Drag a gold corner to move it. “Grow” pushes all three corners further apart — watch the angle sum fall and the area climb towards its ceiling of π.
Drag anywhere to swing a new line through P; let go to keep it. Green lines never meet the gold line L; red ones cross it. Drag P itself to move the point. Dragging is taken by the line tool here, so use the arrow keys to slide the whole plane — the lines you have drawn travel with it.
Drag the disc to slide the whole plane past you (a hyperbolic translation) — tiles swell as they reach the middle and shrink as they leave. Scroll does nothing here: in hyperbolic geometry there is no zoom, only motion.
Where's the maths? This is geometry — but geometry on a surface that is curved away from itself everywhere, which mathematicians call the hyperbolic plane. Every rule you have learned about triangles, angles and parallel lines was proved on a flat sheet of paper, and one of those rules (there is exactly one parallel line through a point) turns out to be an assumption rather than a fact. Change that assumption and everything still works logically, but the pictures change: triangles have angle sums under 180°, and through a point outside a line there are infinitely many lines that never meet it. The circle on screen is the Poincaré disc, a map of that whole infinite plane squeezed into a finite picture — like a world map, it must distort something, and what it distorts is size. Every tile you can see is exactly the same size and shape as every other; they only look as if they shrink because the map crowds infinity onto the rim. The apparently curved "lines" are the straight lines of this world: the shortest routes, which on this map are always circular arcs meeting the rim at a right angle.
Beyond the syllabus: Non-Euclidean geometry is enrichment.
A straight line (a geodesic) in the disc is either a diameter or an arc of a circle that cuts the boundary circle at 90°. Distances are measured with a ruler that stretches as you approach the rim, so the rim is infinitely far away and can never be reached. That one change of ruler is the whole of hyperbolic geometry.
The headline result, due to Gauss, Bolyai and Lobachevsky, is about triangles. If a hyperbolic triangle has angles A, B and C then
area = π − (A + B + C)
with the angles in radians. Because an area must be positive, the angle sum is always less than 180°, and the shortfall (the defect) is the area. Two consequences follow at once. A triangle can never have an area bigger than π, no matter how far apart you push its corners. And there are no similar triangles that are not congruent: the angles fix the area, so they fix the size — enlargement, which you use freely in Sec 2, simply does not exist here.
The tiling shows the same thing from the tiles' side. On flat paper, regular p-gons meeting q at a corner need each interior angle to be exactly 2π/q, and that forces (p − 2)/p = 2/q, so only {3, 6}, {4, 4} and {6, 3} work — the triangle, square and honeycomb tilings. Hyperbolically the angles can be squeezed smaller, and every pair with 1/p + 1/q < 1/2 tiles the plane. Each tile is built from 2p copies of one right-angled triangle whose angles are π/p, π/2 and π/q, reflected over and over across its own sides; the readout under the disc checks that q of the tiles' corner angles still add to exactly 360°, as they must if the tiles are to fit with no gap.
The parallel tool is the original scandal. Given a line L and a point P not on it, Euclid's fifth postulate says exactly one line through P misses L. Here the lines through P that miss L fill a whole fan, bounded by two limiting parallels that head off to the same point on the rim as L does. Everything between them misses; everything outside crosses.
M. C. Escher drew this disc four times as Circle Limit I–IV after seeing a figure of the {6, 4} tiling in a paper by the geometer H. S. M. Coxeter, who then wrote back explaining where Escher's angels and devils came from. The same geometry is the local shape of a lettuce leaf, a coral, and — with the sign of curvature flipped — the surface of the Earth, where triangles have more than 180°.
Try this: switch to Triangle, note the three angles and the sum, then press Grow three or four times. Predict before each press what happens to the sum, and check that π minus the sum (in radians) matches the area shown. Push it as far as it will go: can you make the sum smaller than 5°, and does the area ever get past π ≈ 3.1416? Then switch to Parallels and try to find a line through P that misses L in a way Euclid would allow — you will find far too many.
Controls: drag the disc to slide the plane past you (arrow keys do the same) · in Triangle mode drag a gold corner · in Parallels mode drag to swing a line through P · the panel on the left changes the tiling {p, q}, the colouring and the detail. There is no zoom: in hyperbolic geometry every view is already the same size.