User:David Eppstein/Gallery

Mathematical illustrations[edit]

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[edit]

Suboptimal no-3-in-line placement by the method of Erdős
Dual arrangement to a regular pentagon
Rota's basis conjecture
Rota's basis conjecture
Vámos matroid
Möbius–Kantor configuration
Cremona–Richmond configuration
Reye configuration
Perles configuration
Hesse configuration
Hjelmslev's theorem
Schläfli double six
The Gray configuration
Non-Desargues configuration
Möbius–Kantor configuration
Complete quadrangle and dual complete quadrilateral
Desargues configuration as two mutually inscribed pentagons
No-three-in-line problem
Point sets with fewer than n/2 ordinary lines

Geometric graph theory[edit]

Grid bracing and its graph-theoretic analysis
Flexibility of an unbraced square grid in the grid bracing problem
Convex drawing
Greedy geometric spanner with distance ratio 1.1
Greedy geometric spanner with distance ratio 2
Triangle-free penny graph
Penny graph
The smallest cubic matchstick graph
Integral Fáry embedding of the octahedron
Indifference graph
Forbidden induced subgraphs of indifference graphs
Slope number of the Petersen graph
Slope number of bounded-degree planar graphs
Power diagram
Gabriel graph
Relative neighborhood graph
β-skeleton
Empty regions for the β-skeleton
String graph representation of a planar graph
An Erdős-Diophantine graph
Ageev's 5-chromatic circle graph
A circular-arc graph
A bitonic tour
An intersection graph of rectangles, with boxicity two.
Unit disk graph.
Hadwiger–Nelson problem: 7-coloring of the plane and 4-chromatic unit distance graph. From the junkyard.
A 3-colored triangulation for Fisk's proof of the art gallery theorem. From my WG09 slides.

Cayley graphs and symmetric graphs[edit]

The Johnson graph J(4,2)
$K_{6}$ drawn with three crossings
The smallest zero-symmetric graph with only two edge orbits
The graph of the 3-3 duoprism
Paley graph of order 9 and one of its triangles
7-vertex antihole graph
K₅ subdivision of a crown graph illustrating the Kelmans–Seymour conjecture
Hamiltonian decomposition
Oberwolfach problem
Distinguishing coloring of a 6-key keyring (6-vertex cycle graph)
distinguishing coloring of a hypercube
1-planar 8-crossing Nauru graph
Hamiltonicity of the truncated octahedral graph
18-vertex zero-symmetric graph
The unique edge coloring of G(9,2)
12-vertex crown graph
Perfect matchings in a complete graph, or chord diagrams
K₅ and K_3,3 as minors of the Petersen graph
K_2,2,2,2 as a 1-planar graph
Kuratowski subgraph
Cycle and its complement
Matchings in a complete graph
Butterfly network as a multitree
Johnson graph J(5,2)
The Clebsch graph labeled as in a construction for Keller's conjecture
The path formed by the Steinhaus–Johnson–Trotter algorithm
K_3,3 with each of its three color classes drawn as parallel line segments on distinct lines
Edge coloring of a complete graph
The Clebsch graph
Construction of the Clebsch graph from a hypercube
One-crossing drawing of K_3,3
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
Pappus graph
Brinkmann graph
3-crossing drawing of the Heawood graph
The Petersen graph and its complement
The Nauru graph drawn as a unit distance graph
The intersection graph of the faces of a cube
Toroidal Nauru graph
Genus-4 Nauru graph
Crown graphs
The Wagner graph
The Petersen graph as a Kneser graph
The Shrikhande graph embedded on a torus.
The Gray graph.
The Folkman graph.
A permutohedron.
The Möbius–Kantor graph as a unit distance graph.
The Möbius–Kantor graph embedded symmetrically on a torus.
The cube-connected cycles of order 3.
The Paley graph of order 9 as a perfect graph.
The Petersen graph as a Moore graph.
Two views of the Möbius ladder graph M₁₆
The De Bruijn graph as a line graph
Three-dimensional binary De Bruijn graph
Paley graph, of order 13, as a circulant
Desargues graph

Dessins d'enfants[edit]

A dessin d'enfant
Transforming a dessin d'enfant into gluing instructions for a Riemann surface
The dessins d'enfants for the Chebyshev polynomials
The dessin d'enfant for the sextic monomial p(x)=x⁶
Two conjugate dessins d'enfants

Miscellaneous graph theory and graph drawing[edit]

A graph of cutwidth 2
K₅ triangle packing and covering
The 2-core of a graph
The diamond graph and its line graph
Petersen minor in the flower snark
A bipartite graph and its line graph
Complementary perfect graphs
The Fritsch graph and its dual map
26-fullerene graph with distance labels
Unit distance graph with 16 vertices and 40 edges
Giant component
Universal vertex
Dominated vertex
Tree with 480 symmetries, example for Jordan–Pólya number
Feedback arc set NP-completeness
Feedback arc set
3-colorings of a square, up to symmetry
Shuffle-exchange network
3-colorings of a path graph
Hamiltonian path in the Herschel graph
Medial graph of the Herschel graph
Linear arboricity
Example for Grötzsch's theorem
Barnette–Bosák–Lederberg graph
King's graph
The densest possible planar locally linear graphs
Sudoku graph
Layers of a planar drawing of the graph of a rhombic dodecahedron
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
Szymanski's conjecture
Sumner's conjecture
Sum coloring
Zero-divisor graph
Lombardi drawing of the Golomb graph
The extension property of the Rado graph
Xuong tree
Graphic matroid parity
Graph-encoded map
Ribbon graph
Half graph
Hypertree
Tangled Kempe chains in the Errera graph
Tangled Kempe chains in the Poussin graph
Poussin graph
3-leaf power
Clique-width construction
A Halin graph without any 8-cycles
Map graph
Turán's brick factory problem
Acyclic orientations
Topological ordering
Transitive closure
Condensation
Pagewidth
Book crossing number
Split decomposition
An equilibrium stress
Scorpion graph
Self-dual graph
Isospectral enneahedra
Bidirected graph features
A 2-degenerate graph and its 2-core
Crossing number and 1-planarity of Tietze's graph
Tietze's graph on a Möbius strip
RAC drawing
Rainbow coloring of a wheel graph
Book embedding
Nonisomorphic planar duals
A line perfect graph
Partition of a line graph into cliques
Zarankiewicz problem
strangulated graph
dominance drawing
An upward planar drawing
A planar DAG that is not upward planar
Tutte embedding
Nested triangles graph
Grid drawing of nested triangles graph
Linear embedding
Euler tour of a tree
Unordered binary trees with four leaves
Fleischner's theorem
Square of a graph
Vertex-face coloring
Henneberg construction
Skew partition
Tree-depth
Gallai–Hasse–Roy–Vitaver theorem
A bramble in a 3x3 grid graph
Grinberg's non-Hamiltonian 5-cyclic-edge-connected graph
List coloring for K_3,27
Moser spindle as a pseudotriangulation
A planar Lombardi drawing of the Frucht graph
A Hamiltonian cycle in the Dürer graph
A 3-regular 1-connected well-covered graph
3-regular 3-connected well-covered graphs
A well-covered graph
A multigraph that has degree six, edge multiplicity three, and requires nine colors in any edge coloring
A 3-regular planar graph that requires four colors in any edge coloring
Hajós construction
An Apollonian network
Graph toughness
The Frucht graph in Lombardi drawing style
The Chvátal graph in Lombardi style
A block graph
A trivially perfect graph
An apex graph
Robertson's irreducible apex graph
The Petersen family
The Herschel graph
Planar separator for a grid graph
Forbidden minors for branchwidth three
Branch-decomposition
Chvátal graph
A caterpillar tree
Forbidden minors for pathwidth 1
Maximal independent sets
Domination in a product of stars, for Vizing's conjecture
Subdivision of K₅
The Goldner–Harary graph
The Herschel graph
A Halin graph
Forming a Schlegel diagram from shadows and light
The Rado graph
An equitable coloring
A cycle double cover of the Petersen graph
A Fibonacci cube
Finding matchings in claw-free graphs
An augmenting path in a claw-free graph
A simplex graph
A forbidden subgraph for comparability graphs
A factor-critical graph
Thomassen's girth-3 hypohamiltonian graph
A hypohamiltonian graph
Hadwiger's conjecture
A clique-sum
A distance-hereditary graph
Construction of a distance-hereditary graph
An implication graph
The median graph representing the set of solutions to a 2-satisfiability instance
A squaregraph
Converting a triangle-free graph into a median 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.
The median of three vertices in a tree.
A cograph described by a cotree.
A forbidden subgraph for the line graphs of hypergraphs
A self-complementary graph
Forbidden minors for partial 3-trees.
A graph and its tree decomposition.
An interval graph.
Partition of the complete bipartite graph K_4,4 into three forests, showing that it has arboricity three.
A functional graph.
The butterfly and diamond, forbidden minors for pseudoforests.
A pseudoforest.
The Erdős–Faber–Lovász conjecture.
A 4x4 grid graph and one of its spanning trees.
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.
Series-parallel graph.
Split graph.
The rooted product of graphs.
The lexicographic product of graphs.
The tensor product of graphs.
The cartesian product of graphs.
Partition of the torus into seven mutually adjacent regions, for Heawood conjecture.
Cactus graph
The Grötzsch graph.
König's theorem proof
König's theorem example
Erdős–Gyárfás conjecture, Markström's 4- and 8-cycle-free graph
The Grötzsch graph as the Mycielskian of a 5-cycle graph.

Miscellaneous geometry[edit]

Pythagorean dense rational distance set
Stereographic projection of an spherical octahedron and its associated circle packing
Illustration for a proof of the Erdős–Anning theorem
Grid points within a convex polygon
Polygonalizations of a 3x3 grid
Boscovich's cardioid
Intersections of a line with a convex curve
Decagon simplicial arrangement
Convex curve
Three circles associated with an antiparallelogram
Tangent circles to the extended sides of an antiparallelogram
Two circles tangent to the extended sides of a non-convex kite
The inscribed circle and ex-tangential circle of a kite
A kite and its dual isosceles trapezoid
Triaugmented triangular prism net
Triaugmented triangular prism
$A fractal rosette of Penrose kites$

A fractal rosette of Penrose kites
Unflippable polygon
16 polygonalizations of a set of six points
Planar double bubbles
Crease pattern for Schwarz lantern
Empty regions for the Euclidean minimum spanning tree
Mrs. Miniver's problem
Two perpendicular quadrisections of a triangle
Geometry of pentagons in the dual snub square tiling
Zone in an arrangement of lines
Maximum in-radius isosceles triangle
Translucent Jessen icosahedron
Greek cross to rectangle dissection
Lifted frustum
Two orthogonal polyhedra
Fractal opaque set for a square
Opaque forests for a circle
7 disks can cover a single disk of twice the radius
Triangulating a grid polygon with primitive triangles
Tiling the plane with primitive triangles
Distance contours in a square grid
81 grid points in a radius-5 circle
Crossed lines method for curves of constant width
Comparison of Blichfeldt's theorem with Minkowski's theorem
Blichfeldt's theorem
Collinear form of Cairo pentagonal tiling
Equilateral form of Cairo pentagonal tiling
Reuleaux triangle and its middle hedgehog
Curve of constant width based on a semi-ellipse
Tresillo rhythm as a polygon
Ring lemma
Relative convex hull
Convex hull of a simple polygon
Net for Jessen's icosahedron
Jessen's icosahedron
Convex curve through 13 grid points
Danzer set
Isosceles set
Partition of a cyclic pentagon into isosceles triangles
4-hexagon net for an octahedron
Binary tiling
Partitions of a square into similar rectangles
Simplicial depth
Fermat's right triangle theorem
Ordinary lines
Moss's Egg
Maxima of a point set
Osgood curve
Ambiguity in network curve-shortening
Self-similar lens shape under curve-shortening
Curve-shortening of a convex curve
The grim reaper curve
Supporting lines of a Reuleaux triangle
Polyhedral Delta-Y transformations
Visual proof of Balinski's theorem
Non-convex pentagonal tiling
Non-convex pentagonal tiling
Farthest-first traversal
The Reuleaux triangle in a cluster of four soap bubbles
Langley’s Adventitious Angles
Orthocentroidal circle
Barrier resilience
Euclid XIII.10
Midpoint-stretching polygon
Heesch's anisohedral tiling
Viète's formula
ε-net
Heilbronn triangle problem
89th stage of toothpick sequence
Biggest little hexagon
Overlaid Pythagorean tilings
de Longchamps point
Coloring argument for De Bruijn's theorem
Non-harmonic packing for De Bruijn's theorem
Reuleaux kite
Equidiagonal quadrilateral
Circles meeting at the isodynamic point
Construction of the isodynamic point
Compound of three octahedra
Automedian triangle
Subdivisions of a pentagon
An orthodiagonal quadrilateral
British flag theorem
Pythagorean dissections
Similarity tiling by Koch snowflakes
Aperiodic section of the Pythagorean tiling
Tiling of the plane by shifted squares
Nearest neighbor graph
Theil–Sen estimator
Kawasaki's theorem
Midsphere
Inner radius
Shapley–Folkman lemma
Skew lines on nested hyperboloids
Tverberg's theorem
Radon's theorem
Helly's theorem
The Hadwiger conjecture in combinatorial geometry
The Schönhardt polyhedron.
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
Intersecting planes
Chao's characterization of tangential quadrilaterals
A tangential quadrilateral
$Convex layers and a halfspace. For an example in fractional cascading.$

Convex layers and a halfspace. For an example in fractional cascading.
A Vietoris–Rips complex.
Happy Ending problem, eight points with no pentagon
Orthogonal convex hull and tight span.
Pseudotriangles.
The Szilassi polyhedron. From the junkyard.
A point quadtree.
The face lattice of a square pyramid.
The Nagel point of a triangle.
The Philo line and its application to doubling the cube.
Heesch's problem, Amman's dented hexagon. From the junkyard.
Convex shelling.
Happy Ending problem, quadrilaterals in five-point sets.
Pitteway triangulation
K-sets
H tree
Erdős–Szekeres theorem, geometric interpretation as monotone path

Number theory and combinatorics[edit]

Gomory's theorem on domino tiling mutilated chessboards
A chessboard region with no domino tiling but equal colors
Lambek–Moser theorem for prime numbers
Lambek–Moser theorem
Dyadic approximation of ${\sqrt {2}}$
Numbers without unusually-accurate dyadic approximations
Salem–Spencer set
Relative error of approximations to the prime-counting function
Primes vs composites
Growth rate of the Moser–de Bruijn sequence
Addition of numbers in the Moser–de Bruijn sequence
Gould's sequence
A cap set
Bijection between binary trees and stack-sortable permutations
Congruum
Lower bound construction for the Erdős–Ko–Rado theorem
Centered octahedral number
Centered cube number
Two right triangles, with one hypotenuse the square of the other
Trees counted by the Wedderburn–Etherington numbers
Sauer–Shelah lemma
Levi's lemma
Dickson's lemma applied to the hyperbola xy ≥ 9
Trees counted by the ordered Bell numbers
Rank of a partition
Pell's equation
Gaussian moat
Stella octangula number
Representation of octahedral numbers as centered square pyramids
Octahedral number
Squares in a square
Calkin–Wilf sequence
Calkin–Wilf tree
15=4+5+6 is a polite number
Dyadic rationals
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.
Square pyramidal number.
Rational approximations to the octagon from Pell numbers.
Integer right triangles with nearly-equal legs from Pell numbers.
Divisibility of regular numbers.

Order theory[edit]

2-dimensional modular lattice
Dedekind–MacNeille completion
A fence
The extreme case for the 1/3–2/3 conjecture
A series-parallel partial order
A semiorder
Free distributive lattice of monotonic Boolean functions.
Young–Fibonacci lattice.
Young's lattice.
Example for Birkhoff's representation theorem.
A Tamari lattice.
The Hasse diagram of a distributive lattice.
Dilworth's theorem, transformation from chain decomposition to bipartite matching.
The 13 possible strict weak orderings on a set of three elements.
A partial order of dimension 4 and its realizer.
The lattice of subgroups of Dih₄.
A concept lattice.
Three views of an antimatroid.

Cellular automata[edit]

Anneal
1d reversible CA defined from a rectangular band
Critters transition rule
Toothpick sequence
Rule 90 gate array
Trees in Rule 90
Critters
Block cellular automaton
Day & Night, Bell's p256 butterfly gun
One-dimensional cyclic cellular automaton.
Two-dimensional cyclic cellular automaton.
puffer train.
The space rake.
Rule 184 as a model of traffic flow.
Rule 184 as a model of deposition.
Rule 184 as a model of ballistic annihilation.
Rule 184

Algorithms and data structures[edit]

Finding the median of 5 values in 6 comparisons
X + Y sorting
Garsia–Wachs algorithm
Boyer–Moore majority vote algorithm
Another SPQR tree
Reduction from 3SAT to 3-coloring
Prefix sum
An SPQR tree.
Consecutive pairs that bracket x in a sequence, for the Cartesian tree article
A Cartesian tree
Using a Cartesian tree for range searching
Johnson's algorithm
Floyd's "tortoise and hare" cycle detection algorithm.
Bloom filter

Topology[edit]

Pentagonal Möbius strip
5-vertex polyhedral Möbius strip
6-vertex polyhedral strip with triangular boundary
Level sets of a cross-cap
Symmetric Whitehead link
Six-component "rubberband" Brunnian link
Algebraic link diagram for the Borromean rings
Hedgehog space
Trefoil with bridge number 2
square ribbon knot
A train track on a triple torus.
A switch in a train track.

Miscellaneous[edit]

Monotone piecewise linear function
Fokker periodicity blocks
Combinatorial equivalence between plane trees, polygon subdivisions, and parenthesizations
A Langford pairing
Pitch class space
Descriptive chess notation
Binary erasure channel
Binary symmetric channel
Diagram for a triangulated category
academic genealogy of Johannes De Groot and his namesake

Photography[edit]

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.