# Chapter ᚓ(E): The cutting
*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T20, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*
[Guide for AI readers](index.md) · [Its glossary cards](ch20-cards.md) · [Its references](ch20-refs.md) · [Previous chapter](ch19.md) · [Next chapter](ch21.md)
*A footnote such as [r-20.3] is reference entry 20.3 and links to it. Section 20 of 28; chapters are named by their Ogham letter, never by number.*
ᚓ – [Eadhadh](ch20-cards.md#g-og-edad): associated with the aspen tree, Section 20
## Chapter ᚓ(E): The cutting
> “Je le vois, mais je ne le crois pas. (I see it, but I do not believe it.)”
> — Georg Cantor, to Richard Dedekind · letter of 29 June 1877, on finding that a line and a square have the same number of points [r-20.21](ch20-refs.md#r-20.21)
Formula: ([λ(𝒞)](ch20-cards.md#g-lamC) + [λ(ℚ ∩ [0, 1])](ch20-cards.md#g-lamQ) + [c′(½)](ch20-cards.md#g-cantorderiv))/(([dimH 𝒞](ch20-cards.md#g-dimH) + [Σ 2n−1/3n](ch20-cards.md#g-removed) + [c(½)](ch20-cards.md#g-cantorfn) + [λ(SVC)](ch20-cards.md#g-fatcantor)) {brace: [Cantor](ch20-cards.md#g-cantor)} + ([dimH Koch](ch20-cards.md#g-koch) + [dimH Sierpiński](ch20-cards.md#g-sierp) + [dimH Menger](ch20-cards.md#g-menger)) {brace: other dimensions} + ([5](ch20-cards.md#g-banachtarski)) {brace: a paradox} ) {brace: [measure theory](ch20.md#ch-T20-measure-theory)} · ([f2(½) − ½](ch20-cards.md#g-logisticfixed))/(2·[δ](ch05-cards.md#g-feig)·|[αF](ch05-cards.md#g-feigA)|) {brace: [dynamical systems](ch20.md#ch-T20-dynamical-systems)}
#### Type of number
Proven transcendental · Status open ·
Key concepts: Measure theory and dynamics. It includes symbols you may not have seen before, such as: [λ](ch20-cards.md#g-lamC), [dimH](ch20-cards.md#g-dimH), [𝒞](ch20-cards.md#g-cantor), and [δ](ch05-cards.md#g-feig). Zero here is size without weight: an uncountable set of total length zero, and a map that settles and stops moving.
That was muddy! Switch off the engine for a moment and look down at the gaps in the exposed ground from the cutting. What would be left if we kept taking pieces away? Our mathematical excavation will use an interval and one repeated instruction. Later, another repeated instruction will let us watch something settle, the way the mud we churned up is settling now—or, under different conditions, behave much less quietly.
Measure theory asks how much space a set occupies. Dynamics asks what repeated application of a rule does. The same interval, [0, 1], gives us a place to try both.
Field notes
### [measure theory](ch20.md#ch-T20-measure-theory) {#ch-T20-measure-theory}
[Lebesgue measure](ch20-cards.md#g-lebesgue) extends familiar length to far more complicated sets, while keeping countable additivity: disjoint measurable pieces have measures that add, even when there are countably many of them. The recurring surprise is that “how many points?” and “how much length?” can give very different answers. Three sets here each defeat a tempting shortcut, from where a set sits or how many points it has to how long it is: [the rationals](ch01-cards.md#g-rats) are [dense](ch20-cards.md#g-dense), turning up in every interval however short, yet have no length; the [Cantor set](ch20-cards.md#g-cantor) has uncountably many points, yet no length; and the fat [Cantor set](ch20-cards.md#g-cantor) below is nowhere [dense](ch20-cards.md#g-dense), containing no interval at all, yet keeps half the interval’s length. [r-20.5](ch20-refs.md#r-20.5) [r-20.8](ch20-refs.md#r-20.8)
#### Numerator
[λ(𝒞)](ch20-cards.md#g-lamC) = 0; the [Cantor set](ch20-cards.md#g-cantor), uncountable yet weightless: the cutting in one set.
Remove the open middle third of [0, 1], then the middle third of each piece left, forever. After n stages the remaining length is (2/3)ⁿ, which tends to zero. [r-20.1](ch20-refs.md#r-20.1) [r-20.2](ch20-refs.md#r-20.2) Yet what survives is far from empty: it is exactly the numbers that can be written in base 3 with only the digits 0 and 2 (including [1/3](ch04-cards.md#g-third), which in base 3 is 0.1 but can also be written 0.0222…). Each survivor is an endless string of two-way choices, 0 or 2, just as a binary expansion is an endless string of 0s and 1s, so there are as many survivors as there were points in the interval we started with. [r-20.4](ch20-refs.md#r-20.4)
[λ(ℚ ∩ [0, 1])](ch20-cards.md#g-lamQ) = 0; [the rationals](ch01-cards.md#g-rats), [dense](ch20-cards.md#g-dense) everywhere and weighing nothing.
The rationals can be listed, first, second, third, and so on. Pick any length ε, however tiny: cover the first rational with an interval shorter than ε/2, the next with one shorter than ε/4, and so on, and the whole cover is shorter than ε. Since ε can be as small as we like, the rationals have no length at all. Yet every interval, however short, contains one. [r-20.5](ch20-refs.md#r-20.5)
[c′(½)](ch20-cards.md#g-cantorderiv) = 0; the devil’s staircase, which climbs while being flat [almost everywhere](ch20-cards.md#g-aeverywhere).
The Cantor function rises continuously from 0 to 1 but is constant on every removed interval; across the middle third (⅓, ⅔) [it sits at ½](ch20-cards.md#g-cantorfn), so its slope at ½ is 0. At every point off the weightless Cantor set its slope is zero, so it is flat [almost everywhere](ch20-cards.md#g-aeverywhere), and still it climbs. Back in Chapter ᚉ(C), the gallery of motion, [the fundamental theorem of calculus](ch11-cards.md#g-ftc) recovered a change by integrating a derivative; here that would give a climb of 0, yet the staircase rises by 1. It escapes because it lacks the hypothesis the theorem needs, [absolute continuity](ch20-cards.md#g-abscont): roughly, that pieces of tiny total length can move the function only a tiny total amount. The Cantor set fits inside pieces as short as we like in total, and the whole climb happens there. [r-20.2](ch20-refs.md#r-20.2) [r-20.6](ch20-refs.md#r-20.6) [r-20.25](ch20-refs.md#r-20.25)
#### Denominator
Length, dimension, a staircase’s height, and a failure of ordinary volume: these examples keep different questions about size apart.
Cantor
[dimH 𝒞](ch20-cards.md#g-dimH) = ln 2/ln 3 ≈ 0.6309; the [Cantor set](ch20-cards.md#g-cantor)’s fractional dimension.
Cut a segment into thirds and you get three copies at one third the size, and 3 × (1/3)¹ = 1; cut a square into thirds each way and you get nine copies, and 9 × (1/3)² = 1. The dimension is the power that makes the shrunken copies add back up to the whole. Each stage of the Cantor set keeps two copies at one third the size, so its dimension d solves 2 × (1/3)ᵈ = 1: more than a scatter of points, less than a line. [r-20.7](ch20-refs.md#r-20.7) [Counting the boxes](ch20-cards.md#g-boxdim) needed to cover the set at finer and finer scales gives the same number. [r-20.18](ch20-refs.md#r-20.18)
[Σ 2n−1/3n](ch20-cards.md#g-removed) = 1; the length we cut away, which is why nothing is left to weigh.
[One third](ch04-cards.md#g-third), then two ninths, then four twenty-sevenths, and so on: the removed middle thirds add up to the whole interval.
[c(½)](ch20-cards.md#g-cantorfn) = ½; the staircase’s height where its slope is zero.
A nonzero value at the very point where the numerator took a zero slope.
[λ(SVC)](ch20-cards.md#g-fatcantor) = ½; the “fat” [Cantor set](ch20-cards.md#g-cantor), a look-alike with a different answer.
The Smith–Volterra–[Cantor set](ch20-cards.md#g-cantor) cuts smaller gaps: first the middle quarter, then a sixteenth from the middle of each piece left, then a sixty-fourth, and so on. Every interval, however short, still meets a gap, so the set is still nowhere [dense](ch20-cards.md#g-dense); yet the gaps together remove only half the length: a pattern of gaps alone does not force zero length. [r-20.8](ch20-refs.md#r-20.8)
other dimensions
[dimH Koch](ch20-cards.md#g-koch), [dimH Sierpiński](ch20-cards.md#g-sierp), [dimH Menger](ch20-cards.md#g-menger) = ln 4/ln 3, ln 3/ln 2, ln 20/ln 3; three famous fractals, answering the same scaling question.
Each comes from the number and size of its smaller copies: four at [one third](ch04-cards.md#g-third) for [the Koch curve](ch20-cards.md#g-koch) (≈ 1.2619), three at one half for [the Sierpiński triangle](ch20-cards.md#g-sierp) (≈ 1.585), twenty at [one third](ch04-cards.md#g-third) for [the Menger sponge](ch20-cards.md#g-menger) (≈ 2.727). All are positive. [r-20.9](ch20-refs.md#r-20.9) [r-20.10](ch20-refs.md#r-20.10) [r-20.11](ch20-refs.md#r-20.11)
a paradox
[5](ch20-cards.md#g-banachtarski) = 5; the pieces of the Banach–Tarski paradox, measure theory’s most famous shock.
A solid ball can be cut into five pieces that, moved only by rotations and slides, reassemble into two balls each the size of the original. The “pieces” are not chunks but sets built with [the axiom of choice](ch27-cards.md#g-ac), and some of them are [non-measurable sets](ch20-cards.md#g-vitali): not sets of volume zero, but sets to which no volume can be assigned at all, so adding up their volumes is not an available operation. [r-20.12](ch20-refs.md#r-20.12)
The first fraction is 0/(Cantor + other dimensions + a paradox): every value below the line is positive, so the sum is nonzero.
**Fine print.** [The Banach–Tarski pieces](ch20-cards.md#g-banachtarski) cannot be cut from material, and countable additivity is not contradicted.
### [dynamical systems](ch20.md#ch-T20-dynamical-systems) {#ch-T20-dynamical-systems}
The [logistic map](ch20-cards.md#g-logisticmap) fᵣ(x) = rx(1 − x) is one line of arithmetic used as a population model: x is this year’s population as a fraction of its ceiling, fᵣ(x) is next year’s, and [r](ch20-cards.md#g-logisticr) is the growth rate. [r-20.13](ch20-refs.md#r-20.13) A *fixed point* is an input the rule returns unchanged.
#### Numerator
[f2(½) − ½](ch20-cards.md#g-logisticfixed) = 0; the population rule, placed at a tame resting point.
f₂(½) = 2 × ½ × ½ = ½, so subtracting ½ leaves zero. At this parameter every starting value strictly between 0 and 1 settles toward ½. [r-20.13](ch20-refs.md#r-20.13) [r-20.14](ch20-refs.md#r-20.14)
#### Denominator
2 · [δ](ch05-cards.md#g-feig) · |[αF](ch05-cards.md#g-feigA)| ≈ 2 × 4.6692 × 2.5029; the same parameter, times the constants that govern the road to chaos.
Raise r and the resting point loses stability: a two-cycle appears, alternating between two values, then a four-cycle, then an eight. [Feigenbaum’s δ](ch05-cards.md#g-feig) says how fast these doublings crowd together: the stretch of r between one doubling and the next ends up about 4.669 times shorter than the stretch before it. |[αF](ch05-cards.md#g-feigA)| is the same kind of ratio for the population values rather than for r: each new fork ends up about 2.5 times narrower than the fork it split from. They return from Chapter ᚃ(F), where the question was what kind of numbers they are; here we see what they describe. The product is positive. [r-20.14](ch20-refs.md#r-20.14) [r-20.15](ch20-refs.md#r-20.15)
The second and final fraction is 0/(2δ|αF|): 2δ|αF| > 0.
**Fine print.** The convergence to ½ concerns the interior: 0 stays at 0, and 1 maps to 0. Feigenbaum’s constants are universal only for a specified class of smooth one-humped maps with a quadratic maximum, not for every iterative rule. The bars around [αF](ch05-cards.md#g-feigA) are there because it is conventionally negative; the formula uses its magnitude.
What this chapter adds to the formula: 0 / (Cantor + other dimensions + a paradox) · 0 / (2δ|αF|) = 0
We placed the population rule at a tame fixed point and kept its more complicated behavior nearby in the denominator. What is left of the interval after endless removals has more points than we could ever list, yet no length at all.
From here I can see the launchpad. Let’s go park and meet by the boarding ladder. I’ll race you there!
Watch and read
- [The Devil’s Staircase | Infinite Series](https://www.youtube.com/watch?v=dQXVn7pFsVI) PBS Infinite Series: builds the Cantor set and then the function that climbs it
- [Cantor set](https://en.wikipedia.org/wiki/Cantor_set) Wikipedia
- [Lebesgue measure](https://en.wikipedia.org/wiki/Lebesgue_measure) Wikipedia