Paper 18
Self-similar fractals: the Koch curve and the Cantor set
A whitepaper on the two constructions that make fractal dimension concrete: a dust with more points than the integers yet zero length, and a curve of infinite length bounding a finite area. Both are built by one rule repeated forever, and both carry a dimension that is not a whole number.
Why these two figures
The companion paper "Strange attractors and fractal dimension" (14) argued that the geometry a chaotic system settles onto is a fractal, and it used one figure in passing to make the idea of non-integer dimension believable: the Cantor set. This paper slows down on that figure and its sibling the Koch curve, because they are the cleanest place to see what a fractal is before the extra difficulty of a strange attractor is added.
A strange attractor is a fractal you have to earn: it is produced by stretching and folding a flow, its self-similarity is only approximate and statistical, and its dimension has to be estimated numerically. The Koch curve and the Cantor set are fractals you can draw by hand. Their self-similarity is exact, their dimension is a short calculation, and they were invented decades before the word "fractal" existed, as deliberate counterexamples to the assumption that any reasonable set has a whole-number dimension. They are the controlled experiment. Once they are clear, the attractor is the same idea let loose in a dynamical system.
The Cantor set: a dust of measure zero
Georg Cantor described the set in 1883. The construction could not be simpler. Take the closed interval from 0 to 1. Remove the open middle third, the piece from 1/3 to 2/3, leaving two closed intervals. Remove the middle third of each of those. Remove the middle third of each of the four that remain. Continue forever. What survives every removal is the Cantor set.
Two facts about what is left seem to pull in opposite directions, and holding them together is the whole lesson.
The set has length zero. At each step you keep two-thirds of the length you had, so after n steps the total length of the surviving pieces is (2/3) raised to the n. That sequence marches to zero. The removed pieces, added up, account for the entire original length of 1. In the language of measure, the Cantor set has Lebesgue measure zero: it is negligible as far as length is concerned.
And yet the set is uncountably infinite. It is not a scattering of a few leftover points. Write each number in the interval in base 3. A point survives every middle-third removal exactly when it can be written using only the digits 0 and 2, never a 1. There are as many such numbers as there are numbers in the whole interval, an uncountable infinity, the same size as the entire line you started from. So the Cantor set has as many points as the line it is carved out of, while occupying none of its length.
That is the paradox a single dimension resolves. Length, a one-dimensional measure, calls the set nothing. Counting, a zero-dimensional notion, calls it as big as the line. The truth is in between, and it needs a dimension between 0 and 1 to state.
The Koch curve: finite frame, infinite edge
Helge von Koch published his curve in 1904, looking for a curve that has a tangent nowhere, a shape so crinkled that at no point can you draw the single straight line a smooth curve would locally look like. The construction is again one rule repeated. Take a straight segment. Replace its middle third with the two other sides of an equilateral triangle built on that third, so a triangular bump juts out where the flat middle used to be. One segment becomes four, each one-third as long. Apply the same replacement to all four. Continue forever.
The bookkeeping runs opposite to the Cantor set. Every step multiplies the number of segments by 4 and the length of each by 1/3, so the total length is multiplied by 4/3 at every step. That grows without bound: the Koch curve has infinite length. But it never escapes a small region of the page. Each bump is strictly smaller than the one before, and the whole curve stays inside a thin band around the original segment. Assemble three Koch curves into a triangle and you get the Koch snowflake, whose boundary is infinitely long yet encloses a finite area, exactly 8/5 of the area of the triangle you started from.
Infinite length, finite frame, and a tangent nowhere. As with the Cantor set, length is the wrong ruler. A curve is supposed to be one-dimensional, but this one is too rough for a one-dimensional length to stay finite. It needs a dimension above 1.
Similarity dimension: counting the copies
For a shape built from scaled copies of itself, there is a direct way to read off its dimension, and it agrees with the box-counting recipe from paper 14 without any grids.
Look at how ordinary shapes behave under scaling. Take a line segment and shrink it by a factor of 2; it takes 2 of those to rebuild the original. A square shrunk by 2 needs 4 copies to rebuild; a cube needs 8. The pattern is that a D-dimensional object shrunk by a factor r is rebuilt from r-to-the-D copies. Line: 2 = 2 to the 1. Square: 4 = 2 to the 2. Cube: 8 = 2 to the 3. Turn that around. If a shape is made of N copies of itself each scaled down by a factor of r, its dimension is the exponent that makes N equal r-to-the-D:
D = log N / log r
Now apply it. The Cantor set is 2 copies of itself, each scaled by 3 (each half is a one-third-size replica of the whole). So D = log 2 / log 3, which is about 0.6309. Between a point and a line, exactly as the length-versus-counting paradox demanded. The Koch curve is 4 copies of itself, each scaled by 3. So D = log 4 / log 3, about 1.2619. Between a line and a plane, exactly as its infinite length demanded.
| Figure | Copies N | Scale r | Dimension log N / log r | Sits between |
|---|---|---|---|---|
| Line segment | 2 | 2 | 1.0000 | (a line) |
| Cantor set | 2 | 3 | 0.6309 | point and line |
| Koch curve | 4 | 3 | 1.2619 | line and plane |
| Filled square | 4 | 2 | 2.0000 | (a plane) |
The formula degrades gracefully to the answers you already trust for the line and the square, and returns non-integers exactly for the two figures that broke the whole-number assumption. That is the sense in which fractal dimension is not a trick but a genuine extension of dimension.
A note on rigor. For strictly self-similar sets that do not overlap, this similarity dimension coincides with the Hausdorff dimension that Felix Hausdorff defined in 1918, the measure-theoretic notion that made non-integer dimension precise. Benoit Mandelbrot gathered these scattered objects, long treated as a gallery of pathological curiosities, under the name "fractal" in 1975 and argued they were the right language for the roughness of real coastlines, clouds, and turbulence. The interactive figure at /fractals lets you set the iteration count and watch both the copies and the dimension.
The bridge back to strange attractors
Return to the slice through the Lorenz attractor described in paper 14. Cut the attractor with a plane and look at the points where trajectories cross it. You do not see a smooth curve. You see a curve that, magnified, splits into several near-parallel strands, and each strand under further magnification splits again, without end. That cross-section is, in its structure, a Cantor set. The stretching-and-folding that builds the attractor is doing to a sheet of trajectories exactly what the middle-third removal does to an interval: separating it into scaled copies of itself at every level.
This is why the Lorenz attractor's dimension comes out near 2.06 rather than exactly 2. It is a two-dimensional sheet in the folding direction, times a Cantor set in the transverse direction, and the Cantor factor adds its fractional 0.06 or so of extra dimension. The non-integer part is the fingerprint of the fold. The difference between the exact fractals here and the attractor is only that the attractor's self-similarity is approximate and generated by dynamics rather than exact and generated by a hand-applied rule. The dimension means the same thing in both.
The connection to memory, kept honest
The corpus uses dynamical systems as a disciplined metaphor for memory, and this paper adds nothing new to the mechanism, only to the vocabulary. The Cantor set makes vivid one idea the reconsolidation picture leans on: that a set can be negligible by one measure and enormous by another. A labile trace is a small target in time, a brief window, while carrying the full content of a memory. That is a loose echo of measure-zero-yet-uncountable, and it is offered as an echo, not a claim.
No memory has a measured fractal dimension. No brain circuit has been shown to be a Koch curve or a Cantor set. The strong evidence that a reactivated memory becomes briefly labile is from animal work, established through amygdala studies such as Nader, Schafe and LeDoux (2000); its extension to human memory remains emerging and in places contested, and none of the geometry here touches that gap. What these two figures provide is intuition for a single word, dimension, that the strange-attractor paper needs and that the phrase "memory in motion" borrows. The mathematics is exact. The bridge to memory is a metaphor, and it stays labeled as one.
What to take away
- The Cantor set removes middle thirds forever. What remains has length zero yet contains uncountably many points, as many as the whole line. Its dimension is log 2 / log 3, about 0.63.
- The Koch curve replaces middle thirds with bumps forever. It has infinite length inside a finite region and a tangent nowhere. Its dimension is log 4 / log 3, about 1.26. The Koch snowflake bounds a finite area of 8/5 the starting triangle.
- For a shape made of N copies of itself scaled by r, the similarity dimension is log N / log r. It returns 1 for a line and 2 for a square, and non-integers for these fractals. For exact self-similar sets it equals the Hausdorff dimension.
- A cross-section of the Lorenz attractor is a Cantor set. That transverse fractal is why its dimension is about 2.06 rather than 2. Exact fractals and strange attractors carry the same kind of dimension; only the attractor's self-similarity is approximate and dynamically generated.
- For this corpus the payoff is intuition for one word. Length and counting can disagree about the size of a set, and dimension is the number that reconciles them. The bridge to memory is analogy, not mechanism.
Sources
- Cantor (1883). Ueber unendliche, lineare Punktmannichfaltigkeiten. Mathematische Annalen. Origin of the Cantor set.
- von Koch (1904). Sur une courbe continue sans tangente, obtenue par une construction geometrique elementaire. Arkiv for Matematik. Origin of the Koch curve.
- Hausdorff (1918). Dimension und ausseres Mass. Mathematische Annalen. The measure-theoretic definition of dimension.
- Mandelbrot (1982). The Fractal Geometry of Nature. W. H. Freeman. The unifying account and the term "fractal."
- Falconer. Fractal Geometry: Mathematical Foundations and Applications. For similarity dimension, Hausdorff dimension, and the coincidence between them for self-similar sets.
- Strogatz. Nonlinear Dynamics and Chaos. For the Cantor-set cross-section of strange attractors and box-counting dimension.
- Companion papers in this corpus: "Strange attractors and fractal dimension" (14), for the attractor geometry and box-counting; "The limit of prediction" (02), for the Lorenz system.
- Interactive tool: /fractals, for the Koch and Cantor constructions and the live similarity-dimension readout.