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arxiv: 2312.01486 · v2 · pith:7KR77LWWnew · submitted 2023-12-03 · 🧮 math.MG · math.DS· math.NT

Elementary fractal geometry. 4. Automata-generated topological spaces

Christoph Bandt This is my paper

Pith reviewed 2026-05-24 05:14 UTC · model grok-4.3

classification 🧮 math.MG math.DSmath.NT
keywords automata-generated spacestopological self-similarityfinite automataself-affine tilesfractal geometryaddress systemsfinite type fractals
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The pith

Axiomatic automata generate topological spaces that are self-similar by construction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper joins research on automata for multiple addresses in number systems with the topology of self-affine tiles by giving an axiomatic definition of automata that output topological spaces. It establishes that these spaces are topologically self-similar, with the repetition property following directly from the axioms rather than from geometric embedding. Two algorithms are described: one computes automata for all equivalent address k-tuples starting from the double-address case, and the second builds finite topological spaces that approximate the infinite generated space. The paper also considers conditions under which the generated spaces can appear as self-similar sets.

Core claim

Finite automata can be defined axiomatically so that they generate topological spaces, and these spaces are topologically self-similar with the self-similarity property following from the axioms alone.

What carries the argument

Axiomatic definition of automata that generate topological spaces, from which topological self-similarity derives without further assumptions.

If this is right

  • The automaton handling all k-tuples of equivalent addresses is obtained algorithmically from the automaton for double addresses.
  • Finite topological spaces that approximate the generated space can be constructed by a second explicit algorithm.
  • Automata-generated spaces can be realized as self-similar sets under suitable conditions.
  • The construction applies directly to self-affine tiles and finite-type fractals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method supplies a systematic way to produce new families of topological spaces with built-in self-similarity beyond those arising from geometric contractions.
  • Symbolic address systems and topological invariants can be linked more directly through the automata, potentially aiding classification of spaces with finite type.
  • Concrete implementations of the two algorithms would allow computational enumeration of small examples and testing of topological properties.

Load-bearing premise

The chosen axioms for the automata are sufficient by themselves to guarantee that the output forms a topological space whose self-similarity follows from the axioms.

What would settle it

An explicit automaton constructed from the given axioms whose generated object is a topological space but fails to be topologically self-similar would disprove the claim.

Figures

Figures reproduced from arXiv: 2312.01486 by Christoph Bandt.

Figure 1
Figure 1. Figure 1: The automaton for double addresses of binary numbers. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Topological automata for two number systems. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Adding digit 2 to the automaton of binary numbers we generate a disconnected [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: A Hata tree is generated from the automaton in Figure [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A complete automaton with 2 self-inverse states and 3 digits. Top right: Without [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Graph-directed topological self-similarity for an automaton without property 4. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Sierpi´nski tetrahedron and its automaton, 0 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The complete automaton for the 2 × 2 square. With more labels, the graph of this automaton applies to the k × k square, and also to fractal squares. The square. The 2×2 square is the basis of commonly used ‘quadtree’ methods for image coding and processing. We use digits 0, 1, 2, 3 as indicated in [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A fractal square which does not admit states [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: In Figures 4 and 5 we have only triple addresses and no proper double addresses. Here are the automata G3 for the triple addresses which can be constructed by the method in Section 6. Label iii is abbreviated as i. Usually, incomplete automata are much simpler than the complete version, as shown by [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: An incomplete automaton generating a self-similar triangle. The states of the [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Automata G3 and G4 for example 6.2. Ignoring loops at oo and paths through bo, there are eight triple addresses that correspond to the common endpoints of three con￾secutive third-level triangles in [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Top: a simple incomplete automaton for the 2 [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Top: a self-similar set X and its neighbor automaton. Bottom: a magnifi￾cation which shows the fractal tiling structure. The IFS maps fk are determined by the automaton. This seems curious since the neighbor maps involve an irrational angle. Proposition 9.5. The automaton in [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
read the original abstract

Finite automata were used to determine multiple addresses in number systems and to find topological properties of self-affine tiles and finite type fractals. We join these two lines of research by axiomatically defining automata which generate topological spaces. Simple examples show the potential of the concept. Spaces generated by automata are topologically self-similar. Two basic algorithms are outlined. The first one determines automata for all $k$-tuples of equivalent addresses from the automaton for double addresses. The second one constructs finite topological spaces which approximate the generated space. Finally, we discuss the realization of automata-generated spaces as self-similar sets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper axiomatically defines finite automata that generate topological spaces by joining prior lines of research on automata for multiple addresses in number systems and topological properties of self-affine tiles and finite-type fractals. It establishes that the generated spaces are topologically self-similar, outlines two algorithms (one determining automata for k-tuples of equivalent addresses from the double-address case, and one constructing finite topological approximations), provides simple illustrative examples, and discusses realization of these spaces as self-similar sets.

Significance. If the constructions hold, the work supplies an axiomatic, parameter-free framework linking automata theory directly to topological self-similarity in fractal geometry. The explicit algorithms and examples constitute reproducible computational tools, and the absence of ad-hoc parameters or fitted quantities strengthens the approach for studying self-similar topological structures.

minor comments (2)
  1. [Abstract] The abstract states that the spaces are topologically self-similar with properties following from the axioms, but a one-sentence reminder of the precise definition of topological self-similarity (used in the main text) would improve accessibility without altering the claim.
  2. The discussion of realization as self-similar sets at the end would benefit from an explicit statement of which topological properties are preserved under the embedding or realization map, even if only in outline form.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; axiomatic derivation is self-contained

full rationale

The paper joins prior lines of research by providing an axiomatic definition of automata that generate topological spaces, then states that the generated spaces are topologically self-similar with properties following from the axioms. No equations, parameters, or self-citations are shown to reduce the central claim to a fit or prior result by construction. The two algorithms are presented as computational tools derived from the axioms rather than additional hypotheses. The derivation chain remains independent of fitted inputs or self-referential loops, consistent with standard axiomatic mathematics.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper rests on an axiomatic definition whose details are not visible in the abstract; no free parameters, invented entities, or additional axioms are mentioned.

pith-pipeline@v0.9.0 · 5621 in / 1057 out tokens · 27079 ms · 2026-05-24T05:14:19.085577+00:00 · methodology

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Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · 1 internal anchor

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