A circle unrolled into a sine wave

• the product to 1 vanishes at 0 ×17
• the product to 1 vanishes at -1 ×14
• the 1:1 figure crosses itself 2pq − p − q times ×11
• at phase 1.178 the 1:2 figure has the generic crossing count ×7
• the chords of the regular 3-gon from one corner multiply to 3 ×6
• both vanish at 0 ×4
• both vanish at -3 ×3
• the doubled arc from 0.25 of a quarter ends where Fagnano's formula says ×3
• the gap after 10 terms lies between 1/(N + 1) and 1/N ×3
• 1 + 1/2² + 1/3² + … is π²/6 ×1
• 1 + 1/2⁴ + 1/3⁴ + … is π⁴/90 ×1
• 1 + 1/2⁶ + 1/3⁶ + … is π⁶/945 ×1
• 5 and 17 pass and 7 fails ×1
• a coprime pair up to five ×1
• a generic Lissajous figure crosses itself 2pq − p − q times ×1
• a million terms and the tail 1/N give π²/6 ×1
• a step up to 1 and two to six sectors ×1
• and they stay on the hyperbola ×1
• between three and seven offsets ×1
• cosh and sinh are the even and odd halves of e^t ×1
• every slice between neighbouring points has the same area ×1
• every station lies on the figure eight ×1
• more factors bring the product closer at x = ½ ×1
• sl blows up beside each marked pole ×1
• sl repeats along (1 + i)ϖ, imaginary part ×1
• sl repeats along (1 + i)ϖ, real part ×1
• sl repeats along 2ϖ ×1
• sl vanishes at each marked zero ×1
• the catenary's parameter is between 0.2 and 2 ×1
• the circle's parameter is between 0 and 2π ×1
• the circle's quarter arc is π/2 ×1
• the circular sector's area is half its parameter ×1
• the correct digits roughly double each step ×1
• the drawn arrangement is a zigzag ×1
• the drawn list has the zigzag number of members ×1
• the figure eight, walked as chords, is 2ϖ long ×1
• the figure eight's quarter arc is π over twice the arithmetic–geometric mean of 1 and √2 ×1
• the first station is one equal share of arc from the centre ×1
• the folding phases really fold ×1
• the four families fail at 4, 8, never, never ×1
• the gap after N factors is about π/(8N) ×1
• the grid runs to between 2 and 5 ×1
• the hyperbola's parameter is between 0 and 2.2 ×1
• the hyperbolic sector's area is half its parameter ×1
• the length is between 3 and 6 ×1
• the offset is generic — the curve does not fold back on itself ×1
• the parabola and the catenary really differ ×1
• the point is on the circle ×1
• the point is on the hyperbola ×1
• the point's height is the wave's height there ×1
• the product's x² coefficient is the Taylor series' ×1
• the product's x⁴ coefficient is the Taylor series' ×1
• the product's x⁶ coefficient is the Taylor series' ×1
• the range is between 1 and 3.2 ×1
• the ratio is between one and three ×1
• the relative error is within the next pole's 3^-(n+1) ×1
• the row shows both a folded and an open figure ×1
• the second is not odd ×1
• the steps pass 1 at the seventh and 2 at the fifty-sixth ×1
• the time spent in each cell is close to the product of two arcsine densities ×1
• the two coordinates are a point of the unit circle ×1
• the view is one of sector, squeeze, waves, catenary, lissajous, lissgrid, lissphase, lissdense, lemwalk, lemlength, lemdouble, lemagm, lemcomplex, lemdivide, prodpartial, prodsame, polychords, basel, wallis, prodcoef ×1
• twenty thousand factors give sin(π/2) to four places ×1
• two whole frequencies up to six with no common factor ×1
• Wallis's partial products climb towards π/2 from below ×1
• x + y is e to the parameter ×1
• π divided by the mean is ϖ ×1

The whole library · What the figures prove