8 rectangles under a curve

• the integral gives 0! exactly ×11
• the area under x^0e^(−x) is 0! ×8
• at 1 steps the length is still 2 ×7
• the shadow in the direction (1, 2) keeps its full length at stage 0 ×7
• the area under 1/x^0.9 from 1 to 10 matches its closed form ×6
• the integral of sin to the power 2 is (2 − 1)/2 times the one for power 0 ×5
• the area under 1/x^1.25 beyond 1 is 1/(p − 1) ×4
• the horn to 10 holds π(1 − 1/T) ×4
• and it equals the far-end area of the reflected exponent 1.75 ×3
• on block 1 the integrand gains exactly 1 ×3
• the area under 1/√x from 1 to 10 matches its closed form ×3
• the area under 1/x from 1 to 10 matches its closed form ×3
• the area under 1/x² from 1 to 10 matches its closed form ×3
• the staircase of 2 steps is exactly 2 long ×3
• between 256 and 8192 samples ×2
• the area under 1/x^0.25 between 0 and 1 is 1/(1 − p) ×2
• √π lies inside every pair ×1
• ∫ dx/(x ln x) from e^e to 10^50 is ln ln x between the ends ×1
• ∫ from 1 to ∞ of x^(−3/2) is 2 ×1
• ∮ x dy and the half-sum ½∮(x dy − y dx) give the same area ×1
• ½∮(x dy − y dx) is π(b² + a²/2) for the limaçon ×1
• a circle's perimeter is 2π and a flattened ellipse's tends to 4a ×1
• a fine enough polygon reaches the curve's length ×1
• a known increasing curve over a real stretch ×1
• adding corners never shortens the inscribed polygon ×1
• after 20,000 lines the estimate is within its own error band of the true length ×1
• and (−½)! = √π ×1
• and are within 1/N of it after N factors ×1
• and by this stage it is already small ×1
• and every one of them falls short of the curve's own length ×1
• and is nearly nothing by the end ×1
• and its length is nowhere near the curve's ×1
• and so does the polynomial ×1
• and the fat one keeps a length in the limit ×1
• and the finest is strictly longer than the coarsest, so the family is climbing ×1
• at least one triangle counts negative, so cancellation is visible ×1
• between 20 and 20,000 lines ×1
• between 6 and 40 strips are drawn ×1
• between three and five corner counts, each up to 64 ×1
• between two and six divisions ×1
• both parts only ever increase ×1
• doubling the corners divides the gap by four ×1
• each area is n times the one before ×1
• each curve has the same length ×1
• each curve's mean crossing count is 2L/(πd), whatever its shape ×1
• each divergent integrand ends above the convergent ones at its own level ×1
• each inscribed polygon is shorter than the ellipse, and doubling the corners shortens the gap ×1
• each integral is still growing at the far end ×1
• each panel is at least as close as the one before it ×1
• each stage is met by fewer lines than the one before ×1
• each staircase runs closer to the curve than the last ×1
• every horizontal line crosses the curve going up as often as coming down ×1
• every integrand in the chain still has a finite integral to infinity ×1
• every parametrisation traces the same point set ×1
• every wiggle eventually breaks log-convexity ×1
• for a flat ellipse, P − 4a is (2b²/a)(ln(4a/b) − ½) to within a hundredth of the bracket ×1
• for an ellipse the error falls like 1/n² ×1
• for nearly circular ellipses the error of each formula falls like a power of h: the first power for Kepler, the average and Euler, the third and fifth for Ramanujan's ×1
• for the wavy curve too the error falls like 1/n², with a larger constant ×1
• halving the wiggle doubles how long it survives ×1
• in a general direction the shadow shrinks ×1
• lines is read only by the crofton view ×1
• log of the wiggled curve bends the wrong way over stretches of every unit interval ×1
• log Γ bends one way everywhere ×1
• middle quarter, halving: no cell lies wholly inside the set, so the lower sum is nought ×1
• middle quarter, halving: refining the partition never raises the upper sum ×1
• middle quarter, halving: the upper sum never falls below the set's own length ×1
• middle thirds: no cell lies wholly inside the set, so the lower sum is nought ×1
• middle thirds: refining the partition never raises the upper sum ×1
• middle thirds: the upper sum never falls below the set's own length ×1
• more bars means less error ×1
• no width, no area ×1
• out and back covers the image a whole number of times ×1
• out, back, out again covers the image a whole number of times ×1
• refining a partition never lowers the variation ×1
• seed is read only by the crofton, favbuffon view ×1
• sin p is the fraction's error p − qπ, to first order ×1
• so in the limit the middle-thirds set has no length at all ×1
• so the first upper sum is on its way down to nothing ×1
• stage is read only by the cantorlen view ×1
• straight through, accelerating covers the image a whole number of times ×1
• straight through, once covers the image a whole number of times ×1
• the accumulation only ever climbs, because f is positive ×1
• the area between the two shrinks every time the steps are halved ×1
• the area left of the curve up to b ×1
• the area read from the boundary matches the cells counted inside ×1
• the area under 1/√x between 0 and 1 is 1/(1 − p) ×1
• the area under the curve up to a ×1
• the arithmetic–geometric mean and the trapezoid rule agree to twelve places ×1
• the arithmetic–geometric mean settles to the last digit within seven steps ×1
• the boundary integral equals the area weighted by winding number ×1
• the Cantor, noodle and Koch views draw their own curves ×1
• the counts increase ×1
• the curve is one of quarter, parabola, wiggle ×1
• the curve lies inside the disc the lines are thrown at ×1
• the directions (2, 1) and (1, 2) keep the full shadow ×1
• the drawing keeps a few points per column rather than every term ×1
• the drawn partition has between 24 and 240 cells ×1
• the error shrinks as points are added ×1
• the fat construction's first middle is between a third and an eighth ×1
• the finest one is within a twentieth of the curve everywhere ×1
• the finest partition is a power of two between 64 and 16384 ×1
• the four corners project to 0, 3, 6 and 9 quarters along x + 2y ×1
• the function is one of root, square, smooth ×1
• the function is one the figure knows ×1
• the function moves, so the decomposition is not trivial ×1
• the functions are among root, square, smooth ×1
• the gap shrinks like √π/(2n) ×1
• the inner loop is wound round twice ×1
• the inscribed polygon's length, measured, matches the closed form ×1
• the interval runs from nought to at most 2 ×1
• the limaçon has an inner loop: 0 < b < a ×1
• the maps differ in length or in speed, so the panels are comparing something ×1
• the maps run at different speeds ×1
• the middle-thirds approximation's length is two-thirds to the power of the stage ×1
• the panels really are in increasing order of rectangles ×1
• the parametrisations are among once, back, thrice, slow ×1
• the partial products rise towards π/2 from below ×1
• the polynomial's ratio does not ×1
• the printed total is the area of the bars drawn ×1
• the quoted exact area is the area under this curve ×1
• the series needs a handful of terms for a mild ellipse and hundreds for a flat one ×1
• the set is approximated at between 5 and 14 stages ×1
• the share of random lines that meet the stage agrees with the shadows' average ×1
• the share of the straight line, w, is between 0 and 1 ×1
• the shoelace sum equals the area counted inside ×1
• the signed area is within 1/T of π/2 ×1
• the signed triangles add up to the shoelace sum ×1
• the slope of the accumulated area is the height of the curve ×1
• the slowest divergent drawn is still above the fastest-converging drawn at the far end ×1
• the speed integral and the polygon agree for out and back ×1
• the speed integral and the polygon agree for out, back, out again ×1
• the speed integral and the polygon agree for straight through, accelerating ×1
• the speed integral and the polygon agree for straight through, once ×1
• the squares' shadows along x + 2y tile the interval end to end ×1
• the stage drawn is 1 to 7 ×1
• the stage-10 polygon is within 0.02 of the limit ×1
• the stage-k Koch curve has length 1.8 × (4/3)^k ×1
• the staircase is built against quarter or diagonal ×1
• the staircase of 1 step is exactly 2 long ×1
• the step counts are between 2 and 5 whole numbers up to 512 ×1
• the sum to ten million is 30.3145 ×1
• the sums run to between ten thousand and twenty million terms ×1
• the sweep runs over between three and eight partition sizes ×1
• the sweep runs up to between 4 and 16384 steps, or an interval end for the variation view ×1
• the term at n = 355 is about 24.6 ×1
• the third block ends at a number of a few hundred digits ×1
• the too-well-approximated number's sum races away ×1
• the tracer arm's length times the wheel's roll is the area enclosed ×1
• the tracer is within reach of the arms ×1
• the two areas together are at least ab ×1
• the two regions fill the big rectangle less the small one ×1
• the unsigned area exceeds (2/π) ln T ×1
• the variation of t sin(1/t) climbs by as much at the finest refinement as at the coarsest ×1
• the variation of t² climbs less and less, so it settles ×1
• the variation of t² sin(1/t) climbs less and less, so it settles ×1
• the view is one the family draws ×1
• the volume is below π and the surface above 2π ln T ×1
• the wiggle even keeps the recurrence ×1
• the wiggle keeps every factorial ×1
• the wiggle's ratio to Γ repeats with period one ×1
• their difference is the function ×1
• two to four exponents between 0 and 3 ×1
• w is read only by the cantorlen view ×1
• while the second has already settled on the set's own length ×1
• with equality when b = f(a) ×1
• x runs up to 10 to a power from 20 to 300 ×1
• Γ lies between the two products ×1
• Σ 1/(n (ln n)²) has almost stopped growing ×1
• Σ 1/(n ln n) − ln ln N has settled to a constant ×1

The whole library · What the figures prove