#>hyperbolic_manifolds
non-euclidean poincaré disk manifolds

Software / Mathematical Depth

The Curvature of Space

What happens when the parallel postulate fails? In Hyperbolic Geometry, the sum of a triangle's angles is always less than 180 degrees, and space effectively "expands" as you move outward. This surface explores how we project these infinite manifolds into finite pixels using the Poincaré disk model.

^"poincare_disk"{

The Poincaré Disk Model

A projection of the hyperbolic plane where the entire infinite space is contained within a unit circle. Lines are represented as circular arcs perpendicular to the boundary.

Hyperbolic Disk Projection Origin Infinite space, finite projection.

Möbius Transformations

The automorphisms of the Poincaré disk. In rendering, these allow for smooth "panning" through hyperbolic space while preserving angles.

wiki: Möbius
#>code_block_lab

Matrix Enrichment

How it looks in Spw: representing a hyperbolic transform as a complex matrix projection.

#>hyperbolic_transform
#!physics/non-euclidean

^"mobius"{
  z: point(x, y)
  transform: (a*z + b) / (c*z + d)
  [a, b; c, d]: matrix2x2(1, 0.5, 0.5, 1)
}