Concept

Hyperbolic geometry

The geometry in which, through a point off a line, infinitely many lines miss it, and the angles of a triangle add to less than half a turn. It has constant negative curvature, and most surfaces with two or more handles carry it.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A right triangle on a sphere. A spherical triangle with a right angle where the equator meets a meridian and legs of 50 and 60 degrees; its hypotenuse is shorter than the flat theorem predicts.

The triangle that a globe gets wrong

On a sphere, a right triangle with legs of fifty and sixty degrees has a hypotenuse of seventy-two, not seventy-eight. The theorem is not approximately true there — it is false, and what replaces it says exactly how much room the surface has.

geometry · Pythagoras

A straightedge construction on a circle, and the same construction moved by a map that keeps the circle. Two copies of one straightedge construction on a circle — six points, five chords and their crossings — the second the image of the first under a projective map fixing the circle. Chords and crossings correspond exactly, but the centre (orange) is carried to (0.551, 0.000).

The centre a straightedge cannot find

Give a straightedge one circle and its centre, and it can do everything a compass can. Take the centre away and it cannot even find it again — because to a straightedge a circle has no centre. The maps that keep a circle and its straight lines are the motions of the hyperbolic plane, and in that plane the centre is a point like any other.

computation · Compass-only

Named alongside it

The objects these essays reach for when they reach for this one.

Circle Construction Curvature Geodesic Invariant Operation set Parallel postulate Polar Projective transformation Pythagorean theorem Spherical excess Spherical geometry

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