Friezes

Some (mostly on-line) resources

Books:

H.S.M.Coxeter, Regular complex polytopes, Cambridge University Press, Cambridge, 1991.

Introductory papers:

J.H.Conway, H.S.M.Coxeter, Triangulated polygons and frieze patterns, ---- (Questions about frieze patterns).
J.H.Conway, H.S.M. Coxeter, Triangulated polygons and frieze patterns (continued), ---- (Solutions).
H.S.M.Coxeter, Frieze patterns. ----
H.S.M.Coxeter, J.F.Rigby, Frieze patterns, triangulated polygons and dichromatic symmetry . ----
D. Broline, D.W. Crowe, I.M.Isaacs The geometry of frieze patterns. ----
T. Holm Friezes and tilings.
M. Pressland From frieze patterns to cluster categories.
S. Morier-Genoud, Lecture notes on Integrable Systems and Friezes. ----
E. Smirnov Friezes and continued fractions (in Russian).

Introductory videos:

S. Tabachnikov, Frieze Patterns, Numberphile, 2019. ----
S. Tabachnikov, Frieze Patterns, extra Numberphile, 2019. ----

Survey paper:

S. Morier-Genoud, Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics. ----

Recent results:

S. Morier-Genoud, V. Ovsienko, S. Tabachnikov, SL₂(Z)- tilings of the torus, Coxeter-Conway friezes and Farey Triangulations. ----
C. Bessenrodt, T. Holm, P. Jorgensen, Generalized frieze pattern determinants and higher angulations of polygons. ----
C. Bessenrodt, T. Holm, P. Jorgensen, All SL₂-tilings come from infinite triangulations. ----
F. Bergeron, C. Reutenauer, SL_k-tilings of the plane. ----
I. Assem, C. Reutenauer, D. Smith, Friezes. ----
C. Bessenrodt, Conway- Coxeter friezes and beyond: polynomially weighted walks around dissected polygons and generalized frieze patterns . ----
M. Cuntz, On wild Frieze patterns. ----
M. Cuntz, T. Holm, Subpolygons in Conway-Coxeter frieze patterns. ----
T. Holm, P. Jorgensen, A p-angulated generalisation of Conway and Coxeter's theorem on frieze patterns. ----
V. Ovsienko, Partitions of unity in SL(2,Z), negative continued fractions, and dissections of polygons. ----
I. Short, Classifying SL₂-tilings. ----
A. Felikson, O. Karpenkov, K. Serhiyenko, P. Tumarkin, 3D Farey graph, lambda lengths and SL2-tilings. ----
I. Short, M. Van Son, A. Zabolotski, Frieze patterns and Farey complexes. ----
A. de St. Germain, On upper bounds of frieze patterns (paper and the project formalising the proof in Lean 4).

Friezes and cluster algebras:

K. Baur, R. Marsh, Frieze patterns for punctured discs. ----
J. Propp, The combinatorics of frieze patterns and Markoff numbers. ----
S. Morier-Genoud, V. Ovsienko, S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons. ----
S. Morier-Genoud, Arithmetics of 2-friezes. ----
V. Ovsienko, S. Tabachnikov, Coxeter's frieze patterns and discretization of the Virasoro orbit. ----
B. Fontaine, Non-zero integral friezes. ----
B. Fontaine, P-G. Plamondon, Counting friezes in type D_n. ----
E. Gunawan, R. Schiffler, Frieze vectors and unitary friezes. ----
I. Canakci, A. Felikson, A. Garsia Elsener, P. Tumarkin, Friezes for a pair of pants. ----
A. Felikson, P. Tumarkin, Frizes from surfaces and Farey triangulation. ----

Non-commutative friezes:

M. Cuntz, T. Holm, P. Jorgensen, Noncommutative frieze patterns with coefficients. ----

Y-friezes:

A. de St. Germain, When frieze patterns meet Y-systems: Y-frieze patterns. ----
A. de St. Germain, Y-frieze patterns. ----

Non-integer friezes:

Amy Tao, Zhichun Joy Zhang, Friezes over Z[√2] and Z[√3]. ----
M. Cuntz, T. Holm, P. Jorgensen, Frieze patterns with coefficients. ----
M. Cuntz, T. Holm, C. Pagano, Frieze patterns over algebraic numbers. ----
E. Banaian, L. Farrell, A. Tao, K. Wright, J. Zh. Zhang, Friezes over Z[√2] . ----

Pentagramma Mirificum:

H.S.M.Coxeter, Frieze patterns, ----
V. Schechtman, Pentagramma Mirificum and elliptic functions. ----

Introduction to cluster algebras:

L. K. Williams, Cluster algebras: an introduction, ---- (see Section 2.1)
S. Fomin, N. Reading, Root systems and generalized associahedra, ----

Some slides:

Continued Fractions and SL2-tilings LMS Northern Regional Meeting and Workshop, slides
C. Bessenrodt, Weighted walks around dissected polygons – Conway-Coxeter friezes and beyond, ----