Generator

the Koch curve, after 5 steps

A generator in the dynamics library, called 41 times across 8 essays. Below: what it draws with nothing chosen 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.

With nothing chosen

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.

A carpet whose two dimensions differ by 0.076

Box counts against box size, on logarithmic axes

A mass split 10 times, 0.3 to the left and 0.7 to the right

A spectrum of dimensions from 0.51 to 1.74

Mass-weighted box counts for 4 exponents, against the formula

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.

piece 0's mass is read off its binary digits ×1024
at depth 2 the cover's total agrees with the closed form at exponent 1.037 ×36
0 right-hand choices are shared by C(10, 0) pieces ×19
the variogram estimate is unbiased on fractional Brownian motion at D = 1.9 ×9
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
the midpoint construction and the series agree at 1/2²⁰ ×6
f at q = 0 is the least value of qα + β(q) ×5
at α = 0.5 the measured memory falls as β rises ×4
the measured dimension at α = 0.5 is near 2 − α/2 ×4
the counted slope for q = 0 is β(0) from the formula ×3
the counted slope for w = 0.6 is 2 + log₂ w ×3
the measured Hurst exponent at β = 0.2 is within reach of 1 − β/2 ×3
α is minus the slope of β at q = 0 ×3
after 6 steps there are as many pieces as the rule makes ×2
f at q = -3 is the least value of qα + β(q) ×2
a finite number of splits undercounts ×1
a smaller box never covers the set in fewer boxes ×1
a thousand samples already pin the dimension ×1
and above it the total shrinks to nought ×1
and clearly different memory ×1
and its limiting dimension is the curve's ×1
and none is larger than the support ×1
and the average over eight independent paths is closer ×1
and the point of contact is the information dimension ×1
and they make up the whole support ×1
at q = 0 every piece counts once, which is the support's dimension ×1
at q = 1 the curve touches the diagonal ×1
at q = 1 the moment is the total mass, so β(1) = 0 ×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 mass is split is a whole number between 1 and 12 ×1
how many times the rule is applied is a whole number between 2 and 6 ×1
more splits bring the counts closer to the curve ×1
near w = 1 detail finer than the grid is still large and the count reads low ×1
near w = ½ the count carries a logarithm and reads high ×1
no set of points is larger than its own scaling exponent allows ×1
one to five exponents between −2 and 6, leaving out 1, whose sum is always the whole mass ×1
one to four factors between ½ and 0.9 ×1
one to three increasing split counts, up to 1000 ×1
splitting in two, level times, makes two to that power pieces ×1
the box count moves by more than three tenths across the units ×1
the box count reads low on rough paths and close on smooth ones ×1
the box level marked is a whole number between 2 and 7 ×1
the Brownian path's counted slope is near 1.5 ×1
the chosen cells are distinct positions of the grid ×1
the circulant embedding's eigenvalues are non-negative ×1
the comparison measure is uneven enough to have a curve ×1
the contraction beats the stretch, so the formula's first case applies ×1
the count's exponent is the curve's at the matching q ×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 counted slope for q = -1 is β(-1) from the formula ×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 factor each level's raise shrinks by is between 0.3 and 0.95 ×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 left half is a line plus the whole graph shrunk by w ×1
the local dimension runs from near 1.8 at short lags to near 1.2 at long ones ×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 marked columns are the counted columns ×1
the masses add to one ×1
the measure lives on the interval or on the Cantor set ×1
the number of box sizes is a whole number between 4 and 10 ×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 sampled points is a whole number between 50000 and 1000000 ×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 raises at a fine level have the variance the construction prescribes ×1
the rough, long memory record's measured dimension is near 2 − α/2 ×1
the rough, short memory record's measured dimension is near 2 − α/2 ×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 seed of the Brownian path is a whole number between 1 and 99 ×1
the seed of the records is a whole number between 1 and 99 ×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 share of the mass the left piece of each split carries is between 0.02 and 0.98 ×1
the shares of the measure drawn for comparison is between 0.02 and 0.98 ×1
the smooth, long memory record's measured dimension is near 2 − α/2 ×1
the smooth, short memory record's measured dimension is near 2 − α/2 ×1
the two exponents add to the logarithm of the area factor, which the map fixes ×1
the two records have the same measured dimension ×1
the variogram estimate ignores the units ×1
the view is one the family draws ×1
while the Hurst exponent still spreads widely at a quarter of a million ×1
with equal shares every point scales alike ×1
α is minus the slope of β at q = -2 ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Dynamics

A carpet with two dimensions

For every self-similar set met 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 for every rate of crowding

Spread a unit of mass over an interval by splitting it unevenly, again and again, and the result covers the whole interval while crowding almost all of its weight onto a set of smaller dimension. Every rate of crowding picks out its own set of points with its own dimension, and the whole family is read off one curve.

Dynamics

A dimension from the stretching rates

An attractor has no construction rule, so its dimension has to be counted — which was the whole case for defining dimension by counting. 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.

Dynamics

The room a jagged graph takes up

The graph of a continuous function is a curve, and a curve ought to have dimension one. Build the function by raising midpoints — by an amount that shrinks more slowly than the intervals do — and its graph needs more boxes than any line, at a rate fixed by one ratio. A random path built the same way needs exactly as many.

Dynamics

Two numbers in one jagged record

A measured record comes with no rule, so its dimension has to be estimated from a finite stretch of samples — and two things go wrong that never trouble a graph built from a formula. The popular estimator depends on the units the record is written in, and the dimension, which describes the record up close, turns out to be independent of its memory, which describes it from far away. For a self-affine path the two are tied by D = 2 − H; for a record, they are two numbers.

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