Friezes: resources

Friezes

Some (mostly on-line) resources

Here, we collect the links topicwise (each paper listed once only - for one of its topics)
Click here for the same papers in the alphabetic order

Books:

H.S.M.Coxeter, Regular complex polytopes, Cambridge University Press, Cambridge, 1974.

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. ----
D. Broline, D.W. Crowe, I.M.Isaacs, The geometry of frieze patterns. ----
H.S.M.Coxeter, J.F.Rigby, Frieze patterns, triangulated polygons and dichromatic symmetry . ----
C.-S. Henry, Coxeter Friezes and Triangulations of Polygons.
T. Holm, Friezes and tilings.
M. Pressland, From frieze patterns to cluster categories.
S. Morier-Genoud, Lecture notes on Integrable Systems and Friezes. ----
K. Baur, Frieze patterns of integers.
E. Smirnov, Friezes and continued fractions (see also original version in Russian).

Introductory videos:

S. Tabachnikov, Frieze Patterns, Numberphile, 2019. ----
S. Tabachnikov, Frieze Patterns, extra Numberphile, 2019. ----
E. Faber, A Fascination with Fractured Friezes Numberphile, 2024. ----

Survey paper:

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

Friezes and Farey triangulation:

S. Morier-Genoud, V. Ovsienko, S. Tabachnikov, SL₂(Z)- tilings of the torus, Coxeter-Conway friezes and Farey Triangulations. ----
V. Ovsienko, Partitions of unity in SL(2,Z), negative continued fractions, and dissections of polygons. ----
S. Morier-Genoud, V. Ovsienko, Farey boat I. Continued fractions and triangulations, modular group and polygon dissections . ----
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. ----
S. Benzaira, I. Short, M. Van Son, A. Zabolotskii, Enumerating tame friezes over ℤ/nℤ. ----

Friezes and cluster algebras, categorification:

P. Caldero, F. Chapoton, Cluster algebras as Hall algebras of quiver representations. ----
J. Propp, The combinatorics of frieze patterns and Markoff numbers. ----
I. Assem, C. Reutenauer, D. Smith, Friezes. ----
I. Assem, G. Dupont, Friezes and a construction of the euclidean cluster variables. ----
I. Assem, G. Dupont, R. Schiffler, D. Smith, Friezes, Strings and Cluster Variables. ----
L. Guo, On tropical friezes associated with Dynkin diagrams. ----
K. Baur, B. Marsh, Categorification of a frieze pattern determinant . ----
V. Ovsienko, S. Tabachnikov, Coxeter's frieze patterns and discretization of the Virasoro orbit. ----
T. Holm, P. Jorgensen, Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object . ----
T. Holm, P. Jorgensen, Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object, II . ----
K. Baur, E. Faber, S. Gratz, K. Serhiyenko, G. Todorov, Mutation of friezes. ----
K. Baur, E. Faber, S. Gratz, K. Serhiyenko, G. Todorov, Conway-Coxeter friezes and mutation: a survey . ----
E. Gunawan, G. Muller, Superunitary regions of cluster algebras. ----

Friezes from surfaces:

K. Baur, B.R. Marsh, Frieze patterns for punctured discs. ----
M. Tschabold, Arithmetic infinite friezes from punctured discs. ----
K. Baur, M. Parsons, M. Tschabold, Infinite friezes. ----
K. Baur, K. Fellner, M. Parsons, M. Tschabold, Growth behaviour of periodic tame friezes. ----
B. Fontaine, P-G. Plamondon, Counting friezes in type D_n. ----
E. Gunawan, G. Musiker, H. Vogel, Cluster algebraic interpretation of infinite friezes. ----
E. Gunawan, R. Schiffler, Frieze vectors and unitary friezes. ----
K. Baur, I. Canakci, K. Jacobsen, M. Kulkarni, G. Todorov, Infinite friezes and triangulations of annuli . ----
I. Canakci, A. Felikson, A. Garsia Elsener, P. Tumarkin, Friezes for a pair of pants. ----
K. Baur, L. Bittmann, E. Gunawan, G. Todorov, E. Yıldırım, Infinite friezes of affine type D . ----
A. Felikson, P. Tumarkin, Frizes from surfaces and Farey triangulation. ----

Further results on friezes of finite type

M. Cuntz, Frieze patterns as root posets and affine triangulations . ----
B. Fontaine, Non-zero integral friezes. ----
G. Muller, A short proof of the finiteness of Dynkin friezes. ----
A. de St. Germain, On upper bounds of frieze patterns (paper and the project formalising the proof in Lean 4).

Polygon dissections, infinite triangulations:

T. Holm, P. Jorgensen, SL₂-tilings and triangulations of the strip . ----
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. ----
C. Bessenrodt, Conway- Coxeter friezes and beyond: polynomially weighted walks around dissected polygons and generalized frieze patterns . ----
C. Conley, V. Ovsienko, Counting quiddities of polygon dissections. ----

Weak freizes:

I. Canakci, P. Jorgensen, Friezes, weak friezes, and T-paths . ----
T. Holm, P. Jorgensen, Weak friezes and frieze pattern determinants . ----

SL_k-tilings

F. Bergeron, C. Reutenauer, SL_k-tilings of the plane. ----
S. Morier-Genoud, V. Ovsienko, S. Tabachnikov, R. Schwartz, Linear difference equations, frieze patterns and combinatorial Gale transform. ----
M. Cuntz, On wild Frieze patterns. ----
K. Baur, E. Faber, S. Gratz, K. Serhiyenko, G. Todorov, Friezes satisfying higher SL_k-determinants . ----
Z. Peterson, K. Serhiyenko, SL_k-tilings and paths in Z^k. ----

Friezes with coefficients:

M. Cuntz, T. Holm, P. Jorgensen, Frieze patterns with coefficients. ----
M. Cuntz, T. Holm, Subpolygons in Conway-Coxeter frieze patterns. ----
M. Cuntz, Grassmannians over rings and subpolygons. ----
J. P. Maldonado, Frieze matrices and infinite frieze patterns with coefficients. ----

Non-integer friezes:

M. Cuntz, A combinatorial model for tame frieze patterns . ----
T. Holm, P. Jorgensen, A p-angulated generalisation of Conway and Coxeter's theorem on frieze patterns. ----
M. Cuntz, T. Holm, Frieze patterns over integers and other subsets of the complex numbers . ----
S. Morier-Genoud Counting Coxeter's friezes over a finite field via moduli spaces . ----
Amy Tao, Zhichun Joy Zhang, Friezes over Z[√2] and Z[√3]. ----
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] . ----
B. Böhmler, M. Cuntz, Frieze patterns over finite commutative local rings . ----

2-friezes, q-deformed friezes, superfriezes:

J-Ph. Burelle, G. Dupont, Quantum frieze patterns in quantum cluster algebras of type A . ----
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. ----
S. Morier-Genoud, V. Ovsienko, S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators. ----
S. Morier-Genoud, V. Ovsienko, Quantum real numbers and q-deformed Conway-Coxeter friezes . ----

Symplectic and Non-commutative friezes:

S. Morier-Genoud, Symplectic Frieze Patterns. ----
M. Cuntz, T. Holm, P. Jorgensen, Noncommutative frieze patterns with coefficients. ----
M. Cuntz, T. Holm, P. Jorgensen, Non-commutative friezes and their determinants, the non-commutative Laurent phenomenon for weak friezes, and frieze gluing. ----

Heronian friezes, spherical friezes:

S. Fomin, L. Setiabrata, Heronian friezes. ----
K. Waddle, Spherical friezes. ----
A. Sneperger, Some determinants and relations in Heronian friezes. ----

Y-friezes, additive friezes:

G. C. Shephard, Additive Frieze Patterns and Multiplication Tables. ----
A. de St. Germain, When frieze patterns meet Y-systems: Y-frieze patterns. ----
A. de St. Germain, Y-frieze patterns. ----

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
Frieze patterns in algebra, combinatorics and geometry CIRM 2025, slides
C. Bessenrodt, Weighted walks around dissected polygons - Conway-Coxeter friezes and beyond,

Software:

Triangulations and Friezes by Andrei Zabolotskii
Visual Cluster Algebras by Antoine de Saint Germain
Polygon triangulations and frieze patterns by Karin M. Jacobsen.