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 5 essays across 3 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

Morley's construction on a sphere. A spherical triangle with angles 76.1, 97.3, 48.8 degrees, its angle trisectors as great-circle arcs, and the inner triangle they form, with sides 0.23902, 0.23881, 0.23902.

The trisectors on a sphere almost agree

Draw Morley's construction on a sphere and the triangle the trisectors cut out is not equilateral — but its sides agree to within a tenth of a percent even when the angles add to 222°. Every step of the rotation proof survives on the sphere except one, and that one is the angle sum.

geometry · Morley

Every finite kaleidoscope on up to eight mirrors. 1: A₁; 2: A₂ B₂ H₂ G₂; 3: A₃ B₃ H₃; 4: D₄ A₄ B₄ H₄ F₄; 5: D₅ A₅ B₅; 6: E₆ D₆ A₆ B₆; 7: E₇ D₇ A₇ B₇; 8: E₈ D₈ A₈ B₈.

One determinant for every kaleidoscope

Three mirrors close up into a kaleidoscope when the angles of their triangle add to more than half a turn. In more dimensions there is no triangle to add up, but there is a matrix — the inner products of the mirrors' directions — and the mirrors close up exactly when it is positive definite. Trying every arrangement of up to nine mirrors against that one test finds Coxeter's list: three families in every dimension, a fork that runs out at eight mirrors because its determinant reaches zero at nine, and five exceptions. The same search finds the hyperbolic kaleidoscopes, and finds that they stop at five mirrors.

geometry · Regular polyhedra

The two-handled surface's octagon, eight to a corner. Regular hyperbolic octagon with angles π/4 and 64 neighbouring copies in the Poincaré disc.

An octagon only the hyperbolic plane can hold

Glue an octagon's sides in the pattern that makes a surface with two handles and its eight corners become one point, so their angles must add to a single turn. A flat octagon's add to three. The octagon that fits has corners of 45° and lives in the hyperbolic plane, where its area is forced to be exactly 4π and its four gluing moves are motions of the plane whose commutators multiply to the identity.

topology · Surface classification

Named alongside it

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

Curvature Spherical geometry Angle sum Angle trisection Circle Construction Coxeter diagram Determinant Euler characteristic Fundamental group Genus Geodesic

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