About

The Bagula Double-V is a self-similar fractal constructed by recursive subdivision of line segments into a symmetric double-V motif. Each iteration replaces a single segment with 5 scaled copies arranged at alternating angles ±α, producing a branching structure whose Hausdorff dimension D depends on the scaling ratio r and the number of copies N. Misconfiguring the angle or depth yields either a degenerate line or an unresolvable cloud of points. This tool computes the fractal geometry in real time for depths up to 15 iterations, which can exceed 3¹⁵ = 14,348,907 line segments. Heavy iterations are offloaded to a Web Worker to keep the interface responsive.

The fractal belongs to the broader family of IFS attractors studied in computational geometry. The rendering uses exact trigonometric transformations rather than pixel-based approximation. Note: at depths beyond 12, visual detail saturates at typical screen resolutions. The tool approximates continuous-limit geometry within the discrete pixel grid of your display.

Formulas

The Bagula Double-V fractal replaces each line segment with N = 5 scaled and rotated copies. The Hausdorff dimension is computed from the self-similarity relation:

D = ln(N)ln(1 ÷ r) = ln(5)ln(3) ≈ 1.465

Each recursive step applies 5 affine transformations. Given endpoints (x₀, y₀) and (x₁, y₁), the segment direction vector is:

dx = x₁ − x₀ , dy = y₁ − y₀

The five child segments connect through intermediate points computed by rotating sub-vectors by ±α (default α = 60°):

x′ = x ⋅ cos(α) − y ⋅ sin(α)

y′ = x ⋅ sin(α) + y ⋅ cos(α)

Where N = number of self-similar copies, r = scaling ratio (1/3), D = Hausdorff (fractal) dimension, α = rotation angle per V-branch.

Reference Data

Depth	Segments	Scale Factor	Approx. Hausdorff Dim.	Render Complexity
1	5	0.333	1.465	Trivial
2	25	0.111	1.465	Trivial
3	125	0.037	1.465	Low
4	625	0.012	1.465	Low
5	3,125	0.004	1.465	Moderate
6	15,625	0.001	1.465	Moderate
7	78,125	-	1.465	High
8	390,625	-	1.465	High
9	1,953,125	-	1.465	Very High
10	9,765,625	-	1.465	Extreme
11	48,828,125	-	1.465	Extreme
12	244,140,625	-	1.465	Pixel-saturated

Frequently Asked Questions

At depths beyond 10-12, the individual line segments become smaller than a single pixel on most displays. The fractal's Hausdorff dimension D ≈ 1.465 means it fills more space than a line but less than an area. When segment length drops below 1 pixel, anti-aliasing blends them into a continuous mass. Reduce depth or zoom into a sub-region for detail.

The angle α controls the deflection of the V-branches. However, the Hausdorff dimension D = ln(N)/ln(1/r) depends only on the number of copies N and the scaling ratio r, not on angle directly. Changing α alters the visual appearance and overlap behavior but not the theoretical dimension - unless the angle causes sub-segments to overlap perfectly, which effectively reduces N.

Depth 12 produces approximately 244 million segments. Most browsers can handle rendering up to depth 9-10 smoothly (under 10 million segments). Beyond depth 10, the Web Worker may take several seconds. The tool caps at 15 but recommends staying at 8-10 for interactive use. At depth 15 you would need over 30 billion segment computations.

The classic Bagula Double-V uses r = 1/3 with 5 copies, yielding D ≈ 1.465. This tool allows adjusting the scaling ratio. Note that if r × N > 1, the IFS segments overlap, producing a filled region rather than a fractal curve. If r is too small, the structure collapses to isolated points. The mathematically interesting range is roughly 0.2 ≤ r ≤ 0.45.

This tool uses deterministic recursive subdivision: every segment is computed explicitly and drawn as a line. The chaos game plots random orbits of the IFS and approximates the attractor statistically. Deterministic rendering is exact but O(N^depth) in time and memory. The chaos game is O(iterations) regardless of fractal complexity, but requires millions of points to fill in detail. For the Double-V at low depths, deterministic rendering is superior.