Generator

boxcount

A generator in the dynamics library, called 8 times across 1 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: "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.

`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: "table"`

Dimension from the rule, and dimension from the count. A table of five sets with the dimension their construction rule forces and the dimension obtained by counting occupied boxes at shrinking sizes.

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.

after 0 steps there are two to the power 0 pieces ×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
between one and four sets this family draws ×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
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 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 sample is finer than the smallest box counted ×1
the set is one this family draws ×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 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.

The whole library · What the figures prove