N-flake¶
Generate a family of polygonal self-similar attractors by placing contracted copies of a regular n-gon at its vertices, optionally adding a convention-controlled central copy, and iterating the resulting similarity system.
Core Idea¶
An (n)-flake, also called a Sierpiński (n)-gon or polyflake in qualified usage, is a self-similar polygonal fractal obtained as the attractor of similarity maps that place contracted regular (n)-gons at the vertices of a parent (n)-gon, with any central copy and scale convention stated explicitly. Each retained polygon is replaced by the same vertex-centered configuration at a fixed scale, so finite approximants converge in the Hausdorff metric to the invariant compact set of the iterated function system; under separation, the similarity dimension solves the copy-count scaling equation.
Scope of Application¶
N-flake applies when the analyst can specify a regular (n)-gon with \(n\geq3\), a declared set of similarity maps placing contracted copies at its vertices and possibly its center, and the unique compact attractor of that system and establish that one fixed family of contractive polygonal similarities is iterated indefinitely and its placement, copy count, and central-copy convention determine the limiting attractor. The entry fixes the two-dimensional regular-polygon family and marks extensions explicitly. Images and formulas from one center convention must not be generalized to all n-flakes.
Clarity¶
A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because n-flake, Sierpiński n-gon, and polyflake overlap in usage but do not always fix center inclusion or contraction ratio, so the IFS is the authoritative identity statement. The disciplined statement is that the object counts as N-flake exactly when one fixed family of contractive polygonal similarities is iterated indefinitely and its placement, copy count, and central-copy convention determine the limiting attractor
Manages Complexity¶
The abstraction compresses triangle, square and Vicsek variants, pentaflake, hexaflake, higher n, centered and uncentered rules, star-polygon forms, two- and higher-dimensional analogues, and finite approximants into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.
Compression can hide assumptions. A responsible use therefore declares n, regularity, orientation, contraction ratio, vertex placement, center inclusion, copy count, overlap, separation condition, finite iteration, and limiting dimension and returns to the full diagnostic whenever a convention or boundary case changes.
Abstract Reasoning¶
- Type the carrier. Establish a regular (n)-gon with \(n\geq3\), a declared set of similarity maps placing contracted copies at its vertices and possibly its center, and the unique compact attractor of that system and reject examples from a different problem. 2. Lock the rule. Express that one fixed family of contractive polygonal similarities is iterated indefinitely and its placement, copy count, and central-copy convention determine the limiting attractor independently of one notation or implementation.
Knowledge Transfer¶
Transfer within fractal geometry is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from For n equal to three with one half scale copies at the three vertices and no central copy, the attractor is the usual Sierpiński triangle. to A pentaflake convention places five vertex copies and one central copy of a regular pentagon at a scale chosen so neighboring copies touch without overlapping. demonstrates that continuity.
Relationships to Other Abstractions¶
Current abstraction N-flake Domain-specific
Parents (1) — more general patterns this builds on
-
N-flake is a kind of Fractal Geometry Prime
The proposed strict upward parent is
prime:fractal_geometry.
Hierarchy paths (5) — routes to 5 parentless roots
- N-flake → Fractal Geometry → Scale Invariance → Invariance
- N-flake → Fractal Geometry → Recurrence
- N-flake → Fractal Geometry → Scale
- N-flake → Fractal Geometry → Self-Organization
- N-flake → Fractal Geometry → Scale Invariance → Symmetry
Neighborhood in Abstraction Space¶
N-flake sits in a sparse region of the domain-specific corpus (62nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Extremal & Geometric Combinatorics (13 abstractions)
Nearest neighbors
- Map graph — 0.88
- Lattice graph — 0.86
- Expander graph — 0.85
- Baranyai's theorem — 0.85
- Triangulation (topology) — 0.85
Computed from structural-signature embeddings · 2026-09-08