Farey Sequences, Mediants, and Adjacent Fractions
FareySequence Mediants RationalEnumeration
01:30 Abide by Reason introduces Farey sequences for viewers who know rational numbers but haven’t seen this structured way to list them with deep number-theoretic properties.
Ford Circles as a Geometric Representation of Rationals
FordCircles RationalGeometry TangencyStructure
05:00 This note targets learners who want a visual picture of the rationals in ([0,1]) beyond points on a line.
Tangency Condition for Ford Circles and Farey Adjacency bc − ad = 1
TangencyCondition FareyAdjacency PythagoreanArgument
09:00 Abide by Reason shows how a simple geometric calculation in the Ford circle picture encodes the same adjacency condition seen in Farey sequences.
Alternating Tangent Circles and Convergence to the Golden Ratio
GoldenRatio FibonacciSequence RationalApproximation
14:30 This note is for learners intrigued by how a simple geometric “walk” among Ford circles recovers the golden ratio.
Farey Walks, Ford Circles, and Diophantine Approximation of Irrationals
DiophantineApproximation FareyWalk RationalApproximants
18:30 Abide by Reason hints at a general method for approximating irrationals using walks through Farey sequences and Ford circles.