New Fractals: Variable-Degree Generative Geometry

Abstract

New Fractals merits publication as an experimental geometry instrument. The submitted work is not a single rendered concept or a collection of names attached to familiar images. It implements eight distinct browser-operable generators across affine iterated-function systems, L-systems, three-dimensional recursive replacement, adaptive centroid partition, and polynomial-root direction trees. A separate Web Audio study maps root-tree paths and collisions into sound, while a Python/FastAPI prototype explores spatial reuse with SciPy's cKDTree.

The implementation is substantial and visually coherent. Its strongest contribution is the composition of existing mathematical primitives into designable, inspectable geometry families—especially the variable-degree root tree, where generation n exposes n root directions, produces n! terminal paths through depth n, and uses factorial attenuation to keep the construction bounded.

The public boundary is equally important. The supplied labels sometimes treated similarity estimates as Hausdorff dimensions, described a finite hash-routed path space as cryptographic content-addressable storage, and implied insertion complexity that the current rebuilding index does not provide. The live publication corrects those statements. It presents the work as implemented generative geometry, not as a proof of mathematical novelty, a certified dimension result, or a production CAS.

1. What was implemented

The submitted project contains approximately 5,331 authored lines across TypeScript, JavaScript, Python, HTML, and CSS, excluding generated standalone output. Its main surfaces are:

• a Vite/TypeScript Canvas application with a slide-out design explorer;
• eight selectable geometry studies with live controls and explanatory math panels;
• a browser-only Three.js/Web Audio synthesizer experiment;
• a FastAPI/WebSocket spatial collision simulator backed by NumPy and SciPy cKDTree;
• a standalone dependency-free ES-module build of the primary Canvas instrument.

The eight public studies are:

1. Shattered Lattice — a dense self-affine canopy with an anisotropic compression map;
2. Nexus Trellis — a central transform plus four isolated satellite transforms;
3. Dawless Signal-Fold — a carrier-and-return geometry using rotated and reflected affine maps;
4. Orchestration Cascade — a stepped 3D IFS with one elevated deck and four descending sub-decks;
5. Vitrified Mycelium — recursive curved filaments joining rigid spatial anchors;
6. Cytoplasmic Tessellation — adaptive recursive centroid partition whose child count responds to boundary complexity;
7. Variable-Degree Tree · Chebyshev — generation-specific directions derived from the real roots of T_n;
8. Variable-Degree Tree · Roots of Unity — generation-specific complex roots distributed around the unit circle.

These are implemented mechanisms. The public build permits direct visual inspection and parameter variation without a paid API or network service.

2. The variable-degree root construction

The most distinctive mechanism is the increasing-degree tree.

At generation n, the engine obtains n directions from either:

• the real roots of the Chebyshev polynomial T_n, mapped into branch angles; or
• the complex roots of z^n = 1, used as evenly distributed angular directions.

A complete path chooses one branch at every generation. The number of terminal paths through generation N is therefore:

1 × 2 × 3 × ... × N = N!

The branch length is attenuated as:

L_n = L_0 / (n!)^γ

for an operator-selected γ. This provides a useful counterforce to factorial path growth: combinatorial variety increases rapidly while geometric steps shrink rapidly. The implementation additionally carries a local frame forward through the tree so that splay and orientation remain spatially coherent rather than becoming a flat radial diagram.

The construction is real and operable. “Ascendant-Degree Root Topology” is treated as the project's name for this composition, not as an established mathematical category or a verified novelty claim. No exact-title prior-art result was identified during the publication review, but absence from a limited search is not evidence of invention priority.

3. Dimension claims: corrected from D to D*

For a finite family of contracting similitudes with contraction ratios r_i, the solution of

Σ r_i^s = 1

is the similarity dimension. Under the open set condition, that value agrees with the Hausdorff dimension of the self-similar attractor. Without verified similitudes and separation, the equation is not by itself a Hausdorff-dimension certificate. Self-affine maps, curved replacement, overlap, adaptive child counts, and mixed rendering layers require different analysis.

The live instrument therefore uses D*, meaning a scale or similarity proxy. The review produced these corrections:

The previous 1.83 label for Vitrified Mycelium did not solve its displayed equation. The previous Dawless estimate used 0.3 as the contraction of a matrix whose singular scale is √(0.3²+0.3²) = √0.18. Both are corrected in the public build.

A future dimension study should estimate box-counting or correlation dimensions across increasing render resolution, report confidence intervals, test sensitivity to sampling, and separately examine overlap and separation conditions. Those measurements should not be backfilled from an attractive rendered result.

4. Cytoplasmic partition: a useful adaptive mechanism

Cytoplasmic Tessellation computes a boundary-complexity quantity of the form

Q = P / √A

from polygon perimeter P and area A, then uses Q to influence the next fan count around the centroid. This is a valid adaptive recursive partition rule. It can generate nested cell-like and vascular-looking structures whose branching degree changes with local shape.

It is not a stationary IFS, and it does not currently supply a measured fractal dimension. Its merit is procedural: local morphology alters later subdivision rather than merely coloring a fixed recursion tree.

5. Spatial reuse prototype: valuable, but not a cryptographic CAS

The Python prototype maps SHA-256 bytes into one branch choice at each root-tree generation. It then queries a spatial index for nearby occupied coordinates, emits NEW_NODE or COLLISION, and can report that a complete path was already occupied.

That is a useful hash-routed spatial prefix lattice. It is not content-addressable storage in the conventional cryptographic sense.

At maximum generation N, the number of complete root-choice paths is only N!. With the submitted default N = 5, the terminal path space contains:

5! = 120 paths

Arbitrarily many SHA-256 payloads are therefore projected into 120 full paths. Distinct payloads must collide. During local review, two different test payloads reached the same complete path, while reinserting the same payload predictably reproduced its path. This is deterministic routing and spatial reuse—not a collision-resistant content identity.

The current cKDTree nearest-neighbor query is appropriate for bounded spatial lookup. However, the implementation marks the index dirty after each new point and rebuilds it before the next query. The query may be tree-accelerated, but the complete insertion workflow does not inherit a blanket O(log N) guarantee. A production experiment should use batched rebuilds, a dynamic spatial index, or a two-tier recent-point buffer with explicit complexity measurements.

6. Verification performed for publication

The publication candidate passed the following local checks:

• TypeScript compilation of the dependency-free Canvas surface using TypeScript 5.8.3 with no emitted-on-error output;
• Python bytecode compilation of simulator/engine.py;
• deterministic reinsertion behavior for one payload;
• an explicit distinct-payload full-path collision test at N = 5;
• browser smoke traversal of all eight selectable designs;
• no captured page or console errors during that traversal;
• a 390-pixel mobile-width layout check with no document-width overflow;
• visual inspection of the standalone instrument after replacing unsupported claims.

The uploaded source archive has SHA-256:

dc0c7bc1ccc0d7b411716495df84932a5f37fb6573e8891dbdfcc5d782903ee9

The machine-readable assessment is published at /evidence/new-fractals/assessment.json.

7. Publication decision

Decision: publish.

The work earns a Glyphd Labs surface because it satisfies the relevant threshold:

• it contains multiple implemented mechanisms rather than a concept-only claim;
• the mechanisms are directly inspectable in a deterministic public instrument;
• the strongest compositional idea is legible and reusable;
• incorrect or premature claims can be bounded without destroying the project;
• the remaining uncertainty can be expressed as specific experiments.

The publication classification is:

IMPLEMENTED GEOMETRY INSTRUMENT
+ LOCALLY VERIFIED PUBLIC BUILD
+ MATHEMATICAL HYPOTHESES / PROXIES
- NOT A FORMAL NOVELTY PROOF
- NOT A CERTIFIED HAUSDORFF-DIMENSION RESULT
- NOT A CRYPTOGRAPHIC CONTENT-ADDRESSABLE STORE
- NOT A DEPLOYED SPATIAL BACKEND

The Web Audio synthesizer and FastAPI simulator are meaningful extensions, but they are not included in the live public runtime. The public page represents the primary Canvas instrument only; the note records the extension work and its present limits.

8. Next experiments

The most consequential next work is not more naming. It is evidence production:

1. Dimension suite — multi-resolution box-counting and correlation-dimension estimates with stable sampling, confidence intervals, and known-fractal controls.
2. Separation suite — numerical overlap detection and explicit open-set-condition analysis for the similarity candidates.
3. Variable-degree comparison — compare factorial attenuation against geometric and polynomial attenuation under equal node budgets.
4. Root-family comparison — Chebyshev roots, roots of unity, Legendre roots, and learned directional sets under the same rendering contract.
5. Spatial capacity benchmark — collision rate, prefix-sharing distribution, rebuild cost, and memory use as N, tolerance, and payload count vary.
6. Audio study — define repeatable mappings from path events to timbre and test whether structural differences remain perceptible under blind listening.

These experiments can promote selected statements from plausible mechanism descriptions to measured claims.

Publication boundary

This note evaluates and publishes the supplied implementation. It does not establish priority, originality in the patent sense, mathematical uniqueness, exact Hausdorff dimension, collision-resistant identity, or production scalability. Geometry names are project nomenclature. Scale values marked D* are proxies. The public artifact is evidence of implementation and inspectability, not evidence for every interpretation that could be placed on its images.

Source register

• N19-S01 — Hayden Lindley, Gemini, and Cursor, new-fractals.zip, supplied August 1, 2026; SHA-256 dc0c7bc1ccc0d7b411716495df84932a5f37fb6573e8891dbdfcc5d782903ee9.
• N19-S02 — P. A. P. Moran, “Additive Functions of Intervals and Hausdorff Measure,” Proceedings of the Cambridge Philosophical Society 42 (1946), 15–23.
• N19-S03 — SciPy documentation, scipy.spatial.cKDTree.query, consulted August 1, 2026.
• N19-S04 — Glyphd Labs local verification receipt, docs/receipts/2026-08-01-new-fractals.json.