1 Field-valued patterns
Throughout this document, we will study frieze patterns of finite height. This terminology (as compared to finite width or finite order) is unconventional but more convenient for formalisation. For us, the height of a frieze pattern corresponds to the number of rows, including the rows of ones but excluding the rows of zeros. Throughout this section, we fix an arbitrary field \(F\).
Fix \(n \in \mathbb {N}^*\). A map \(f : \{ 0,1,\ldots , n,n+1\} \times \mathbb {Z} \longrightarrow F\) is called an \(F\)-valued pattern of height n if,
1) for all \(m \in \mathbb {Z}\), \( f(0,m) = f (n+1,m) = 0\),
2) for all \(m \in \mathbb {Z}\), \( f(1,m) = f (n,m) = 1\), and
3) for all \((i,m) \in \{ 1,2,\ldots , n\} \times \mathbb {Z}\), we have
An \(F\)-valued pattern \(f\) of height \(n\) is said to be nowhere zero if \(f(i,m) \neq 0\), for all \(i \in \{ 1,\ldots , n\} \) and for all \(m \in \mathbb {Z}\).
Let \(f\) be a nowhere-zero \(F\)-valued pattern of height \(n\). For all \(m\), we have
We begin by proving the first statement. That is, we prove
for \( i \in \{ 0, \ldots , n-1\} \). We do so by induction on \(i\).
Base case \(P_0\): We have that for all \(m \in \mathbb {Z}\), \(f(2,m) f(1,m) - f(0,m) = f (2,m+1)*1 - 0 = f (2,m)\).
Inductive hypothesis. Suppose that our claim holds for some \(i \in \{ 0,\ldots , n-2\} \) fixed. Then,
Since \(f\) is nowhere-zero, we may divide both sides of the equation by \(f (i+1,m+1)\) to obtain the desired equality.
The second statement is proved almost identically. Namely, we prove
by induction on \(i\), starting with \(i = n-1\) and proving the inductive step \(Q_i \Rightarrow Q_{i-1}\).
Base case \(Q_{n-1}\): for all \(m \in \mathbb {Z}\), \( f (n-1,m) f (n,m-1) - f(n+1,m-2) = f (n-1,m)*1 - 0 = f (n-1,m)\).
Inductive hypothesis. Suppose that \(Q_{i+1}\) holds for some fixed \(i \in \{ 0,\ldots , n-2\} \). Then,
Again since \(f\) is nowhere-zero, dividing by \(f (i+2,m-1)\) on both sides we obtain \(Q_i\).
Let \(f\) be a nowhere-zero \(F\)-valued pattern of height \(n\). Then, for all \(m \in \mathbb {Z}\) and all \(i \in \{ 0,\ldots , n+1\} \), we have
We prove a stronger statement, called the glide symmetry of frieze patterns. First, consider the map \(\rho _n: \{ 0,1,\ldots , n+1\} \times \mathbb {Z} \longrightarrow \{ 0,1,\ldots , n+1\} \times \mathbb {Z}\) given by
We show that every nowhere-zero \(F\)-valued pattern of height \(n\) is \(\rho _n\)-invariant, i.e. satisfies
The proposition will then follow by observing that \(\rho _n^2 : (i,m) \mapsto (i,m+n+1)\). Thus, consider the statement
where \(i \in \{ 0, \ldots , n+1\} \). To prove that \(P_i\) holds for all \(i\), it is sufficient to prove that \(P_0, P_1\) hold, and that \(P_i \wedge P_{i+1} \Rightarrow P_{i+2}\).
\(P_0:\) for all \(m \in \mathbb {Z}, f(0,m) = 0 = f(n+1,m)\).
\(P_1:\) for all \(m \in \mathbb {Z}, f(1,m) = 1 = f(n,m+1)\).
Now suppose we are given \(i \in \{ 0,1,\ldots , n-1\} \) such that \(P_i\) and \(P_{i+1}\) hold. Then, for any fixed \(m \in \mathbb {Z}\), we have
Let \(f\) be a nowhere-zero \(F\)-valued pattern of height \(n\). Then, \({\rm Im}(f) := \{ f (i,m) : i \in \{ 1,\ldots , n\} , m \in \mathbb {Z}\} \) is a finite set.
Consider the finite set \(\mathcal{D} = \{ (i,m) : i \in \{ 1,\ldots , n\} , m \in \{ 0,\ldots , n\} \} \). By Proposition 1.3,
and the right-hand side is obviously finite.