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Ladder

Constant width — the ladder

4 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. A Reuleaux triangle. A curve of constant width on 3 vertices, with 6 pairs of parallel supporting lines drawn across it. Every pair is 180.1 apart.

    Round is not the only way to be the same width

    A shape that measures the same in every direction sounds like a description of a circle. It is not — there are infinitely many others, one of them is on a coin in most people's pockets, and a drill built from one cuts a nearly square hole.

    rung 1 · geometry
  2. The two numbers that make up a width. A Reuleaux polygon with 3 sides, its centre marked as the origin, and the two supporting lines with normals 33° and 213°. The perpendicular distances from the origin to the two lines are marked; they add to the width.

    The shape described from outside

    A convex shape can be given by its boundary or by the family of lines that touch it, and the second description turns the constant-width condition into one line of arithmetic — after which the perimeter falls out, and curves with no corners at all can simply be written down.

    rung 2 · geometry
  3. Area at equal width: the triangle least, the circle most. A bar for each curve of constant width the family draws, all at the same width, with the bar's length its enclosed area and the extremes marked.

    The least area a width can hold

    Barbier's theorem says every curve of constant width has the same perimeter, which removes perimeter as a way of telling the family apart. Area is not like that — the circle holds the most and the Reuleaux triangle the least — and the reason the minimiser has corners is a constraint rather than a preference.

    rung 3 · geometry
  4. The Reuleaux tetrahedron's width runs from 2.83 to 2.9. The largest and smallest width of the intersection of four balls, plotted against the polar angle of the measuring direction, with the constant width it would need to have drawn flat.

    The same question in space

    Intersect four balls at the corners of a tetrahedron and the result is not of constant width — it misses by two and a half per cent, computed exactly. Repairing it gives a body that is, and whether that body is the smallest of its kind has been open for a century.

    rung 4 · geometry

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