This is the Farey tessellation of the upper half-plane. Each pair of Farey neighbours a/b, c/d, with |ad − bc| = 1, is joined by a hyperbolic geodesic: a semicircle, or a vertical line when one end is ∞. Each rational p/q also carries its Ford circle, of diameter 1/q², tangent to the line at p/q. Two Ford circles touch exactly when their fractions are neighbours.
Scroll or pinch to zoom without limit, and drag to pan. Circles and geodesics fade in once they are a couple of pixels across, and labels once the circles are big enough to hold them. The arithmetic is exact (BigInt), so zooming to 10¹⁰⁰ is fine. Click a circle to fly to it. Choosing a constant or typing a number starts a voyage: the view jumps home, slides until that number’s vertical geodesic is at the centre (in the disk, the geodesic from ∞ rotates until it is the vertical diameter), then dives along it. Space or the dive button pauses and resumes. The centre is locked on the exact number, so the dive stays on it however deep it goes; dragging releases the lock. The panel’s continued fraction is always that of the centre of the screen.
Zoom to any number: type an expression in the box and press go. It understands integers, decimals like 1.618 or 2.5e-3, fractions, + − × ÷, ^ for powers, ! for factorials, implicit multiplication (2pi, 3√2), and the constants pi, e, phi and tau. Functions: sqrt, cbrt, root(x, n), exp, ln (or log), log10, log2, log(x, b), the trig functions sin, cos, tan, cot, sec, csc and their inverses asin, acos, atan, atan2(y, x), the hyperbolic functions sinh, cosh, tanh, asinh, acosh, atanh, and abs, floor, ceil. The arithmetic is exact for rationals (1/7, sqrt(9/4) and (−8)^(1/3) are recognised as fractions) and otherwise uses 700-bit fixed point, so the number is known well beyond the deepest zoom. The address bar keeps the expression, so a link reopens the same dive.
Cutting sequence: the dashed line is the vertical geodesic from ∞ down to the centre point x = n + [0; a₁, a₂, a₃, …], which lies between the integers n and n + 1. Each ideal triangle it passes through is shaded by the edge it leaves through: L (blue) for the triangle’s left edge, R (rust) for its right edge. The topmost triangle, with vertices n, n + 1 and ∞, counts as L.
Reading from the top down, the shading comes in runs: maximal blocks of consecutive triangles of the same colour. There are a₁ blue triangles, then a₂ rust, then a₃ blue, and so on. So the run lengths are exactly the continued-fraction terms of x. The panel lists them in matching colours, as many as the current zoom resolves. Once you have zoomed in, the earliest triangles are above the window, so the runs you can see start partway through.
Poincaré disk: the same tessellation seen from the point z₀ halfway up the half-plane window, directly above the centre point x. A Möbius transformation carries the half-plane to the disk, sending z₀ to the centre, ∞ to the top and x to the bottom. Zooming in moves z₀ down the vertical geodesic toward x, so the disk view travels along that geodesic, and the cutting-sequence triangles stream past the centre. Toward a quadratic irrational the picture settles into a repeating cycle: toward φ, each zoom by a factor of φ² gives the mirror image and φ⁴ repeats it exactly. Drag sideways to slide along a horocycle, drag vertically or scroll to move toward or away from x, and click a horocycle to fly to it.
Near a rational point, the fan of geodesics from that cusp is infinite and gets arbitrarily dense. Where neighbouring arcs are less than a pixel apart they are drawn as a shaded region. Keys: + and − zoom, 0 goes home, d dives, m switches between the half-plane and the disk.