V − E + F = 2, five times

• in dimension 1 the sum is 1 − (−1)^d ×8
• the alternating sum of the 5-cell is 0 ×5
• cyclic, 6 corners: cells = edges − corners ×4
• cyclic, 6 corners: faces = 2 × (edges − corners) ×4
• cyclic, 6 corners: the alternating sum is 0 ×4
• 5-cell: cells = edges − corners ×3
• 5-cell: faces = 2 × (edges − corners) ×3
• 5-cell: the alternating sum is 0 ×3
• V − E + F is 0 for this solid ×3
• with exact readings the 40 by 36 grid counts the targets ×3
• the whole boundary of the 3-cube has alternating sum 1 − (−1)^3 ×2
• ∫ χ(h ≥ s) ds is the pieces' lifetimes less the holes' ×1
• a bipyramid lies on the line F = 2V − 4 ×1
• a convex solid has exactly two critical points in every direction ×1
• a deeper overlap tolerates more smoothing ×1
• a honeycomb on the torus has V − E + F = 0 ×1
• a prism lies on the line V = 2F − 4 ×1
• a pyramid has as many corners as faces ×1
• a spanning tree has one fewer edge than the graph has vertices ×1
• a wide gap separates the targets' features from the mistakes' ×1
• about one sensor's width of smoothing is needed to swallow the mistakes ×1
• and 720° is 360° times V − E + F ×1
• and its twelve edges ×1
• and nothing he forbids is reached ×1
• and six sides to every face ×1
• below the squared field's floor the region is the whole torus, with χ = 0 ×1
• between 20 and 400 surfaces ×1
• between one and three holes are punched ×1
• corner cut off: V − E + F = 2 ×1
• counting and discarding the edge pieces extrapolate to the same constant ×1
• cube: each face has three sides and each corner three edges ×1
• cube: Euler ×1
• cutting a three-edged corner adds two corners and one face ×1
• dodecahedron: each face has three sides and each corner three edges ×1
• dodecahedron: Euler ×1
• each ring has χ = 0 ×1
• each shape follows its own exact curve ×1
• each unfolded face closes back on the corner ×1
• every count Steinitz allows is reached by a pyramid and the two moves ×1
• every detection region is a disc, χ = 1 ×1
• every detection region is a ring, χ = 0 ×1
• every edge lies between exactly two faces ×1
• every edge of the cube is shared by two faces ×1
• every edge of the dodecahedron is shared by two faces ×1
• every edge of the icosahedron is shared by two faces ×1
• every edge of the octahedron is shared by two faces ×1
• every edge of the surface is shared by exactly two faces ×1
• every edge of the tetrahedron is shared by two faces ×1
• every game stays within the bounds ×1
• every partial boundary before the last facet sums to 1 ×1
• every random game of Brussels sprouts on n crosses lasts exactly 5n − 2 moves ×1
• four levels ×1
• gluing on a tetrahedron adds one corner and two faces ×1
• icosahedron: each face has three sides and each corner three edges ×1
• icosahedron: Euler ×1
• lives remaining are 3n minus the number of moves ×1
• no leftover edge closes a loop among the faces ×1
• no mistake lives longer than twice the largest change smoothing left in the readings ×1
• octahedron: each face has three sides and each corner three edges ×1
• octahedron: Euler ×1
• on a surface with no edges, maxima minus saddles plus minima is zero ×1
• once divided by the formula's prefactor, every scale gives the same curve ×1
• pentagonal bipyramid: each face has three sides and each corner three edges ×1
• pentagonal bipyramid: Euler ×1
• pentagonal prism: each face has three sides and each corner three edges ×1
• pentagonal prism: Euler ×1
• pentagonal pyramid: each face has three sides and each corner three edges ×1
• pentagonal pyramid: Euler ×1
• pieces minus holes matches the exact Euler characteristic ×1
• pieces minus holes, counted separately, follows the exact density ×1
• rounding the lifetimes counts the targets ×1
• sixteen times the sensors, about sixteen times the error ×1
• smoothed, with each lifetime rounded, it counts them again ×1
• tetrahedron glued on: V − E + F = 2 ×1
• tetrahedron: each face has three sides and each corner three edges ×1
• tetrahedron: Euler ×1
• tetrahedron: V − E + F = 2 ×1
• the 5-sided bipyramid has V − E + F = 2 ×1
• the 5-sided prism has V − E + F = 2 ×1
• the 5-sided pyramid has V − E + F = 2 ×1
• the alternating sum is not 2, on a solid with flat faces and straight edges ×1
• the alternating sum of the tesseract is 0 ×1
• the average Euler characteristic follows the formula ×1
• the blind rules tilt the balance of signs and the exact rule keeps it level ×1
• the constant is below the percolation prediction by more than two standard errors ×1
• the counted pieces minus holes agree with the exact density ×1
• the counts obey Euler's formula ×1
• the crossing curves steepen as the square grows ×1
• the cube's faces fall short of hexagons by exactly 12 ×1
• the defects on the cube come to 720° ×1
• the defects on the dodecahedron come to 720° ×1
• the defects on the icosahedron come to 720° ×1
• the defects on the octahedron come to 720° ×1
• the defects on the tetrahedron come to 720° ×1
• the dodecahedron's faces fall short of hexagons by exactly 12 ×1
• the error follows p/2 times the sensors that see nothing ×1
• the faces at a corner of the cube do not close up flat ×1
• the faces at a corner of the dodecahedron do not close up flat ×1
• the faces at a corner of the icosahedron do not close up flat ×1
• the faces at a corner of the octahedron do not close up flat ×1
• the faces at a corner of the tetrahedron do not close up flat ×1
• the faces short of six sides fall short by 12 in all ×1
• the flattened cube keeps its eight corners ×1
• the flattened cube still gives 2 ×1
• the game ends with 4n regions, one free arm in each ×1
• the hexagonal prism's faces fall short of hexagons by exactly 12 ×1
• the index matches the table ×1
• the indices add up to the Euler characteristic counted directly ×1
• the large pieces follow a power law near percolation's exponent ×1
• the leftover edges are what the tree did not use ×1
• the leftover edges join every face to every other ×1
• the leftover edges number one fewer than there are faces ×1
• the length runs from 2n to 3n − 1 ×1
• the level is between −3 and 3 ×1
• the median gives way first; the rounded lifetimes hold to three in ten ×1
• the mistaken readings alone throw the integer count far off ×1
• the mode is one the family draws ×1
• the option is one this view reads ×1
• the rings overlap in two separate pieces ×1
• the rounded count holds until the overlaps' lifetimes fall below a half ×1
• the search agrees with the pattern ×1
• the solid is one of the five regular ones ×1
• the solid is one the family builds ×1
• the squared field breaks the symmetry the Gaussian one keeps ×1
• the sum never changes ×1
• the sum over the level sets equals the sum of the regions' own χ ×1
• the sweep ends at the torus's Euler characteristic ×1
• the targets' regions are discs or rings ×1
• the tesseract has 32 edges ×1
• the tesseract's counts are the 16-cell's, reversed ×1
• the tetrahedron's faces fall short of hexagons by exactly 12 ×1
• the tree is grown from one of the graph's own vertices ×1
• the tree reaches every vertex ×1
• the triangular prism's faces fall short of hexagons by exactly 12 ×1
• the truncated icosahedron's faces fall short of hexagons by exactly 12 ×1
• the truncated octahedron's faces fall short of hexagons by exactly 12 ×1
• the truncated tetrahedron's faces fall short of hexagons by exactly 12 ×1
• the two trees together use every edge exactly once ×1
• the unrounded integral only falls as the smoothing widens ×1
• the window closes as the overlap gets shallow ×1
• three edges at every corner ×1
• three faces meet at every corner of the cube ×1
• three faces meet at every corner of the dodecahedron ×1
• three faces meet at every corner of the hexagonal prism ×1
• three faces meet at every corner of the tetrahedron ×1
• three faces meet at every corner of the triangular prism ×1
• three faces meet at every corner of the truncated icosahedron ×1
• three faces meet at every corner of the truncated octahedron ×1
• three faces meet at every corner of the truncated tetrahedron ×1
• up to one reading in ten, both repairs count exactly ×1
• V − E + F = 2 for the cube ×1
• V − E + F = 2 for the dodecahedron ×1
• V − E + F = 2 for the hexagonal prism ×1
• V − E + F = 2 for the tetrahedron ×1
• V − E + F = 2 for the triangular prism ×1
• V − E + F = 2 for the truncated icosahedron ×1
• V − E + F = 2 for the truncated octahedron ×1
• V − E + F = 2 for the truncated tetrahedron ×1
• V − E + F is -2 for this solid ×1
• V − E + F is 2 for the cube ×1
• V − E + F is 2 for the dodecahedron ×1
• V − E + F is 2 for the icosahedron ×1
• V − E + F is 2 for the octahedron ×1
• V − E + F is 2 for the tetrahedron ×1
• which is V − E + F = 2 ×1
• with every region a disc, the integral counts the targets ×1
• with every region a ring, the integral sees nothing ×1
• with exact readings the integral counts the targets ×1
• χ above u is maxima minus saddles plus minima above u ×1
• χ of a disc is 1 ×1
• χ of a disc with two holes is -1 ×1
• χ of a ring is 0 ×1
• χ of two discs is 2 ×1
• χ(A ∪ B) = χ(A) + χ(B) − χ(A ∩ B) ×1

The whole library · What the figures prove