Generator

the Koch curve, after 5 steps

A generator in the dynamics library, called 23 times across 5 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

boxcount is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

the Koch curve, after 5 steps. the Koch curve drawn from its own rule: replace the middle third of every segment with two sides of a triangle.

`show: "affine"`

A carpet whose two dimensions differ by 0.076. A self-affine carpet built by keeping 5 cells of a 4 by 2 grid and repeating 4 times. Its box dimension is 1.6610 and its Hausdorff dimension 1.5850.

`show: "slope"`

Box counts against box size, on logarithmic axes. For each set, the logarithm of the number of occupied boxes plotted against the logarithm of one over the box size, with a straight line fitted and its slope reported.

`show: "lyapunov"`

A dimension of 1.2576, from two stretching rates. The running averages of the Hénon map's two Lyapunov exponents, settling at 0.4177 and -1.6217. Kaplan and Yorke's formula turns them into a dimension of 1.2576 without counting a single box.

`show: "hausdorff"`

Infinite below 1.2619, nought above it. The total of the s-th powers of the diameters in the natural cover of the Koch curve, plotted against s for 4 depths. Every curve passes through one at s = 1.2619 and they separate either side of it.

`show: "boxes"`

the Koch curve in boxes of side 0.037. A set with a grid of boxes laid over it, every box containing part of the set shaded, and the number of such boxes counted.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

at depth 2 the cover's total agrees with the closed form at exponent 1.037 ×36
after 0 steps there are two to the power 0 pieces ×7
at the dimension itself the total is exactly one, at depth 2 ×6
after 6 steps there are as many pieces as the rule makes ×2
a smaller box never covers the set in fewer boxes ×1
and above it the total shrinks to nought ×1
below the dimension the total grows without bound as the cover is refined ×1
between 20 and 400 thousand steps ×1
between one and four sets this family draws ×1
between two and six depths, each between 1 and 14 ×1
each count is exactly the number of pieces at that depth ×1
how many box sizes are counted is a whole number between 3 and 6 ×1
how many times the box size is divided is a whole number between 1 and 5 ×1
how many times the rule is applied is a whole number between 2 and 6 ×1
the chosen cells are distinct positions of the grid ×1
the contraction beats the stretch, so the formula's first case applies ×1
the counted box dimension closes on the closed form ×1
the counted dimension closes on the one the rule forces ×1
the counted dimension lies strictly between a curve's and a region's ×1
the curve is drawn finer than the smallest box counted ×1
the drawn boxes are the counted boxes ×1
the exponent is swept between 0.05 and 1 either side of the dimension ×1
the first exponent is positive, which is what makes the orbit sensitive to its start ×1
the formula returns a value between a curve's dimension and a region's ×1
the Hausdorff dimension never exceeds the box dimension ×1
the horizontal contraction is the stronger one, which is what makes the carpet self-affine ×1
the map contracts area but does not collapse it ×1
the map's parameter is in the range that has an attractor ×1
the number of columns is a whole number between 2 and 6 ×1
the number of pieces and the ratio give the dimension the rule forces ×1
the number of rows is a whole number between 2 and 5 ×1
the number of times the rule is applied is a whole number between 1 and 7 ×1
the orbit stays on the attractor rather than escaping ×1
the rows carry different numbers of cells, so the two dimensions differ ×1
the rows carry equal numbers, so the two dimensions agree ×1
the sample is finer than the smallest box counted ×1
the second is negative, which is what collapses the attractor onto a set of no area ×1
the set is one built by a rule, or there is no exponent to pivot about ×1
the set is one this family draws ×1
the two exponents add to the logarithm of the area factor, which the map fixes ×1
the view is one the family draws ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Dynamics

A carpet with two dimensions

For every set on this ladder so far the two definitions of dimension agree, and the agreement is a theorem about sets built from copies of themselves scaled equally. Stretch one direction more than the other and the two numbers come apart, by an amount that can be computed exactly.

Dynamics

A dimension from the stretching rates

An attractor has no construction rule, so its dimension has to be counted — that was the rung below's argument for defining dimension by counting at all. Kaplan and Yorke's formula computes it instead, from two numbers that describe the map and never look at the set.

Dynamics

A dimension that is not a whole number

Cover a set with boxes of side ε and count how many are needed. For a line the count grows like 1/ε, for a region like 1/ε². For the Koch curve it grows like 1/ε to the power 1.26, and that exponent is as good a definition of dimension as the other two.

Logic

A line with as many points as a square

Interleave the decimal places of two numbers and one number comes out; take every other place back and the two return. The square has no more points than the segment, and dimension turns out to be invisible to counting.

Dynamics

Infinite on one side and nought on the other

Box counting returns a growth rate. Hausdorff's definition returns a measure — a quantity that is infinite for every exponent below the dimension and zero for every exponent above it, and the dimension is the one place where it is neither.

The whole library · What the figures prove