Overview

This package contains two standalone Geometry Nodes node trees implementing Boy’s surface—a smooth, non-orientable immersion of the real projective plane ℝℙ² with 3-fold rotational symmetry:

1. Math_Visualization (Apéry Formulation): Explicit trigonometric evaluation over a parameterized domain grid $(u,v)$.
2. Math_Visualization.001 (Bryant–Kusner Formulation): Rational complex polynomial mapping over a polar disk $(r,\theta )$, followed by ℝ³ spherical inversion.

Accompanying PDF reference sheets map each mathematical equation directly to its corresponding colored frame inside the node editor.

What is Boy's Surface:
Discovered by Werner Boy in 1901, it is an immersion of the real projective plane (ℝℙ²) in 3D space — topologically equivalent to sewing a Möbius strip to the boundary of a disk without any pinch points or sharp singularities.

How it works (Two Complete Implementations Included):

• Apéry Parameterization: Generates the surface from a parametric (u, v) grid using François Apéry's explicit trigonometric equations, organized into a shared denominator branch D(u, v) = 2 − √2·sin(3u)·sin(2v) and dedicated X, Y, and Z coordinate branches.
• Bryant–Kusner Parameterization: Maps a polar grid (r, θ) on the complex unit disk (w = r·e^(iθ)) into Robert Bryant and Rob Kusner's Willmore-energy-minimizing immersion. The node tree explicitly separates the Real and Imaginary parts of the complex denominator D(w) = w⁶ + √5·w³ − 1 and coordinate numerators, performs ℝ³ spherical inversion (P = g / ||g||²), and constructs a dual-offset shell (±0.001) with flipped faces to eliminate shading seams across the non-orientable Möbius twist.

Geometry Nodes Setup & Controls

• Resolution (Value node in Frame Resolution): Controls grid density ($X\times Y$). Higher values produce smoother self-intersection curves and tunnel silhouettes.
• Domain Grid (Grid node in Frame Base Grid):
  • Size $X=2\pi$ ($\approx 6.283185$), Size $Y=1.0$.
  • Shifted internally to map $X\in [-\pi ,\pi ]$ (polar angle $\theta$) and $Y\in [0,1]$ (radial distance $r$ on the complex disk).
• Shape Calibration (Bryant–Kusner Scale Frames):
  • $X$ & $Y$ Branches (Math.064 & Math.043): Multiplier $k$ (default: -1.650). Governs the radial span and thickness of the 3 outer underbranching lobes.
  • $Z$ Branch (Math.079): Offset $c$ (default: 0.850). Shifts the center of the spherical inversion, governing the scale of the central triangular cap and the aperture of the 3 internal tunnels.
• Non-Orientable Dual Shell:
  • Uses dual $\pm 0.001$ offsets with face flipping and side-wall stripping (Delete Geometry: Not Top). This resolves the normal-inversion boundary artifact across the Möbius twist without requiring heavy post-boolean operations.
• Surface Attributes:
  • Outputs FlowUV ($r^2$), a normalized scalar field running smoothly from 0.0 at the triple point to 1.0 along the outer perimeter seam, accessible in the Shader Editor via an Attribute node.

Shader & EEVEE 5.2 Recommended Settings

For translucent/X-ray styling with internal tunnel visibility:

• Render Engine: EEVEE (Blender 4.2+ / 5.2)
• Surface Settings (Material Properties):
  • Render Method: Blended (avoids dithered noise across gradient alpha)
  • Backface Culling: Camera: ON (prevents double-rendering of the $\pm 0.001$ dual shells)
  • Transparency Overlap: ON (ensures all 3–4 internal overlapping layers composite smoothly)
  • Shadows / Ray Visibility: Uncheck Shadow on the object to prevent the inner shell from self-shadowing the outer skin.