The collection

Every essay — page 33

Page 33 of 40, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Logic

What can be said in a system, what follows from it, and what it cannot settle about itself.

A diagonal walk through countably many listed sets. A grid of six rows and seven columns, each cell numbered by Cantor's diagonal order, with the path of the first twenty-one cells drawn.

The choice inside a countable union

Building a spanning tree by keeping every edge that closes no loop. Four stages of a greedy pass over the fourteen edges of an eight-point graph, ending with a spanning tree of 7 edges.

A spanning tree for every graph

Four proofs about numbers, and what they compute. λn f x. n (λy. f y) (f x) computes n + 1; λn f x. n (λy. f (f y)) x computes 2n; λn f x. n (λy. n (λz. f z) y) x computes n²; λn f x. n (λy. x) (f x) computes 1 at 0, else 0.

What a proof about numbers can compute

Raising the input's type, and what becomes computable. N[o]: 299 functions, 0 not extended polynomials; N[o → o]: 163 functions, 42 not extended polynomials; N[(o → o) → o]: 128 functions, 46 not extended polynomials; N[o → o → o]: 72 functions, 33 not extended polynomials.

One type up

Fixing two letters of a sixteen-leaf formula leaves four. Parity of four letters as a 16-leaf formula; fixing x3 = 0, x4 = 1 leaves 4 live leaves.

What a formula keeps when letters are fixed

6 figures · new

Three countable models of one theory, different only above the constants. Models A (cₙ = n), B (cₙ = −1/(n+1), supremum 0), C (cₙ = 1, 7/5, 41/29, … rising to √2, no supremum) of the theory of a dense order with increasing constants.

Three models and never two

6 figures · new

Computation

A fixed set of operations, and the exact question of what it can and cannot build.

Two points, and everything one round of compass and straightedge adds. Two starting points with the line and circles they permit, and the four points where those objects cross.

What two points can build

The tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √3). A tower of field extensions with the degree of each step, beside the multiplication table of the basis.

Every step is a square root

Every rational number that could be a root of x³ − 2. A table of the candidate rational roots allowed by the rational root theorem, with the polynomial's exact value at each.

The cube that will not double

Which angles with a rational cosine can be cut in three. A dial of angles marked trisectable or not, beside the cubic whose rational roots decided each one.

The angle that will not divide by three

Which regular polygons a compass and straightedge can draw, up to 100. A grid of the integers with the constructible ones filled in, each verdict computed two independent ways.

Which polygons can be drawn

The circle that will not square

A code on the 3-cube, and the balls around its words. The corners of a hypercube with the chosen codewords marked and the words within one error of each shaded.

Distance is a picture

The sixteen words of the [7,4] Hamming code. A table of sixteen seven-bit codewords with their data bits, parity bits and weights.

Sixteen spheres that fill a cube

The syndrome of 1011010, and the bit it names. A parity-check matrix over a received word, with the resulting syndrome matched against the table of single-error syndromes.

Finding the error without reading the message

A degree-2 polynomial over GF(11), and the 7 values sent. A grid of the finite field with the polynomial's value at each point marked, and the transmitted symbols picked out.

A polynomial through the gaps

The arithmetic of GF(4), and of the integers mod 4. Addition and multiplication tables of a finite field, optionally beside the table of a ring of the same kind of size.

The field with four elements

The non-zero elements of GF(16) as the powers of one of them. A ring of the field's non-zero elements in the order the powers of a primitive element produce them, beside a table of exponents.

Every element is a power of one of them

The Fano plane, and the incidence table behind it. Seven points joined by six straight lines and one circle, beside the seven-by-seven table of which point lies on which line.

Seven points, seven lines

A schedule on 9 points where every pair meets exactly once. Points around a circle with the triples of a Steiner system drawn between them, beside the list of triples.

A schedule where every pair meets once

Transversals of the cyclic square of order 6. A cyclic Latin square with a transversal marked if it has one, beside a count of transversals at neighbouring orders.

The thirty-six officers

Every word once, around a cycle

A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

Equal area is enough, and equal volume is not

The midpoint of a segment, drawn with a compass and no straightedge. A segment with the arcs that step its length three times round one end to reach the point twice as far away, and the further arcs that send that point back to the midpoint, every one of them a circle.

The straightedge buys nothing