Introduction to Frieze Patterns

• Frieze patterns: arrays of positive integers arranged in rows
• Structure: infinite row of 1s at top and bottom, with w non-trivial rows between (forming sandwich pattern)
• All entries must be positive integers (no zeros allowed)
• Diamond rule: for any diamond configuration with values a, b, c, d, must satisfy a×d - b×c = 1

Conway-Coxeter Theorem (1972)

• Discovered ~50 years ago by Conway and Coxeter
• Establishes bijection between frieze patterns of width w and triangulated polygons with w+3 vertices
• Published in unusual format: first paper posed questions, second paper provided answers
• Initially overlooked until cluster algebra theory emerged in early 2000s
• Rediscovered due to connections with total positivity and representation theory

Example: Hexagon Triangulation to Frieze (w=3)

• Hexagon with vertices numbered 1-6
• Fan triangulation chosen from 14 possible triangulations (Catalan number)
• Count triangles incident to each vertex: 4,1,2,2,2,1
• First row of frieze: 4,1,2,2,2,1 (with periodic continuation)
• Diamond rule determines all other entries automatically
• Results in symmetric pattern with same sequence appearing shifted in multiple rows
• Final row returns to all 1s, confirming valid frieze pattern

Arc-Based Construction Method

• Alternative approach using polygon arcs instead of vertex counting
• Arcs: all possible straight lines between vertex pairs (i,j where i < j)
• Total arcs: n choose 2 for n-gon
• Boundary arcs: consecutive vertices (always part of triangulation)
• Diagonal arcs: non-consecutive vertices (may or may not be in triangulation)
• Frieze coordinates correspond to arc positions
• Place 1s for all arcs in chosen triangulation, complete using diamond rule
• Produces identical frieze as vertex-counting method

Cutting Frieze Patterns and Polygons

• Demonstrates correspondence between cutting triangulated polygons and frieze patterns
• Example: octagon (n=8) with width 5 frieze
• Cut polygon along diagonal 1-4 creates quadrilateral + hexagon
• Corresponding frieze cut: identify 1-4 arc positions in frieze pattern
• Cutting path: from 1-4 position up to boundary, down to opposite 1-4, repeat
• Red rectangles contain entries corresponding to deleted arcs
• Result: two smaller friezes (width 1 for quadrilateral, width 3 for hexagon)
• Diamond rule preserved in resulting sub-friezes

Research Context and Applications

• Conway-Coxeter originally used triangle cutting for inductive proof
• Recent work extends to general cutting operations (collaboration with Bethany Marsh, Matt Pressland)
• Emerged from research on "triangulated categories" - byproduct of categorical work
• Cutting along non-triangulation arcs produces "friezes with coefficients"
• Active area of current research
• Connections to cluster algebras, representation theory, and categorical interpretations
• Geometric models help prove complex theoretical results in pure mathematics