Dimension Lab
The circle and the sphere are already inside the zeta function. The factor that makes zeta mirror-symmetric is the surface area of a sphere whose dimension is the complex variable itself; the "trivial" zeros are the dimensions where a sphere has no surface. This lab draws that, then the two other places your questions lead: how a sphere's surface piles up at the equator as the dimension climbs, and what the square-root error looks like when you count grid points in a circle. The last tab is a line bouncing inside a circle, because you asked.
Unit ball volume V(n) and unit sphere surface area S(n), any real n exact
Both formulas are true for every real n, not just whole numbers, because the Gamma function fills in the factorials between the integers. The volume of the unit ball peaks near n ≈ 5.26, the surface area near n ≈ 7.26, and both shrink toward zero as n grows — a unit cube's volume stays 1 while the ball inside it evaporates. To the left, 1/Γ vanishes at every non-positive integer, so S(n) = 0 at n = 0, −2, −4, … and V(n) = 0 at n = −2, −4, …. Riemann's completed function is ξ(s) = ζ(s) · s(s−1) / S(s) with the dimension replaced by the complex number s; ξ is finite and non-zero at s = −2, −4, …, so ζ must vanish there. The trivial zeros of zeta are literally the dimensions in which a sphere has no surface.
How the surface of a sphere divides by angle from a pole, and how long its chords are exact
On the circle (n = 2) the surface is spread evenly by angle, so the cap fraction is a straight line. On the sphere (n = 3) it follows Archimedes: the share within angle θ of the pole is (1 − cos θ)/2, which depends only on the cap's height. As n grows the surface abandons the poles and crowds into a belt around the equator: at n = 100, almost everything lies within a few degrees of 90°. The right-hand plot is the consequence for chords: pick two surface points at random and measure the straight line between them. On the circle every length is common; on the 100-sphere nearly every chord has length √2 · r, because nearly every pair of directions is nearly perpendicular. This is the same equator belt seen from inside. A segment bouncing inside a sphere, by the way, never leaves the plane through the center that contains its first chord — so a bouncing line in any dimension is the circle picture in tab 5, tilted.
Zeta on the critical line, before and after the sphere factor exact
On the critical line ζ is a complex number that spins as t increases (cyan and amber). Multiply it by the phase of π−s/2Γ(s/2) — the sphere factor's direction — and the spin is cancelled exactly: the result Z(t) (green) is a plain real number. That is the mirror symmetry made visible; the sphere factor is precisely what straightens zeta out. The zeros of ζ are the places where the real curve crosses the axis. Hardy used exactly this function in 1914 to prove that infinitely many zeros lie on the line: a real function that keeps changing sign must keep crossing zero. What he could not show — what nobody has — is that all of them do.
Count the integer grid points inside a circle of radius r finite
Draw a circle of radius r on graph paper and count the grid points inside. The count is about the area πr², and the leftover — the error — wanders. Gauss showed it is at most proportional to r; the conjecture, still open, is that it is at most about √r (times any tiny power), which is exactly the shape of the prime-count conjecture with r in place of √x. The best proven exponent today is a little under 0.63, so this is another case where a square-root error is plainly visible in every picture and unproven for all time. In four dimensions the same count is governed by sums of four squares, and Jacobi's theta function — the very tool Riemann used to prove the functional equation — gives it exactly. Circle, grid, theta, mirror: one road.
A line segment bouncing inside a circle exact
A segment launched from the rim at angle α to the tangent bounces forever at that same angle, every chord has the same length 2r sin α, and each bounce advances the hit point by the same arc 2α. So everything is decided by whether α is a rational fraction of 180°. If α = (a/q) × 180° the path closes after q bounces and the hit points are exactly q equally spaced marks on the rim — the circle divided into q parts, the same division that gave the spikes at a/q in the overtones lab. If α is irrational the path never closes; it fills the ring between the rim and an inner circle of radius r cos α (the caustic) evenly and forever. Inside a sphere the story is identical, because a bouncing segment can never leave the plane through the center that holds its first chord: the sphere billiard is this picture on a tilted disc. Rational angles give closed polygons and star shapes; irrational ones give rings. This is a real and old piece of geometry, and it is worth knowing that it is "integrable" — completely predictable — which is the opposite of the chaotic billiards physicists use to model heavy nuclei, whose energy levels are the ones that statistically resemble the zeta zeros.