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I've placed most of the following mathematical illustrations in the public domain. If you see an illustration I've drawn for one of my other web sites, and want it to be uploaded to the commons for whatever reason, please ask by email: if I upload it myself, it can be more properly licensed, and it's likely that I still have the vectorized source instead of the rasterized version I use for web images.
Configurations and point-line incidences
Complete quadrangle and dual complete quadrilateral
Desargues configuration as two mutually inscribed pentagons
Point sets with fewer than n/2 ordinary lines
Geometric graph theory
Grid bracing and its graph-theoretic analysis
Flexibility of an unbraced square grid in the grid bracing problem
Greedy geometric spanner with distance ratio 1.1
Greedy geometric spanner with distance ratio 2
Integral Fáry embedding of the octahedron
Slope number of bounded-degree planar graphs
Empty regions for the β-skeleton
Ageev's 5-chromatic circle graph
A 3-colored triangulation for Fisk's proof of the art gallery theorem. From my WG09 slides.
Cayley graphs and symmetric graphs
The smallest zero-symmetric graph with only two edge orbits
The graph of the 3-3 duoprism
distinguishing coloring of a hypercube
1-planar 8-crossing Nauru graph
Hamiltonicity of the truncated octahedral graph
18-vertex zero-symmetric graph
12-vertex crown graph
K5 and K3,3 as minors of the Petersen graph
K2,2,2,2 as a 1-planar graph
Butterfly network as a multitree
Johnson graph J(5,2)
The path formed by the Steinhaus–Johnson–Trotter algorithm
Edge coloring of a complete graph
The Clebsch graph
One-crossing drawing of K3,3
A generalized Petersen graph with only three Hamiltonian cycles
The Shrikhande graph in Lombardi style
The Folkman graph in Lombardi style
The octahedron as pancyclic graph
The odd graph O4
3-crossing drawing of the Heawood graph
Toroidal Nauru graph
Genus-4 Nauru graph
The Shrikhande graph embedded on a torus.
The Gray graph.
The Möbius–Kantor graph embedded symmetrically on a torus.
The cube-connected cycles of order 3.
Two views of the Möbius ladder graph M16
Three-dimensional binary De Bruijn graph
Paley graph, of order 13, as a circulant
Transforming a dessin d'enfant into gluing instructions for a Riemann surface
The dessin d'enfant for the sextic monomial p(x)=x6
Two conjugate dessins d'enfants
Miscellaneous graph theory and graph drawing
Tree with 480 symmetries, example for Jordan–Pólya number
Example for Grötzsch's theorem
The densest possible planar locally linear graphs
Illustration for the proof of Ore's theorem
Hamiltonian cycle in which nonadjacent pairs of vertices have degrees summing to at least n, for Ore's theorem
Lombardi drawing of the Golomb graph
Graphic matroid parity
Tangled Kempe chains in the Errera graph
Tangled Kempe chains in the Poussin graph
A Halin graph without any 8-cycles
Bidirected graph features
A 2-degenerate graph and its 2-core
Tietze's graph on a Möbius strip
Partition of a line graph into cliques
A bramble in a 3x3 grid graph
List coloring for K3,27
Moser spindle as a pseudotriangulation
A planar Lombardi drawing of the Frucht graph
A Hamiltonian cycle in the Dürer graph
The Frucht graph in Lombardi drawing style
The Chvátal graph in Lombardi style
An apex graph
The Petersen family
Forbidden minors for branchwidth three
Domination in a product of stars, for Vizing's conjecture
Subdivision of K5
Forming a Schlegel diagram from shadows and light
The Rado graph
Finding matchings in claw-free graphs
An augmenting path in a claw-free graph
A forbidden subgraph for comparability graphs
Construction of a distance-hereditary graph
The Buneman graph, a median graph representing maximum-parsimony evolutionary relationships.
The retraction of a cube onto a median graph.
The median of three vertices in a median graph.
A cograph described by a cotree.
A forbidden subgraph for the line graphs of hypergraphs
A graph and its tree decomposition.
An interval graph.
The butterfly and diamond, forbidden minors for pseudoforests.
Fáry's theorem, induction step of proof
An aperiodic graph.
A graph that is not aperiodic as all cycles are divisible by three.
The Turán graph T(13,4).
Thue number of the 5-cycle is four
Wheel graphs with 4 to 9 vertices.
Beineke's nine forbidden line graphs.
Partition of the torus into seven mutually adjacent regions, for Heawood conjecture.
The Grötzsch graph.
König's theorem proof
König's theorem example
Erdős–Gyárfás conjecture, Markström's 4- and 8-cycle-free graph
Opaque forests for a circle
Collinear form of Cairo pentagonal tiling
Equilateral form of Cairo pentagonal tiling
Tresillo rhythm as a polygon
Curve-shortening of a convex curve
Visual proof of Balinski's theorem
The Reuleaux triangle in a cluster of four soap bubbles
Heesch's anisohedral tiling
89th stage of toothpick sequence
Overlaid Pythagorean tilings
Coloring argument for De Bruijn's theorem
Non-harmonic packing for De Bruijn's theorem
Circles meeting at the isodynamic point
Construction of the isodynamic point
Similarity tiling by Koch snowflakes
Aperiodic section of the Pythagorean tiling
A cube dissected into orthoschemes.
A Davenport–Schinzel sequence from a lower envelope of line segments
The Fermat–Apollonius circle of an ellipse.
The trisected perimeter point of a 3-4-5 right triangle.
Four levels of the Z-curve
Z-curve via interleaved binary coordinates
The Brocard point of a triangle
Chao's characterization of tangential quadrilaterals
Convex layers and a halfspace. For an example in fractional cascading.
Happy Ending problem, eight points with no pentagon
The Szilassi polyhedron. From the junkyard.
The Nagel point of a triangle.
Heesch's problem, Amman's dented hexagon. From the junkyard.
Happy Ending problem, quadrilaterals in five-point sets.
Erdős–Szekeres theorem, geometric interpretation as monotone path
Number theory and combinatorics
Growth rate of the Moser–de Bruijn sequence
Addition of numbers in the Moser–de Bruijn sequence
A cap set
Bijection between binary trees and stack-sortable permutations
Lower bound construction for the Erdős–Ko–Rado theorem
Trees counted by the Wedderburn–Etherington numbers
Dickson's lemma applied to the hyperbola xy ≥ 9
Trees counted by the ordered Bell numbers
15=4+5+6 is a polite number
Graphical demonstration of a solution to Znám's problem.
Graphical demonstration that 1 = 1/2 + 1/3 + 1/7 + 1/43 + ...; see Sylvester's sequence.
Divisibility of regular numbers.
2-dimensional modular lattice
The extreme case for the 1/3–2/3 conjecture
Example for Birkhoff's representation theorem.
Dilworth's theorem, transformation from chain decomposition to bipartite matching.
The 13 possible strict weak orderings on a set of three elements.
Three views of an antimatroid.
Critters transition rule
Rule 90 gate array
Trees in Rule 90
Day & Night, Bell's p256 butterfly gun
One-dimensional cyclic cellular automaton.
Two-dimensional cyclic cellular automaton.
The space rake.
Algorithms and data structures
An SPQR tree.
Using a Cartesian tree for range searching
Floyd's "tortoise and hare" cycle detection algorithm.
Symmetric Whitehead link
Six-component "rubberband" Brunnian link
Algebraic link diagram for the Borromean rings
A train track on a triple torus.
A switch in a train track.
Diagram for a triangulated category
Most of my photos here are licensed under Creative Commons Attribution Share-Alike. See Flickr for some of my other photos, and my web site for almost all of them; if it's not listed here, it's probably not yet publically licensed, but I may be willing to share anyway.
Bridge across North Lake, Woodbridge, Irvine, California
Theoretical computer scientists
Wall poems in Leiden
"Balancing", by John Hooper