Sturmian lattices and aperiodic tile sets

Basic object	Three Sturmian line families
Parameters	κ, η, α, ρ
Main theorem	Finite aperiodic set for each quadratic irrational α
Not claimed	A single aperiodic monotile

From balanced words to lines

A Sturmian word is the least complicated binary sequence that is still aperiodic. Its number of distinct length-n factors is n+1, and it is balanced: two equal-length factors differ in their number of 1s by at most one. Lower and upper mechanical words obtain the symbols by coding an irrational rotation; the slope gives the frequency of long gaps and the intercept gives their phase.^[13]^[73]

Akiyama, Hamada, and Ito turn this one-dimensional order into a planar lattice using three families of parallel lines at angles 0, 2π/3, and 4π/3. Coordinates take the mechanical-word form a(i)=iκ+η₀+⟦iα+ρ₀⟧, with analogous formulas for b(j) and c(k), subject to the zero-sum constraints η₀+η₁+η₂=ρ₀+ρ₁+ρ₂=0. Whenever i+j+k=0, one has |a(i)+b(j)+c(k)|=1/2, so relevant triples bound a uniformly small equilateral triangle instead of meeting at one point. Consecutive gaps are at least one. Irrational systems admit only the trivial period; rational cases are classified separately and are eventually or fully periodic except for explicitly described singular choices. The discontinuous parameter set is dense but has two-dimensional Lebesgue measure zero.^[13]^[73]

This line system was motivated by structure extracted from Smith Turtle tilings, but the construction stands as a separate finite-tile-set theory. Ref. 73 is a seven-page 2026 announcement whose full proofs live in ref. 13; cite both for completeness, not as independent confirmations.

The four parameter classes

Four kinds of data must be kept separate. κ fixes the minimum passage or gap scale; the vector η translates the three line families; the irrational α is the density of long gaps; and the intercept vector ρ arranges those gaps. The components of η and ρ obey zero-sum constraints. Changing α changes the long-range frequency, while changing ρ changes phase or boundary termination without changing that frequency.

Finite evidence can falsify a proposed Sturmian coding but cannot prove the infinite property by itself. Compute factor complexity, balance, long-gap frequency, and return words over growing windows. A single complexity value other than n+1, or a symbol-count discrepancy above one for equal-length factors, rules out Sturmian behavior for that coding.^[13]

Nuts, Bolts, and the density equation

The construction uses three annular Nuts, labelled S, M, and L, carrying Ammann bars that force the three line families. Disk-like Bolts constrain how frequently the three local classes occur. Their center densities must satisfy

δ(S) : δ(M) : δ(L) = (1−α)² : 2α(1−α) : α².

Thus α is recoverable from frequencies rather than assumed from a drawing. The proof organizes Bolt centers as Delone sets and uses bounded displacement: matched points may move, but by a distance bounded uniformly over the infinite set. Many-to-many bounded-displacement correspondences divide centers into bounded groups, which unfold into finitely many patch-tile shapes.^[73]

The theorem and a 29-tile example

The theorem states that for every quadratic irrational α there is a finite aperiodic tile set 𝒜(α) enforcing α or its Galois conjugate. Its expansion constant is a unit of the real quadratic field ℚ(α). A stronger cardinality estimate is Card(𝒜_λ)≤2λ+O(1). The mechanism is frequency plus bounded displacement; it does not assume that every tiling displays an obvious self-similar inflation. Self-similarity is not required by the proof: ref. 13 also constructs a tiling space with positive topological entropy.^[13]^[73]

For α=√6−2, Nuts enforce Ammann bars and Bolts enforce the density ratio (1−α)²:2α(1−α):α². A class containing 3S+2L, M, and the three Nuts forces (1−α)²:α²=3:2. The fundamental unit is 5+2√6, and unfolding the bounded groups gives 29 patch-tiles. These may be decorated, colored, or disconnected and use matching information. The result is therefore a finite aperiodic tile set, not a Hat- or Spectre-like monotile. The Nuts / Bolts / density ratio are combinatorial matching constraints in the papers, not a physical fastener kit, so this page keeps the quantitative statement rather than a decorative redraw.

Evidence and limits

Ref. 73 is a 2026 preprint note summarizing and extending the fuller treatment in ref. 13. The definitions, classification, and theorem are mathematical claims; no manufacturing, mechanical, or wave advantage follows. The construction is valuable because it shows how symbolic balance, planar line geometry, density invariants, and aperiodic matching rules fit together without collapsing those layers into “looks quasiperiodic.”