The room a jagged graph takes up

A continuous function on an interval has a graph with no gaps in it, drawn without lifting the pen. That makes it a curve, and a curve is the standard example of a set whose box count grows like one over the box size: dimension one.

For a smooth function that is right. For a function with a corner at every point it is not, and the reason can be read off how the function was built.

The graph never leaves a band of height one and a half, and it has no gaps. It still needs far more boxes than a line of the same length would, and the count grows with its own exponent — 1.585 for this function, and any number between one and two for a function built the same way with a different shrinking factor.

Raising midpoints #

The construction is the simplest one that produces roughness at every scale on purpose.

Start with the flat function on the interval from nought to one. Raise the midpoint of the whole interval by one half and join it to the ends, which draws a tent. Now take the two halves: raise the midpoint of each by $w/2$ above the line joining its own ends. Then the four quarters, each by $w^2/2$, and so on — at stage $k$, every one of the $2^k$ intervals of width $2^{-k}$ gets its midpoint raised by $w^k/2$.

The limit is the function

$$T_w(x) = \sum_{k=0}^{\infty} w^k \, s(2^k x),$$

where $s$ measures the distance from a number to the nearest whole number — a sawtooth of tents. Teiji Takagi described the case $w = \tfrac12$ in 1903 as a simple example of a continuous function with no derivative anywhere, and Otto Landsberg studied the whole family in 1908. The figure computes it both ways, by raising midpoints level by level and by summing the series, and requires the two to agree to twelve decimal places at scattered points.

The function is continuous for any $w$ below one, because the raises form a geometric series and so the stages converge uniformly — the kind of convergence that keeps continuity. What $w$ controls is not whether the limit exists but how rough it is.

Why the exponent is two plus a logarithm #

Cover the graph with columns of width $\varepsilon = 2^{-k}$. Inside one column, the stages coarser than $k$ are straight lines — every earlier tent has its corners at wider spacing — and the stages finer than $k$ add wiggles whose heights are $w^k/2$, $w^{k+1}/2$ and so on. So the graph rises and falls across the column by an amount proportional to $w^k$, give or take a constant and a straight-line trend.

A column whose graph rises and falls by $h$ needs about $h/\varepsilon$ boxes of side $\varepsilon$ to cover it. That is $w^k / 2^{-k} = (2w)^k$ boxes per column, and there are $2^k$ columns:

$$N(\varepsilon) \approx (4w)^k = \left(\frac{1}{\varepsilon}\right)^{\log_2(4w)} = \left(\frac{1}{\varepsilon}\right)^{2 + \log_2 w}.$$

The dimension is two, minus how fast the vertical detail shrinks when the horizontal detail halves, measured in halvings. If the raises did not shrink at all, $w = 1$, the graph would try to fill a region and the exponent would be two. If they shrank as fast as the intervals, $w = \tfrac12$, each column would need a bounded number of boxes and the exponent would be one.

The shaded bands in the figures are exactly that argument, drawn. Each band is one column’s rise and fall; the count the figure fits is the sum of the band heights divided by the band width, at seven widths from an eighth to a five-hundred-and-twelfth.

The same graph inside itself #

The formula’s reason has a sharper form. Squeeze the whole graph horizontally by a half and vertically by $w$, add the straight line from $(0,0)$ to $(\tfrac12, \tfrac12)$, and the result is exactly the left half of the graph:

$$T_w\!\left(\tfrac{x}{2}\right) = \tfrac{x}{2} + w\, T_w(x).$$

The figure checks that identity on the grid. It says the graph is self-affine: made of copies of itself shrunk by different factors in the two directions, like the carpet whose rows and columns contract differently. The carpet showed that such sets can have a box dimension and a Hausdorff dimension that differ. For these graphs the two are expected to agree, and for many values of $w$ they are known to, but it is the self-affinity rather than any similarity that sets the count.

The factor $w$ also measures smoothness in the ordinary sense. Two points a distance $\delta$ apart have values that differ by at most a constant times $\delta^{H}$, with $H = -\log_2 w$ — the function is Hölder continuous with exponent $H$, and not with any larger exponent. So the dimension is $2 - H$. A function whose values change like the square root of the step has a graph of dimension one and a half; one whose values change like the step itself — a function with bounded slopes — has a graph of dimension one.

Takagi’s function, at the edge #

$w = \tfrac12$ is Takagi’s original function, and the formula gives exactly one for it.

This graph has dimension one and still has no tangent anywhere, and it still has infinite length. Those are consistent. Across a column of width $\varepsilon$ the raises finer than the column add up to about $\varepsilon/2$ each — at $w = \tfrac12$ the heights shrink exactly as fast as the widths — and there are about $\log_2(1/\varepsilon)$ of them that matter, so a column’s rise and fall is about $\varepsilon \log_2(1/\varepsilon)$ rather than $\varepsilon$. Dimension counts powers and ignores logarithms; length does not.

The measured slope of 1.099 is that logarithm, seen over column widths from an eighth to a five-hundred-and-twelfth. Over any finite range a factor of $\log(1/\varepsilon)$ is indistinguishable from a small extra power, and it only reveals itself as a logarithm by failing to settle as the range widens. The figure that measures several graphs at once requires the direction of that bias rather than hiding it inside a looser tolerance.

Below one half the raises shrink faster than the widths, the slopes of all the tents add up to a finite total, and the function has bounded slopes. Its graph has finite length and dimension one with no logarithm, and it is differentiable at almost every point — rough only at the dyadic corners.

A random path built the same way #

Replace each raise with a random one. At stage $k$, instead of raising every midpoint by $w^k/2$, move it up or down by a Gaussian amount whose typical size is $2^{-k/2}/2$.

This is Paul Lévy’s construction of Brownian motion, the continuous limit of a random walk with smaller and smaller steps. The figure checks that the random raises at a fine level have the variance the construction prescribes.

The typical size shrinks by $\sqrt{\tfrac12}$ each level, which is the same factor as Takagi–Landsberg with $w = 1/\sqrt2$. The column argument does not care whether the raises are fixed or random, only how fast their size shrinks, so it predicts $2 + \log_2(1/\sqrt 2) = 1.5$ for both. A deterministic function and a random path, built by the same rule with the same shrinking factor, have graphs of the same dimension. S. James Taylor proved in 1953 that a Brownian graph has Hausdorff dimension exactly one and a half, with probability one.

A single random path is one sample, and its count carries the randomness of that sample: 1.465 for the path drawn. The figure below also builds eight independent paths and requires their average to lie within 0.03 of one and a half; it comes out at 1.494.

Why the random raises shrink by a square root #

The factor $\sqrt{\tfrac12}$ in Lévy’s rule is not a choice made to match Takagi–Landsberg; it is forced, and the reason is the most familiar fact about sums of random steps.

A path that moves by independent random amounts has a displacement whose variance adds up over time. Over an interval of length $t$ the variance is proportional to $t$, so the typical displacement is proportional to $\sqrt t$ — the wobble grows like the square root of the number of steps, and it is that growth which makes the displacement settle into a bell curve of steadily widening spread. Halve the interval and the typical displacement across it shrinks by $\sqrt{\tfrac12}$, not by a half.

The midpoint of an interval, given its two ends, departs from the straight line between them by an amount with exactly that scaling, and the construction’s raises at stage $k$ have typical size $2^{-k/2}$ for that reason. Any other factor would describe a different kind of path. Brownian motion has dimension one and a half because its steps are independent, and independence is what makes variances add and displacements grow like square roots.

Change the independence and the factor changes with it. A path whose increments are positively correlated — a rise tending to be followed by a rise — spreads faster than a square root, like $t^H$ with $H$ above one half, and is smoother; one whose increments are negatively correlated spreads more slowly and is rougher. Benoit Mandelbrot and John Van Ness introduced the stationary paths of this kind in 1968 as fractional Brownian motion, and their graphs have dimension $2 - H$ with probability one — the same formula as the deterministic family, with the Hölder exponent now set by how the randomness is correlated rather than by a fixed $w$.

That makes the dimension of a measured record a statement about its correlations. A graph of dimension noticeably below one and a half says that its rises tend to persist; one above says they tend to reverse. The number is read off the same count of boxes either way, and it is one of the few ways to see a correlation that runs across every scale at once.

Four graphs, counted together #

On these axes a dimension is a slope, and the four lines are four straight lines. The middle two nearly coincide, and they should: $w = \sqrt{\tfrac12}$ and Brownian motion have the same shrinking factor, and the lines differ only by the randomness in one path and by a constant in front. The graph with the fixed raises and the graph with the random ones are two different curves that occupy the plane at the same rate.

The three Takagi–Landsberg slopes agree with the formula to within 0.02, and that agreement was measured before the figure was built rather than tuned after. At the two ends of the range the count is not so faithful, and each end fails for its own reason.

Both errors are in the direction their causes predict, and the figure requires the direction rather than a tolerance wide enough to swallow them. Near $w = \tfrac12$ the logarithm adds slope; near $w = 1$ the grid’s finest detail has height $w^{20}$, which at $w = 0.85$ is still four per cent of the whole, and the counts at the finest columns come out too small. A wider tolerance would make both warnings vanish, and it would also make the check unable to see a formula that was simply wrong.

Where the count needs its hypotheses #

The function has to be continuous. A graph with jumps can need any number of boxes in a column, and the rise-and-fall count stops meaning anything. Every function here is a uniform limit of continuous stages, which is what guarantees there is a graph to count.

The rise and fall is sampled. Each column’s height is the largest value minus the smallest over the grid points in it, which is never more than the true rise and fall and is close to it only when the detail below the grid spacing is small. That is the hypothesis the $w = 0.85$ count violates.

A dimension is a power law, and a logarithm is not one. The count at $w = \tfrac12$ grows like $\varepsilon^{-1}\log(1/\varepsilon)$, whose box dimension is exactly one, and whose fitted slope over any finite range is larger. A dimension estimated from a slope cannot tell the two apart without a much wider range of scales than a picture can hold.

And one random path is one sample. The Brownian slope of 1.465 is a measurement on one path; the theorem is about almost every path, and the tolerance for a single sample has to be wider than the tolerance for an average of eight.

The cosine sums, and a proof that took a century #

Weierstrass’s original function is built from cosines rather than tents, $\sum a^k \cos(b^k \pi x)$ with $ab > 1$, and the column argument applies to it in the same way: detail of width $b^{-k}$ with height $a^k$ gives a box dimension of $2 + \log a/\log b$. James Kaplan, John Mallet-Paret and James Yorke proved that formula for the box dimension in 1984.

The Hausdorff dimension is harder, because it allows covers of every shape and size and the column argument only bounds it from above. For the cosine sums the equality of the two dimensions was an open problem for more than three decades after the box dimension was settled. Krzysztof Barański, Balázs Bárány and Julia Romanowska proved it for a range of parameters in 2014, and Weixiao Shen proved it in 2018 for every $a$ and every whole number $b$ with $ab > 1$.

That gap is the same one the Hausdorff measure opened between a count on a grid and the smallest possible cover. For a self-similar set the two agree by a general theorem. For a self-affine graph there is no such theorem, and each family has needed its own argument.

Roughness drawn as thickness, over two and a half decades #

The graphs are drawn as envelopes, one vertical stroke per pixel column spanning the lowest and highest values in it. That is the honest way to draw a function whose detail continues below a pixel, and it means the drawing shows the roughness as thickness rather than as a line. Past about a thousand points no drawing of these graphs can do better.

The counts use seven column widths from an eighth to a five-hundred-and-twelfth, on a grid of $2^{20}$ points. That is two and a half decades of scale, which is enough to measure a slope and not enough to see a logarithm for what it is. The exactness of 1.585 belongs to the formula; the figure’s 1.582 is a measurement consistent with it.

And the Brownian path in the figures is one path drawn from a seeded pseudo-random sequence. Another seed draws a different path with a slope a few hundredths different, and the statement that the dimension is one and a half is a theorem about the typical path rather than about the one on the page.

Still open: a dimension for a graph nobody built #

Every graph here comes with a rule, and the rule is what makes the count trustworthy: it says the scaling holds at every level, including the levels below the grid. A record of something measured — the height of a coastline along a straight line, a temperature over years, a price over days — comes with no rule. It has a finest scale, set by how often it was sampled, and a coarsest, set by how long it runs, and in between it can be counted exactly the way the figures count.

What the count cannot say is whether there is a single exponent to find. The same rise-and-fall calculation applied to such a record produces a slope over whatever range of scales is available, and the slope usually looks convincing on logarithmic axes. Whether the record has a dimension at all — a power law holding across scales the data do not reach — is not something any count of it can establish, and for most records it is an open question about the thing measured rather than about the measuring.

Roughness is a rate, not an amount #

The habit is about what “rough” means.

The graph in the first figure never strays more than about one and a half from the axis; the Brownian path strays further, and Takagi’s function strays less. None of those heights predicts the dimension. The dimension is set by how the size of the detail shrinks from one scale to the next, and not by how big the detail is at any scale. Doubling every raise doubles the graph’s height and leaves its dimension exactly where it was; changing the factor by which the raises shrink, even slightly, changes it.

That is worth carrying to any comparison of rough things. Two curves can look equally jagged in one picture and have different dimensions, because the picture shows one scale; two curves can look nothing alike — a fixed sawtooth sum and a random path — and have the same dimension, because the dimension compares scales. Ask how the detail scales, not how much of it there is.