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Concept

Fractal dimension

An exponent saying how a set's content scales with the scale it is measured at, which need not be a whole number.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

the Lorenz system at ρ = 28, projected onto the x–z planepast the critical ρ of 24.74: two fixed points, neither attracting, and a trajectory that settles on neither

Two lobes and no cycle

Three equations, three variables, and a trajectory that never crosses itself, never repeats, and never leaves a region of zero volume. The set it settles onto is not a point, not a loop, and not a surface.

dynamics · strange attractor -0.4-0.20.20.4-0.50.511.52xthe sum1 term2 terms3 terms6 termseach term has 0.5 times the height of the one before and 3 times the frequency, so ab = 1.5 — above one, and thatis the condition6 terms are drawn, the last at 1,215 samples for its 122 oscillations; the sum converges because the heights area geometric series, and the slopes do not because they are multiplied by 1.5 at each step

A curve with a corner at every point

Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.

analysis · the derivative the interval1.0000 leftstage 10.6667 leftstage 20.4444 leftstage 30.2963 leftstage 40.1975 leftstage 50.1317 leftstage 60.0878 leftat stage 6 the 64 surviving intervals have total length 0.08779, and 0.91221 has been taken outthe removed pieces are a geometric series — a third, then two ninths, then four twenty-sevenths — which adds to exactly one,so what is left has no length at all

Almost none of it left, and still uncountably many

Remove the middle third of an interval, then the middle third of each piece left, and keep going. The lengths removed add to exactly the whole interval, so nothing measurable survives — and what survives can be paired off one for one with every point of the interval that was started with.

analysis · measure

Named alongside it

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

Self similarityAttractorBifurcationBijectionCantor setCardinalityContinuityConvergenceCounterexampleDifference quotientDifferentiabilityDissipation

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