3D IFS ENGINE
MATHEMATICAL SPECIFICATION
ITERATED FUNCTION SYSTEM
The 3D Sierpinski Gasket is constructed by taking a regular solid tetrahedron, scaling it by factor
r = 1/2, and arranging 4 identical sub-tetrahedra at the four vertices of the original geometry.
Hausdorff Dimension Calculation:
Even though the Tetrix is embedded in 3-dimensional Euclidean space, its fractal dimension is exactly 2.0. In the infinite limit n → ∞, the enclosed volume approaches zero (Vn = (1/2)n V0 → 0) while the total surface area remains completely constant.
VOXEL SUBDIVISION
The 3D Menger Sponge recursively partitions a solid cube into 27 sub-cubes (3 × 3 × 3) and removes the 6 center face cubes plus the 1 central core cube, leaving N = 20 cubes of scaling factor r = 1/3.
Hausdorff Dimension Calculation:
At the infinite recursion limit, the Menger Sponge possesses zero volume and infinite surface area.