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Doing infinitely many things — page 2

Sums that never end, subdivisions that never stop, and the care required to make either of them mean something.
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. 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.

 Whole-number points on x² − 2y² = 1. The branch of the hyperbola x² − 2y² = 1 in the first quadrant, with the whole-number points on it marked and labelled, and the lattice drawn faintly behind. Number

One solution that makes all the others

The equation x² − 2y² = 1 has infinitely many whole-number solutions, and every one of them is a power of the smallest. The multiplication that produces them is what multiplying two numbers of the form a + b√2 comes to when the √2 terms are collected — so an equation about a hyperbola turns out to carry a group.

 Every way of choosing one thing from each of 4 pairs. A table with one row per choice function on a small family of pairs, each row giving what it takes from each pair, with the row a stated rule names picked out. Logic

The choice nobody can write down

Given finitely many pairs, picking one thing from each is a finite list of decisions and needs no justification. Given infinitely many, the list cannot be finished — and whether one exists anyway is an axiom, independent of everything else, whose consequences include a theorem most people refuse to believe.

 The tail that would have to be a whole number. For each denominator, the value of q! times the tail of the series for e, plotted against the band between zero and one where no whole number lies, with the bound 1/q above it. Number

A tail too small to be a whole number

If e were a fraction with denominator q, then q! times e would be a whole number. It splits into a whole part and a tail, the tail is squeezed strictly between nothing and one, and there is no whole number there.

 The fractions, put in a line. A grid whose rows are numerators and columns denominators, walked by antidiagonals, with the place each fraction takes in the list written in its cell and the repeats left blank. Logic

The arithmetic that loses subtraction

Adding one to an infinite collection changes nothing, and neither does doubling it, or squaring it. What that costs is the two operations that were doing the work — an equation between infinite sizes cannot be cancelled, and how many are left stops being a question.

 A square's worth of points, on a line. A unit square with a point marked, the decimal places of its two coordinates woven into one number, and that number marked on a line beneath. 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.

 The algebraic numbers, arriving in finite batches. A stretch of the number line with the roots of integer polynomials marked, each at the height of the smallest polynomial that catches it, and the count of polynomials at each height. Logic

Countable, and everywhere

The numbers a polynomial can catch arrive in finite batches, so they can be listed. They are also in every interval, however short. Being listable turns out to say nothing whatever about being sparse.

 The tower of sizes, and the gap in it. A tower of infinite sizes, each the number of sub-collections of the one below, with the space between the first two marked as the one no proof decides. Logic

The size that cannot be pinned down

There is no largest infinity, because no collection has as many members as it has sub-collections. What is not settled is whether anything sits between the first two — and that is not an open problem but a proved absence of an answer.

 Ordinal sums and products, in normal form. A table of ordinal expressions with their Cantor normal forms and whether the two sides of each pair are equal, above two tick lines drawing one such pair. Logic

One step in front of infinitely many

Put one step before an infinite run of them and nothing has changed; put it after and something has. Ordinal addition records that difference, which is why it is not commutative — and why it keeps information that counting throws away.

 The numbers to 18 in hereditary base 2, and their ordinals. A table of small whole numbers written in hereditary base notation beside the ordinal obtained by replacing the base with omega. Logic

Every ordinal in base omega

Every ordinal below a certain point is a descending sum of powers of ω, in exactly one way. That notation makes comparison mechanical, it is what hereditary base notation becomes when the base is replaced, and it stops at the first ordinal it cannot name.

 Limit ordinals and the sequences that approach them. Several ordinals with the first terms of their fundamental sequences, and the successors marked as having a predecessor instead. Logic

Reached from below, or not at all

Every limit ordinal anybody meets is the end of an increasing sequence — ω, ω·2, ω^ω, all of them approached one step at a time. The first uncountable ordinal is not, and the reason it is not constrains the size of the continuum.

 The fast-growing hierarchy at its first few ordinals. A table of the fast-growing hierarchy: one row per ordinal index, one column per argument, with the cells too large to evaluate marked as such. Logic

An ordinal as a growth rate

Index a family of functions by the ordinals, each one iterating the last, and the index becomes a measure of how fast a function grows. The point where the index leaves what arithmetic can prove is exactly where the Goodstein sequence became unprovable.

 The partition product's coefficients to q¹². A row of series coefficients computed by expanding a product, beside the same numbers obtained another way. Number

Every partition, hidden in a product

Multiply out one factor for each part size and the coefficient of q to the n is the number of partitions of n. Nothing is being approximated: the product is a bookkeeping device that does the counting by multiplying.

 p(n) to 60, against the Hardy–Ramanujan estimate. The number of partitions of each number up to sixty on a logarithmic scale, with the asymptotic estimate drawn over it and the ratio of the two tabulated. Number

The size of a number with no formula

There is no closed expression for the number of partitions of n. There is an expression for how large it is — with a square root in the exponent and a π in front — and it is accurate enough that rounding a few terms of its refinement gives the exact count.

 Alexander's horned sphere at stage 3. A tree of clasped pairs of horns, 7 of them, each pair's two circles passing once through the other's disc; the horns shrink geometrically and their tips converge. Topology

A ball whose outside is not one

Alexander's sphere separates space into two pieces, exactly as the theorem promises. Its inside is an ordinary ball. Its outside is not, and the obstruction is a tree of clasped horns whose tips never stop.

 An Apollonian gasket, 125 circles in. The Apollonian gasket generated from four mutually tangent circles of curvature −1, 2, 2 and 3, drawn to 4 generations; every curvature in it is a whole number. Geometry

Curvatures that stay whole

Four circles touching one another satisfy an equation in their curvatures. Read it as a quadratic and the second solution is the first subtracted from something — so a packing that starts with whole numbers stays whole forever.

 A set with no interval in it and half its length left, after 6 stages. Stages of removing a shrinking middle from every surviving interval, with the total length left printed at each stage, and the middle-thirds construction of the same depth drawn beneath for comparison. Analysis

No interval in it, and length to spare

The middle-thirds set has no length because the removed pieces add to one. Remove shrinking middles instead and they add to a half — leaving a set that still contains no interval anywhere, and still has half the length it started with.

 The rationals covered by intervals of total length 0.1800. Intervals of rapidly shrinking length placed around the rationals of the unit interval in the order they are listed, with the union of them drawn as a single band beneath. Analysis

Covering a set from outside

To say how long a set is, cover it with intervals and add their lengths, then take the smallest total any covering achieves. That definition is short, obviously right for an interval, and gives the rationals a length of nothing.

 The classes, a selection from them, and the translates that cannot have a length. Points of several classes of the unit interval under translation by rationals, drawn one class per row, above rows showing rational translates of a selection that never overlap. Analysis

A set that has no size at all

Slide the unit interval along itself by every rational and the points fall into classes. Choose one point from each and the resulting set has no length — not zero, not positive, none: countably many disjoint copies of it would have total length nought or infinity, and the union needs something in between.

 One walk on the whole numbers, three chances, three different fates. The relative weight of each state for three step-up chances, drawn as bars, with the running total of those weights and what each case means beneath. Probability

Where the shares have nowhere to go

On finitely many states, a chain that can reach everywhere and is not forced into a rhythm settles down. Give it infinitely many and both conditions can hold while the walk leaves and never returns — or returns with certainty and takes an unbounded average time about it.

 Stretch, fold, and what is left. 5 stages of the horseshoe map's surviving set: one square, then two strips, then four, up to 16, each narrower than the last by a factor of 3. Dynamics

Stretch, fold, and what is left

A system that pushes every pair of nearby points apart and keeps them all inside a bounded region has only one option, and it is the one a baker uses. Stretching and folding is the mechanism, and what survives infinitely many folds is a Cantor set.

 The same banding at every magnification. The Hénon attractor drawn from 26000 points, followed by 2 magnifications of one part of it. Each magnification resolves what looked like a single curve into several parallel ones. Dynamics

Neither a surface nor a solid

The attractor has no volume, because the flow shrinks volumes at a rate that can be read off the equations. It is also not a surface, because a surface cannot carry chaotic dynamics. What is left is an object of dimension a little over two, and that number is measurable.

 A staircase with no steps. The Cantor function drawn to several stages: a continuous non-decreasing curve from nought to one which is constant on every interval of the complement of the middle-thirds set, so its whole rise happens on a set of measure zero. Analysis

A staircase with no steps

A function that rises from nought to one, is continuous everywhere, and has derivative zero at almost every point. All of its climbing happens on a set of no length at all, which is possible because that set has uncountably many points.

 Infinite below 0.6309, nought above it. The total of the s-th powers of the diameters in the natural cover of the middle-thirds Cantor set, plotted against s for 4 depths. Every curve passes through one at s = 0.6309 and they separate either side of it. 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.

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