A phase-time plot stacks a pendulum's states (θ, ω) up a time axis, so each swing is one turn of a curve, not a loop retraced. Here θ″ = −(g/L) sin θ − 2cθ′, L = 1.7 m, c = 0.15 s⁻¹, from rest at 1 rad. The swing shrinks as e^(−ct), losing 95 % of its energy in 10 s, so the helix narrows to a cone. The small-angle θ₀e^(−ct)cos(ω_d t) predicts a 2.62 s period; at 57° it is 2.74 s. Undamped, the curve would coil round one closed loop: a tube.

A source with a steady tone f moves at speed v through still air, and each wavefront spreads at sound speed c from where the source stood when it left. The pitch heard is f′ = f · c/(c − v cos θ), θ measured at emission. At f = 1200 Hz, c = 340 m/s and v = 100 m/s, crests arrive 0.20 m apart ahead and 0.37 m behind: 1700 Hz, then 927 Hz. The pitch passes 1200 Hz just after closest approach, late by the sound's travel time.

An electric dipole p(t) = p₀ sin(ωt) ẑ oscillating at a point radiates a field that, far away, falls off only as 1/r. Its radiation term is E = (μ₀/4π) p̈(t − r/c) sin θ / r along θ̂, with B = r̂ × E / c along φ̂, so E, B and the direction of travel r̂ are mutually perpendicular. The intensity goes as sin²θ / r²: zero along the axis, strongest at the equator. Every sphere around the source carries the same mean power, p₀²ω⁴ / (12πε₀c³).

Each curve starts as a plus sign, and each step replaces every segment with a bent copy of itself. The Lévy C curve bends each segment into two at 45°, always to one side; the dragon alternates sides; the twindragon is a dragon closed by its own half turn. The terdragon follows F ↦ F+F−F at 120°, pieces 1/√3 long. So length grows by √2 or √3 a step, and at N = 8 an arm has 256 segments, or 6561. Every limit has dimension 2.

A bar magnet spinning at angular speed Ω is a magnetic dipole m turning in its own plane, and it radiates at f = Ω/2π. In that plane E points straight out of it, E ∝ m̈ × r̂ at the retarded time t − r/c, so each far crest lies on θ = φ(t − r/c) + π/2: two spiral arms of pitch λ = 2πc/Ω. The whine is 1024f, ten octaves up. In empty space J = 0, so in ∇ × B = μ₀ε₀ ∂E/∂t + μ₀J only the changing E keeps B alive. The radiated power grows as Ω⁴.

A twist is a rotation whose angle grows along its own axis: Tₓ(s) turns the point (x, y, z) about the x axis through the angle sx. The face x = 0 stays put, each slice x = c turns rigidly through sc, and edges along the axis bend into helices. With s = cos θ and side 2, the far face swings through ±2 radians, about ±115°. The map is not linear, yet its Jacobian has determinant 1 everywhere, so the twist keeps volume.

A parametric rose is one surface wound nearly thirteen times about an axis, θ running from −5π/3 to 24π and x ∈ [0, 1] running out along each petal. The opening angle φ = (π/2)e^(−θ/8π) falls from 111° on the drooping outer petals to 4.5° at the bud. X = 1 − ½(5/4(1 − (3.6θ mod 2π)/π)² − ¼)² swells from ½ at each notch to 1 across a petal, 3.6 petals to a turn. Change 3.6 and the petal count changes; change 8π and it opens at a different rate.

The point e^(iθ) = cos θ + i sin θ, carried along a third axis by θ itself, winds a helix of radius 1 that makes one full turn every 2π. Seen down the θ axis the helix is the unit circle. Seen from the side its height is the real part, cos θ. Rolled a quarter turn about the axis, the height becomes the imaginary part, sin θ. Every odd multiple of π lands on the same point of the circle, so e^(iπ) = −1, which is e^(iπ) + 1 = 0.

A point runs along y = 7000 e^(−x²/2) cos²(πx) from x = −3.2 to 3.2, and the number above it is its height. The Gaussian envelope pulls each side peak towards the origin: they sit at x ≈ ±0.952, ±1.906 and ±2.864, with heights 4349, 1042 and 96. Only x = 0 reaches 7000, and the point crosses it at about 1.9 units a second. Pause it: what is the highest number caught? 7000 is there for one frame in 625, with 6927 on either side.

Ten elementary functions y = f(x), each on a window fitted to its shape, from the cusp of √|x| at the origin to x/(1 + x²). sin x / x tends to 1 at x = 0 and its integral over the whole line is π. x/(1 + x²) peaks at x = 1 with height ½. e⁻⁰·³ˣ sin 2x loses 61 % of its amplitude every period π, since e⁻⁰·³π ≈ 0.39. The logistic curve and tanh are one shape, rescaled: 1/(1 + e⁻ˣ) = (1 + tanh(x/2))/2.

A square wave flipping between ±1 at multiples of π is the limit of f_N(t) = (4/π) ∑ sin((2k+1)t)/(2k+1) for k < N. Only odd harmonics appear, since f(t + π) = −f(t). Term k is a circle of radius (4/π)/(2k+1) turning 2k+1 times as fast as the first; the chain ends at height f_N(t). The sum overshoots each jump. Its first peak, at t = π/(2N), is 1.182 for N = 5 and 1.179 for N = 100, about 9 % of the jump. More terms narrow it, never remove it.

A polyrhythm is two even pulses sharing one span. One hand turns every 3 s and sounds each corner it crosses: the square's four, every 0.75 s, against a regular n-gon's n, every 3/n s. The two pulses meet gcd(n, 4) times a turn: once for 3:4, 5:4, 7:4, 9:4 and 15:4, twice for 6:4 and 10:4, and on every square beat for 8:4 and 20:4.

A disk of radius a rolls on a table at tilt θ, its contact point circling at rate Ω. Gravity against the gyroscopic torque fixes Ω² = 4g/(a sin θ), with energy (3/2)Mga sin θ. Rolling resistance μ drains it as dθ/dt = −(2/3)μΩ, so near the end θ√θ falls linearly to zero: a finite-time stop. With a = 0.40 m and μ = 0.0086, θ drops from 70° to 1.4° in 15 s while Ω climbs from 1.63 to 10.08 Hz. The disk itself turns ever slower, at Ω sin θ.

A rose curve is r = sin(kθ). Here k = b/a, with a = 1, 3, 7 down the rows and b = 2, 4, 8 across. With k = p/q in lowest terms and pq even, the rose has 2p petals and closes after θ = 2qπ. Each a is odd and each b a power of 2, so p = b and q = a: each column has 2b petals, 4, 8 and 16, and a only sets how far they wind: 2π, 6π, 14π. Swept over θ ∈ [0, 2aπ] in equal times, kθ = 2πb·t/T, so a column's three pens reach their petal tips together.

Balls released from rest on the line y = H fall under gravity into the bowl y = Ax² and bounce off it elastically. At height y a ball moves at √(2g(H − y)), so none can rise above H, and every flight between bounces is a parabola whose directrix is y = H. The drop is symmetric, so the balls cross in mirror pairs. With A = 2, g = 1.2 and H at the rim, all 1000 are back above 93 % of H at 3.8 s, and again at 7.6s.

A three-digit number with every digit equal to n, divided by the sum of its digits: nnn/(n + n + n). The numerator is 111·n and the denominator 3n, so n cancels and the quotient is 111/3 = 37 for every digit from 1 to 9. With k repeated digits the quotient is Rₖ/k, where Rₖ = 11…1 is the repunit. Below k = 27 it is whole only for k = 1, 3 and 9, and R₉/9 = 12345679.

Japanese line multiplication draws each digit as that many parallel lines: 413 as groups of 4, 1 and 3 lines, and 3 as three lines across them. A group of a lines crossing b lines meets them in a·b points, so each group is one partial product: 4·3 = 12 hundreds, 1·3 = 3 tens, 3·3 = 9 units. By place value, 413 × 3 = 12·100 + 3·10 + 9 = 1239. It is long multiplication with counting in place of the times table; a count above 9 carries, as the 12 does.

Forty-seven symbols of written mathematics fall into six families: relations, logic, sets, number systems, operations and calculus. The number systems nest, ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ, and ∅ is a subset of every set. n! counts the orderings of n objects and ⁿCᵣ = n!/(r!(n − r)!) the ways to choose r. The sign = dates from Robert Recorde in 1557 and ∫ from Leibniz in 1675, a long s for summa. ⌊x⌋ is Kenneth Iverson's 1962 replacement for Gauss's [x].

A gallery of elementary functions, sorted by how the graph behaves rather than by the formula that produces it. Powers and roots: y = xⁿ, where n alone sets steepness and symmetry. Growth and decay: y = eˣ and y = ln(x), inverse to one another. Waves: y = A sin(bx + δ), bounded and repeating. Curves that settle down: y = tanh(x) and e⁻ˣ², each with a horizontal asymptote. The last of them belong to two families at once. cos(5x)/(1 + x²) is a wave inside a decaying envelope.

A Lissajous figure is the path traced by x = sin(at + δ) against y = sin(bt), two perpendicular oscillations and nothing more. Here a runs across and b runs down, forty nine cells driven by one shared phase δ. The path closes after finite time exactly when a : b is rational, and the lobes along each edge count out that ratio. Reading those lobe counts against a known frequency is how an unknown one was measured.