The collection

Every essay — page 9

Page 9 of 39, 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

Algebra

Structure — what stays true when the numbers change.

A parallelepiped of volume 2.94. The image of the unit cube under a three-by-three matrix, beside the six signed products whose sum is its volume.

The only function that behaves like a volume

A lattice of determinant 3, and the ellipse that must hold a point. A lattice with the parallelogram its basis spans, an ellipse centred at the origin, and the nearest non-zero lattice point it contains.

One point in every big enough shape

A determinant counting the 16 spanning trees. A small graph, the minor of its Laplacian, and every one of its spanning trees drawn as thumbnails.

A determinant that counts trees

A determinant of −5 and a permanent of 23 from the same six products. The six products of a three-by-three matrix listed once, added with signs to give the determinant and without signs to give the permanent, with a row operation applied to both.

The same sum without its minus signs

A rotation of four-space, and the two angles it turns through. The four by four matrix of the map sending x to p x conjugate q, beside two dials showing the angle it turns through in each of its two invariant planes.

A rotation of four-space takes two of them

The four-square identity, and how few squares a number needs. A product of two whole quaternions with both sides of the four-square identity evaluated, above a strip colouring every number by the fewest squares that add to it.

The identity that multiplies sums of squares

The twenty-four unit quaternions, at the corners of a four-dimensional solid. The twenty-four units of the Hurwitz quaternions drawn as the vertices of a 24-cell projected into three-space, with the ninety-six edges joining units at distance one.

The integers among the quaternions

What is lost at eight

Two hundred and forty roots, built and counted. An eight by eight grid whose upper cells stand for the pairs of coordinates a root can use, beside bars counting how many roots stand at each angle from a given root: 1, 56, 126, 56 and 1.

Two hundred and forty directions

The half of 24 permutations that commutators reach. A block of 24 squares, one per permutation, with the 12 generated by commutators shaded, beside bars counting the homomorphisms to each cyclic group.

The only bit that survives

A 2×3 sliding puzzle: 360 arrangements of 720 can be reached. Two arrangements of a small sliding puzzle side by side, the solved one and the one with two tiles exchanged, with the count of positions reachable by sliding found by walking every move.

The puzzle that is exactly half solvable

The character table of the permutations of 4 places, computed from traces. A table with one row per irreducible representation and one column per conjugacy class, giving the trace of the matrix each representation assigns, with the dimension column marked.

When the label may be a matrix

The parabola that proves |a·b| ≤ |a||b|. The squared length of a − t b plotted against t. It is a parabola opening upward whose least value is 7.118; that this is never negative is exactly the Cauchy–Schwarz inequality.

The square that cannot be negative

One subtraction clears a direction. Gram–Schmidt on two planar vectors, in 3 panels: the pair as given, the shadow of the second on the first, and the perpendicular pair that is left when the shadow is removed.

One subtraction clears a direction

The nearest point of the plane the columns span. A target vector in space, the plane spanned by two columns, the point of that plane nearest the target, and the residual joining them, which meets the plane at a right angle.

The nearest point of a flat thing

The map moved to the other side of the product. Two panels over the unit curve of the product u₁v₁ + u₂v₂. The left applies A to u and measures it against v, giving 2.328; the right applies the adjoint to v and measures it against u, giving the same number.

Moving a map across a product

Perpendicular to both, and as long as their parallelogram. The vectors (2, 0.4, 0.2) and (0.6, 1.8, 0.3), the parallelogram they span, and their cross product (−0.24, −0.48, 3.36), drawn perpendicular to both with a length of 3.403, which is the parallelogram's area.

A plane disguised as an arrow

The coefficients of (1 + x)¹² sorted by remainder mod 3. The binomial coefficients of the 12th power coloured by the remainder of their index on division by 3, beside the 3 points one plus a root of unity, whose powers averaged pick out each colour's total.

Every third coefficient

The first 60 powers of (3 + 4i)/5. The powers of (3 + 4i)/5 marked on the unit circle, each a further turn by the same angle, with the power that comes closest to returning to 1 marked.

On the circle and never home

The ball around the identity, in 3 groups. A table of the number of group elements within each distance of the identity, one row per group, with the growth type each row exhibits beside it.

How fast the ball fills

The edge that is as big as the ball

2 independent rows and 2 independent columns. An array of 3 rows and 4 columns beside its transpose, with the independent rows of each shaded, showing the same count on both.

Counted across and counted down

2 independent cycles and 3 independent cuts, on 5 edges. A small graph beside its incidence matrix, with the matrix's rank and nullity given and shown to be the number of independent cuts and the number of independent cycles.

The cycles and the cuts

Seven powers of √2 + ∛3 in a space of six. A table of the powers 1 to (√2 + ∛3)⁶ as coordinate vectors over a six-element basis, with the coefficients of the dependency among them: the minimal polynomial x⁶ − 6x⁴ − 6x³ + 12x² − 36x + 1.

Seven powers in a space of six