Ford Circles and Farey Sequences

Video Description

Ford Circles form an interesting mathematical structure. No matter how far down you go, the fractal pattern never ends. Every single circle is tangent to 2 other circles and at the same time is tangent to the number line below. The most amazing thing about this is that each circle always touches the line at one and only one rational point. There is no circle that touches the line at any irrational point. This means that this set of circles gives us a geometric representation of the set of rational numbers. In this video, I will show you exactly how this beautiful image arises from the simple act of slightly changing the rules of fraction addition. _____ Join my Patreon community: https://www.patreon.com/abidebyreason Support the channel with a one-time donation: https://ko-fi.com/abidebyreason _____ Related Videos: Why the Cantor Set is Perfect: https://youtu.be/c4oKdtUtJOk Why the Rationals Have Measure 0: https://youtu.be/1qbjjgesp-c The Connection Between Measure Theory, Set Theory, and Banach-Tarski: https://youtu.be/SvfATfaL2qc Banach-Tarski Paradox Explained: https://youtu.be/R--iM5KbDEg Intro to Measure Theory: https://youtu.be/1BhSQiHTNbg Intro to Topology: https://youtu.be/B-Y3-XpAdMU Animations created using Manim: https://www.manim.community/