger (2018).
Figure 2 gives the diagram of . Striking are the tip-shaped local maxima.
Fig. 2: Diagram
Figure 3 gives the link between the Figures 1 and 2.
Fig. 3: Link
Just for fun some more examples.
Fig. 4.1: n = 1
Fig. 4.2: n = 2
Fig. 4.3: n = 3
Fig. 4.4: n = 4
Fig. 4.5: n = 5
Fig. 4.6: n = 6
Fig. 4.7: n = 7
Fig. 4.8: n = 8
Fig. 4.9: n = 9
Fig. 5: Zigzag lines
Figure 5 depicts two zigzag lines. The vertices of the red zigzag line are the centers of adjoining Ford circles, the vertices of the blue zigzag line peaks of the function diagram.
The x-coordinates of the vertices of both zigzag lines are:
(2)
We recognize in the numerators and denominators of (2) the Fibonacci numbers (Walser 2012). Hence the limit of these x-coordinates is the Golden ratio (Walser 2001, Walser 2013):
(3)
Figure 6 shows the situation.
Fig. 6: Golden ratio
Fig.
7: Pythagorean triangles
The
yellow triangles in figure 7 have the sides:
Tab.
1: Sides and ratios
The
triangles are Pythagorean triangles with:
(4)
References
Steinerberger,
Stefan (2018): An Amusing Sequence of Functions. Mathematics
Magazine, Vol. 91, No. 4, October 2018, p 262-266.
Walser, Hans (2001): The Golden Section. Translated
by Peter Hilton and Jean Pedersen. The Mathematical Association
of America 2001. ISBN 0-88385-534-8.
Walser, Hans (2012): Fibonacci. Zahlen und Figuren. Leipzig, EAGLE, Edition am Gutenbergplatz. ISBN 978-3-937219-60-8.
Walser, Hans (2013): Der Goldene Schnitt. 6., bearbeitete und erweiterte
Auflage. Edition am Gutenbergplatz, Leipzig. ISBN 978-3-937219-85-1.
Links
Francis
Bonahon: Funny Fractions and Ford Circles
(30.11.2018):
https://www.youtube.com/watch?v=0hlvhQZIOQw&t=604s
Hans
Walser: Ford-Kreise (30.11.2018):
http://www.walser-h-m.ch/hans/Miniaturen/F/Ford-Kreise/Ford-Kreise.htm