FORD : � THE PATTERN THAT LIES WITHIN CAITLYN KAY

Kuncoro Kohar, S.T., B.Eng., M.Sc.

Cita Hati Christian Senior School West Campus

INTRODUCTION RESULT AND ANALYSIS CONCLUSION

A fractal is a never-ending pattern that are placed to a horizontal line It has been proven that a set of Ford can be found in almost every part of the to form a Ford Circle. Each circle is labelled Circles has an infinite pattern. However, world. Fractals are infinitely complex (n) with numbers from 1 and so on (ascending its total area is a geometric series which patterns that are self-similar across 1 1 value of n, descending diameter). converges into a certain number (finite). different scales. In this manually constructed Ford Circle, As the pattern continues, each circle 2

5 3 4 A set of Ford Circles is considered as a 5 6 there are circles of 17 different sizes. It can decreases in size, shown in the series fractal because no matter how much it is Figure 1. Ford Circles (Keynote) be seen that n is showing properties of the below: enlarged, there will always be a copy (or Fibonacci sequence. But instead of each d d d d continuation) of smaller circles within the value of n being the sum of two preceding + + + . . . + 12 22 32 n2 pattern. Each Ford Circle is tangent to numbers, it is the sum of the value of n of the two larger circles and one horizontal line. 2 larger adjacent circles. It is possible to calculate the total This research is to uncover the hidden number of circles in a set of Ford Circles patterns and equations within this For instance, the purple circle (5) is the sum with the equation below: geometrical fractal. of n of the green (2) and red circle (3), or blue m (1) and yellow (4). 1 + ∑ φ(n) Figure 2. Ford Circles (zoomed in 3x) n=1 Each of the circle’s diameter were then recorded and analysed. Equations regarding The equation above sums up the total METHODOLOGY the diameter and area of Ford Circles were formulated: number of fractions — within a given Table 1. Diameter boundary (m) — which its numerator and The main focus of this research is the of Cn in Figure 2 Diameter of Area of each Total area of C1 to denominator have a greatest common properties of each circle in a set of Ford Cn Dn (pts) each circle circle Cn in a Ford Circle C1 1000 divisor of 1. Circles. C2 250 2 d d m 2 C3 111 D = d n 2 An = π π The two equations that have been found C4 63 n 2 2 Cn Cn = circle number n ( 2n ) ∑ ( ) n=1 2n C5 40 in this research can be combined On = centre of circle Cn C6 28 On Dn together to form an equation to calculate Dn = diameter of circle Cn C7 20 Dn C1 C2 C3 C4 the total area of a Ford Circle within a d = diameter of circle C1 1 2 3 given boundary (m): An = area of circle Cn Table 2. Values of 10 100 1000 The function is tested with 9 2 m 2 The equations 1 πd d The goal of this research is to: ns ∑ ∑ ∑ different values of s ∈ ℤ+ (Table A = + φ(n) ⋅ π found above can n=1 n=1 n=1 4 ∑ ( 2n2 ) ζ(2) 1.54976 1.63498 1.64393 n=1 1. Find any possible patterns within a be associated with 2). It can be analysed that the set of Ford Circles ζ(3) 1.19753 1.20200 1.20205 the Riemann zeta ζ(4) 1.08203 1.08232 1.08232 series converges into a certain 2. Correlate Ford Circles with existing function ζ(5) 1.03690 1.03692 1.03692 number. theorems ζ(6) 1.01734 1.01734 1.01734 ∞ FUTURE WORK 1 ζ(7) 1.00834 1.00834 1.00834 In the case of Ford Circles, the ζ(s) = ζ(8) 1.00407 1.00407 1.00407 ∑ s n=1 n ζ(9) 1.00200 1.00200 1.00200 value of s is always 2. For further improvements, I would like to Arranging a ζ(10) 1.00099 1.00099 1.00099 explore further about the properties of set of Ford Ford Circles as a fractal, and analyse Circles in other patterns within Ford Circles even Keynote The sum of circles Table 4. Euler’s totient function A set of Ford Circles consisting �(n) Fractions Value more. For instance, the distance in a Ford Circle of circles of 7 different sizes will �(1) 1/1 1 between centres of each circle, or the can be determined (2) 1/2 1 have a total of 19 (1+18) circles. � quadratic functions passing through the Analysing Measuring by Euler’s totient �(3) 1/3. 2/3 2 (4) centres of Ford Circles. And perhaps patterns diameters function � 1/4, 3/4 2 �(5) 1/5, 2/5, 3/5, 4/5 4 correlating this research to other 1 �(6) 1/6, 5/6 2 theorems in . φ(n) = n 1 − ∏( ) �(7) 1/7, 2/7, 3/7, 4/7, 5/7, 6/7 6 p|n p total 18 Finding the Formulating relationship equations REFERENCES between the regarding the The Ford Circle is plotted on a cartesian Table 3. Explanation of each term of the equation Term Description and properties Measures, K. E. (2018). Moments of Distances Between diameter and diameter and graph, with the equation: 2 k • Determines the position of On on the x- Centres of Ford Spheres. Retrieved from Riemann Zeta area of Ford x − axis ( n ) etheses.whiterose.ac.uk/23120/1/ 2 2 2 • k/n = x-coordinate of On Function Circles k 1 1 KayleighMeasuresThesis.pdf. 2 x − + y − = • Determines the position of On on the y- 2 2 1 ( n ) ( 2n ) ( 2n ) y − axis ( 2n2 ) Paquin, D. 2011. Farey Sequences and Ford Circles. • 1/2n2 = y-coordinate of On where 0 ≤ k ≤ n; n, k ∈ ℤ Retrieved from 2 1 • Determines the size of Cn marinmathcircledotorg.files.wordpress.com/2015/12/ Plotting Ford Analysing the • 1/2n2 = radius of circle Cn ( 2n2 ) mmcint-20110511-danapaquin-fareysequences.pdf Circles in a properties of each cartesian term of the Veisdal, J. 2019. The Riemann Hypothesis, explained. The coordinate of the centre (On) of Cn is: Retrieved from graph equation medium.com/cantors-paradise/the-riemann-hypothesis- k 1 explained-fa01c1f75d3f. , ( n 2n2 )

Finding the When plotting Ford Circles, k and n must Conclusions relationship have only one common divisor of 1. This

and further between Euler’s Figure 3. Ford Circles on a cartesian graph (Desmos) prevents any circle from overlapping. improvements Totient Function None of the circles overlap, forming an This concept is depicted in Euler’s and Ford Circles infinite pattern. totient function (φ).