Nonperiodic Infinite Frieze Patterns

• Nonperiodic infinite frieze patterns are two-dimensional arrays of numbers defined by local recurrence relations and lack translational symmetry.
• They are realized via bijections with triangulations of infinite surfaces such as the infinity-gon, with quiddity sequences uniquely determining the pattern.
• Their study connects cluster theory, Penrose tilings, and wild higher-rank friezes, revealing rich combinatorial, geometric, and categorical structures.

Nonperiodic infinite frieze patterns are two-dimensional infinite arrays of numbers—typically integers or elements of a commutative ring—characterized by local recurrence relations generalizing those of classical finite frieze patterns, but lacking any translational (periodic) symmetry. Their combinatorial, geometric, and categorical structure forms a nexus between cluster theory, tiling models such as Penrose tilings, and representation theory of certain curve singularities. Central to their analysis are quiddity sequences, recurrence relations, and bijections with triangulations of infinite surfaces such as the infinite strip or the completed infinity-gon.

1. Structure and Defining Relations

An infinite (fountain) frieze is specified by an array $$\{m_{i,j} \in R\setminus\{0\} \mid i \leq j,\, i,j \in \mathbb{Z}\}$$, with $R$ a commutative ring or integral domain, subject to the following defining conditions (Esentepe et al., 10 Nov 2025, Maldonado, 2022):

• $m_{i,i} = 0$,
• $m_{i,i+1} = 1$ (or nonzero coefficient $c_{i,i+1}$ in generalized settings),
• the unimodular (diamond) relation:

$$m_{i,j}\,m_{i+1,j+1} - m_{i+1,j} m_{i,j+1} = 1 \quad \text{for all } i < j.$$

More generally, for patterns “with coefficients,” the diamond relation becomes

$$m_{i,j} m_{i+1, j+1} - m_{i+1, j} m_{i, j+1} = x_{i,i+1} x_{j,j+1}$$

for invertible $x_{i,i+1} \in R$.

A frieze is called periodic if $\exists\, p > 0$ such that $m_{i+p, j+p} = m_{i,j}$ for all $i \leq j$; nonperiodic otherwise. Nonperiodicity is often detected via the quiddity sequence $\left( m_{i, i+2} \right)_{i \in \mathbb{Z}}$: if this sequence is not eventually periodic, the frieze is nonperiodic (Esentepe et al., 10 Nov 2025).

2. Geometric Realizations: Triangulations and the Infinity-Gon

Nonperiodic infinite friezes admit geometric interpretation via bijections with triangulations of infinite surfaces. For SL$_2$ friezes, the seminal correspondence is with admissible triangulations of the infinite strip $\mathbb{V}$, whose lower and upper boundaries have marked points indexed by $\mathbb{Z}$ (and possibly special points at infinity) (Baur et al., 2015, Smith, 2015). Each triangulation $T$ gives rise to a quiddity sequence,

$$a_i = |\{\text{triangles of } T \text{ incident with } (i,0)\}|$$

which uniquely determines the frieze via recurrence

$m_{ij} = a_j\, m_{i, j-1} - m_{i, j-2}.$

A triangulation is admissible if each lower-boundary marked point is incident to only finitely many arcs, and, for the “infinity-gon” case ($M_2 = \varnothing$), these coincide with friezes having “enough ones.”

In higher categorical settings, notably for the completed infinity-gon (a topological disc with boundary points labeled by $\mathbb{Z} \cup \{\infty\}$), triangulations correspond to indecomposable objects in the Frobenius category $$\mathcal{C}_2 = \mathrm{CM}_{\mathbb{Z}}(k[x,y]/(x^2))$$, the category of $\mathbb{Z}$-graded maximal Cohen–Macaulay modules over the $A_\infty$ singularity (Esentepe et al., 10 Nov 2025). The combinatorics of noncrossing arcs encode both frieze data and categorical structure.

3. Classification: Periodic vs. Nonperiodic Patterns

A crucial result is the complete classification of (possibly nonperiodic) infinite frieze patterns in terms of both their quiddity sequences and geometric realizations:

• A doubly-infinite integer sequence $(a_i)_{i \in \mathbb{Z}}$ is the quiddity row of an infinite frieze if and only if there are no two consecutive 1’s (to avoid zero entries from recurrence) and a finite reduction algorithm terminates in either an everywhere $\geq 2$ sequence or a finite sequence (Baur et al., 2015).

Periodic friezes arise only when the quiddity row is periodic, which corresponds to triangulations with a finite number of upper boundary points (i.e., they descend to triangulations of an annulus), and can be realized as patterns on finite cyclic surfaces (Smith, 2015). Nonperiodic friezes, by contrast, correspond to triangulations with infinitely many upper marked points and no translation symmetry, and cannot be realized in any finite annulus. Such nonperiodic examples include both patterns with non-repeating but structured quiddity (e.g., quasiperiodic, combinatorially aperiodic) and those with more chaotic growth.

4. Nonperiodic Infinite Friezes with Coefficients and Matrix Theory

Infinite frieze patterns with coefficients generalize the classical setting by incorporating variable edge coefficients; the key frieze relations and nonperiodicity extend naturally. For such patterns, the “frieze matrix” formalism is essential: every finite slice of an (even nonperiodic) infinite frieze pattern with coefficients yields a symmetric matrix $M$ whose entries obey generalized Ptolemy relations (Maldonado, 2022). Each such matrix $M$ has a closed formula for its determinant: $$\det M = -(-2)^{n-2} f_{k,\,k+n-1} \prod_{r=k}^{k+n-2} x_r,$$ where $f_{i,j}$ are the frieze entries and $x_r$ the local coefficients. This formula holds regardless of periodicity, showing that nonperiodic infinite friezes retain structural rigidity at the level of finite sections.

A notable structural corollary is that the entries of the frieze are uniquely determined by the infinite sequence of nearest-neighbor coefficients $(x_i)$ and two initial rows, explicitly showing the freedom available in generating nonperiodic patterns (Maldonado, 2022).

5. Combinatorics, Cluster Categories, and Penrose Tilings

The categorical view, as surveyed by Esentepe–Faber, associates nonperiodic infinite friezes (especially those derived from Penrose tilings) with cluster-tilting subcategories of $\mathcal{C}_2$. Fountain triangulations—maximal noncrossing arc sets with a fountain at some integer—are in bijection with such subcategories. The extended cluster character $X^S$ assigns variables to modules/objects so that the resulting array $m_{a,b} = X^S((a,b))$ recovers an infinite (possibly nonperiodic) fountain frieze with coefficients (Esentepe et al., 10 Nov 2025): $$m_{i,i} = 0, \quad m_{i,i+1} = x_{i,i+1}, \quad m_{i,j} m_{i+1, j+1} - m_{i+1, j} m_{i, j+1} = x_{i,i+1} x_{j,j+1}.$$

Penrose tilings, which are themselves nonperiodic and can be represented by double-infinite $0$–$1$ sequences without consecutive $1$’s (up to tail-equivalence), correspond naturally to special right-fountain triangulations of the completed infinity-gon. This creates a bijection:

• Penrose tilings (up to isometry) $\longleftrightarrow$ special class of nonperiodic infinite frieze patterns (Esentepe et al., 10 Nov 2025).

The categorical machinery extends to generalize frieze relations, exchange sequences, and to provide for nonperiodic cluster combinatorics.

6. Higher Rank and "Wild" Infinite Friezes

The theory extends to higher-rank frieze patterns (SL$\,_k$-friezes). For $k>2$, the situation is more flexible: while tame SL$\,_k$ friezes are periodic and governed by Coxeter–Conway rigidity, wild nonperiodic friezes exist even with positive integer entries (Cuntz, 2015). A wild frieze is defined by the vanishing of certain $(k-1)\times(k-1)$ minors and the non-uniqueness of the extension from fixed initial data. Nonperiodicity arises via the freedom in choosing sequences of successive rows within allowed more general compatibility graphs $\Gamma_n$. This produces SL$_k$ friezes with unbounded, nonrepeating entries, and no periodic closure. For example, explicit SL$_3$ wild friezes have been constructed with infinitely many distinct positive integer entries, growing according to recurrences tied to eigenvalues of associated transfer matrices.

These wild patterns are structurally and combinatorially richer than the SL$_2$ case, and a complete classification is out of reach for $k\geq 3$ (Cuntz, 2015).

7. Open Problems and Directions

Outstanding questions focus on:

• Classification of nonperiodic infinite friezes with coefficients and their automorphism groups;
• Extensions to categories beyond $A_\infty$ (e.g., $D_\infty$, $E_\infty$), and to noncommutative singularities;
• Explicit combinatorial and algebraic characterization of higher rank (SL$_k$) wild nonperiodic friezes;
• The structure of automorphisms and symmetry-breaking in nonperiodic cases;
• Connections with the noncommutative geometry of Penrose tiling spaces and their algebraic invariants (Esentepe et al., 10 Nov 2025, Cuntz, 2015).

A plausible implication is the ongoing exploration of cluster structures and categorical models in nonperiodic tiling spaces, providing new frameworks for aperiodic order in algebra and geometry.

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Key References:

• "Penrose tilings, infinite friezes, and the $A_\infty$-singularity" (Esentepe et al., 10 Nov 2025)
• "Infinite friezes" (Baur et al., 2015)
• "On wild frieze patterns" (Cuntz, 2015)
• "Infinite friezes and triangulations of the strip" (Smith, 2015)
• "Frieze matrices and infinite frieze patterns with coefficients" (Maldonado, 2022)