The four ways to glue a square's edges in pairs

• every cuff of every pair of pants is used once, genus 0 with 3 boundary ×21
• enumeration and recurrence agree at n 1, genus 0 ×15
• country 0 borders six different countries ×7
• a genus-2 decomposition cuts along 3g − 3 circles ×3
• a sphere with 3 cross-caps has characteristic -1 ×3
• at most 2g − 3 circles separate, genus 2 ×3
• the enumeration agrees with Harer–Zagier at genus 0 ×3
• a surface with 2 handles has characteristic -2 ×2
• a two-sheeted cover of a sphere with 3 cross-caps has twice its characteristic ×2
• all 12 are one triangulation relabelled ×2
• the classification is drawn up to 3 of them in this view ×2
• the decompositions of genus 2 come in the counted number of types ×2
• a drawn random gluing has 6 to 60 edges ×1
• a move never leaves the list of types ×1
• a non-orientable surface and an orientable one share the characteristic and differ ×1
• a pairing makes a sphere exactly when no two chords cross ×1
• a sphere has characteristic 2 ×1
• a sphere with 1 cross-cap has characteristic 1 ×1
• a sphere with 2 cross-caps has characteristic 0 ×1
• a sphere with k cross-caps is covered by a surface with k − 1 handles ×1
• a torus has characteristic 0 ×1
• a two-sheeted cover of a Klein bottle has twice its characteristic ×1
• a two-sheeted cover of a projective plane has twice its characteristic ×1
• a two-sheeted cover of a surface with 2 handles has twice its characteristic ×1
• a two-sheeted cover of a torus has twice its characteristic ×1
• aba⁻¹b⁻¹ gives the characteristic of a torus ×1
• aba⁻¹b⁻¹ has the Euler characteristic a torus has ×1
• aba⁻¹b⁻¹ is orientable ×1
• abab gives the characteristic of a projective plane ×1
• abab has the Euler characteristic a projective plane has ×1
• abab is not orientable ×1
• abab⁻¹ gives the characteristic of a Klein bottle ×1
• abab⁻¹ has the Euler characteristic a Klein bottle has ×1
• abab⁻¹ is not orientable ×1
• abb⁻¹a⁻¹ has the Euler characteristic a sphere has ×1
• abb⁻¹a⁻¹ is orientable ×1
• an orientable pairing has a whole, non-negative genus ×1
• and every face to a face ×1
• and it is the one that crosses on exactly the letters used twice the same way ×1
• and no edge to itself ×1
• and no face to itself ×1
• and no vertex is its own opposite, so the map moves every point ×1
• and not one Klein bottle ×1
• and one move on the dumbbell's middle circle gives the theta back ×1
• and one side: the projective plane ×1
• and so is the sidedness ×1
• and ten faces ×1
• and the graph has as many independent cycles as the surface has handles ×1
• and the quotient's is a projective plane's ×1
• and thirty edges ×1
• and twenty faces ×1
• and two sides: a torus ×1
• at least one move applies to the word given ×1
• at most four letters, so at most fifteen covers are listed ×1
• between one and five cross-caps are drawn ×1
• between one and five gluings are drawn side by side ×1
• between three and twelve steps are drawn ×1
• between two and four words, each of at least three edges ×1
• capping the boundaries gives a closed surface with characteristic at most two ×1
• decompositions are enumerated for genus 2 to 4 ×1
• each drawn pairing is glued head to tail ×1
• each lattice triangle is one of the fourteen ×1
• each sliver is one of the ten triangles ×1
• each type is met as often as its symmetries predict ×1
• edge a is glued to exactly one other edge ×1
• edge b is glued to exactly one other edge ×1
• edge c is glued to exactly one other edge ×1
• edge d is glued to exactly one other edge ×1
• edge e is glued to exactly one other edge ×1
• edge f is glued to exactly one other edge ×1
• eight corners do make a Klein bottle ×1
• every pairing is drawn for polygons of 4 to 8 edges ×1
• every sampled vertex count has the parity the genus formula allows ×1
• every type is reachable from every other by moves ×1
• every vertex has its opposite among the vertices ×1
• every vertex of the surface has exactly two vertices above it, so nothing is pinched ×1
• exactly one connected double cover is two-sided ×1
• few enough vertices to colour each one differently ×1
• fifteen edges ×1
• genus 0 is the Catalan number ×1
• genus four has seventeen ×1
• genus three has five ×1
• genus two has two types ×1
• Heawood's bound allows this many vertices ×1
• no letter appears more than twice ×1
• no move raises the measure ×1
• no two of the surfaces share a characteristic ×1
• none of them two-sided ×1
• one move on the theta gives the dumbbell ×1
• one vertex, for even n, is 1/(n + 1) ×1
• one-vertex gluings are 1/(n + 1) of all ×1
• only letters of the word can be glued across ×1
• opposite points of the rim carry the same label ×1
• seven labelled vertices carry a hundred and twenty tori ×1
• six labelled vertices carry twelve projective planes ×1
• the antipodal map carries every edge to an edge ×1
• the bound is the least n with n² − 7n + 6χ ≥ 0 ×1
• the camera angles are numbers ×1
• the census runs to between 6 and 24 edges ×1
• the characteristic is unchanged by the cancel move ×1
• the characteristic is unchanged by the collapse move ×1
• the characteristic is unchanged by the crosscap move ×1
• the characteristic is unchanged by the pair move ×1
• the cover's characteristic is twice the surface's, because every cell is doubled ×1
• the cycle count agrees with the pooled corners ×1
• the drawing scale is sane ×1
• the drawn range reaches at least one characteristic that both sides hold ×1
• the drawn vertex counts hold nearly all of the distribution ×1
• the first decomposition has no separating circle ×1
• the fourteen triangles close up into a surface ×1
• the genera share out all (2n − 1)!! pairings ×1
• the icosahedron has twelve vertices ×1
• the icosahedron's characteristic is a sphere's ×1
• the mean vertex count sits just above the harmonic number H(2n), by less than 1/n ×1
• the name follows from the number ×1
• the name follows from the two numbers ×1
• the orientable row is on or off ×1
• the pairings number (2n − 1)!! ×1
• the pants' characteristics, −1 each, add up to the surface's ×1
• the polygon has an even number of edges ×1
• the polygon's corners all become one vertex, so every choice of letters to cross gives a cover ×1
• the quotient has six vertices ×1
• the recurrence divides exactly ×1
• the reduction reaches a standard form ×1
• the sample mean sits within four standard errors of the exact mean ×1
• the seven-vertex torus has 42 ×1
• the six-vertex plane has 60 symmetries ×1
• the sphere's share is Catalan over (2n − 1)!! ×1
• the spheres are the Catalan number ×1
• the standard form names the surface the original word named ×1
• the surface is one-sided, so it has an orientation cover to single out ×1
• the table runs to between 3 and 10 pairs ×1
• the ten triangles close up into a surface ×1
• the two copies join up exactly when some letter is used twice the same way round ×1
• the two numbers name the surface without anything else being consulted ×1
• the two surfaces with the same characteristic are told apart by orientability ×1
• the type census is drawn for eight corners ×1
• the types are drawn side by side for genus 2 and 3 ×1
• the view is one the family draws ×1
• the word aab leaves at least one edge unpaired ×1
• the word abab⁻¹c leaves at least one edge unpaired ×1
• the word abc leaves at least one edge unpaired ×1
• the word abcb⁻¹ leaves at least one edge unpaired ×1
• the word glues every edge to exactly one other ×1
• the word has at least four edges ×1
• the words drawn give different triples, so the figure is separating them ×1
• three fresh letters are available for the collapse ×1
• using all fifteen pairs of six vertices as edges ×1
• using all twenty-one pairs of seven vertices as edges ×1
• with characteristic one ×1
• with characteristic zero ×1
• with the copies given opposite senses of turning, every pair glued in the cover meets with opposite arrows ×1

The whole library · What the figures prove