FareyDiagrams.jl

FareyDiagrams.jl is a Julia library for constructing, exploring, and visualizing Farey diagrams and their associated topographs. It supports plotting using Makie.jl as well as computing triangle centers using both standard and equidistant angular embeddings.

This library can generate both the classic Farey diagram and a variant that reveals extra structure when n = 2.

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Features

• Constructs Farey graphs of any integer order n = {1,2}.
• Supports two graph types:
  • FareyGraph(n): the classic Farey diagram built via mediants.
  • FareyGraph2(n): an extended variant that adds "extra" rational points (e.g. ±2/1, ±1/2) and reveals new symmetries at n = 2.
• Triangle generation and tracking: identify new triangles formed at each level.
• Computes centroids of triangles (used in topographs).
• Embeds vertices on the unit circle using:
  • Standard angle mapping (angle): based on rational number geometry.
  • Equidistant angle mapping (equi_angle): based on continued fractions.
• Arc-based plotting using Luxor and Makie.
• interactive viewer with sliders for level and variant.

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TO-DO

• look for the cases for n>2
• ...

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Example

using FareyDiagrams

# Build the Farey graph of order 2
G, vertex_dict, triangles = FareyGraph(2)

# Or use the extended variant
G2, vertex_dict2 = FareyGraph2(2)

# Plot (requires GLMakie)
using GLMakie

fig = Figure(resolution=(800, 800))
ax = Axis(fig[1, 1], aspect=DataAspect())
plot_farey_graph(ax, 2, 500.0; variant=1)  # change to variant=2 for FareyGraph2
display(fig)