Triangulations in geometry: from Ptolemy to Teichmüller

Snapshots of modern mathematics from Oberwolfach

Triangulations in geometry: from Ptolemy to Teichmüller

Ptolemy’s theorem is a classical result from ancient Greek mathematics, concerning the lengths of sides and diagonals of a polygon drawn in a circle. In this snapshot, I will explain why this theorem is still important today through its role in Teichmüller theory, a subject which seeks to describe all possible “shapes” of a surface with boundary.

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Mathematical subjects

Algebra and Number Theory
Geometry and Topology

Author(s)

Matthew Pressland

License

DOI (Digital Object Identifier)

10.14760/SNAP-2026-004-EN

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snapshot-2026-004.pdf

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snapshots: overview

Matthew Pressland
Triangulations in geometry: from Ptolemy to Teichmüller
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The five Platonic solids and their connection to root systems
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Algebras and quantum games
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Felder und Räume: Symmetrie und Lokalität in Mathematik und theoretischen Wissenschaften
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The periodic tables of algebraic geometry
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Patterns and waves in theory, experiment, and application
Benjamin Dozier
Closed geodesics on surfaces
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Route planning for bacteria
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Characterizations of intrinsic volumes on convex bodies and convex functions
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Fadoua Balabdaoui, Yulia Kulagina
The Enigma behind the Good–Turing formula
Anna Haensch
Reflections on hyperbolic space
Isaac Goldbring
Ultrafilter methods in combinatorics
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Baptiste Rognerud
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The Mathematics of Fluids and Solids
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Martin Buhmann, Janin Jäger
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Topological Complexity, Robotics and Social Choice
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Topological recursion
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Justyna Szpond
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Molecular quantum dynamics
Marco Radeschi
Closed geodesics on surfaces and Riemannian manifolds
Christian Engwer, Markus Knappitsch
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Lukas Pottmeyer
News on quadratic polynomials
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Winkeltreue zahlt sich aus
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Laurent Bartholdi
Profinite groups
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Steven Klee
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Nikolai Nowaczyk
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Lior Bary-Soroker
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Maria Dostert, Stefan Krupp, Jan Hendrick Rolfes
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Ben Schweizer
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Raluca Eftimie
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Tom Solomon
Chaos and chaotic fluid mixing
Thorsten Holm
Friezes and tilings
George Willis
Zero-dimensional symmetry
Christine Breiner
Minimizing energy
Diana Davis
Billiards and flat surfaces
Peter Benner, Hermann Mena, René Schneider
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Gregory Knese
Operator theory and the singular value decomposition
Alain Valette
The Kadison-Singer problem
Ragnar-Olaf Buchweitz, Eleonore Faber
Swallowtail on the shore
Howell Tong
Statistics and dynamical phenomena
Brian Harbourne, Thomas Szemberg
Arrangements of lines
Bruce Reznick
What does ">" really mean?
Harald Helfgott
The ternary Goldbach Problem
Wolfgang Lerche
Matrixfaktorisierungen
John E. McCarthy
Dirichlet Series

Mathematical subjects

Algebra and Number Theory
Analysis
Didactics and Education
Discrete Mathematics and Foundations
Geometry and Topology
Numerics and Scientific Computing
Probability Theory and Statistics

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Chemistry and Earth Science
Computer Science
Engineering and Technology
Finance
Humanities and Social Sciences
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