Matpic
mathematics, in pictures
Concept

Geodesic

The shortest path between two points along a surface, staying on the surface the whole way.

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

Also named here as parallel postulate — the same set of essays touches all of them, so they are one junction rather than several.

Pone line, one point off it, and 4 lines through the point that never meet itevery arc here meets the rim at a right angle, and every miss was checked for crossingsinside the disc

Two worlds that both obey the rules

A statement is independent of a list of axioms when there is a structure satisfying the axioms where it holds and another where it fails. That is not a claim about what nobody has managed to prove — it is a proof that nobody can.

logic · models 50°60°71.3°legs of 50° and 60° give a hypotenuse of 71.25°,against the 78.10° the flat theorem asks for — shortby 6.85°the three angles of this triangle add to more than ahalf turn, and the excess is its area

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

Named alongside it

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

Parallel postulateAxiomConsistencyCurvatureHyperbolic geometryIndependenceInterpretationModelPythagorean theoremRelative consistencySpherical excessSpherical geometry

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