Generator

Multiplying two complex numbers

A generator in the algebra library, called 27 times across 7 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

complex-turn is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Multiplying two complex numbers. In the complex plane, multiplying adds the two angles and multiplies the two lengths.

`show: "image"`

The image of four circles, turning 0 to 3 times. The polynomial applied to circles of four radii, each image drawn as a closed loop with the origin marked, and the number of times the loop goes round it.

`show: "powers"`

Powers of a complex number. The first 12 powers of a complex number, each one a further turn and stretch of the last.

`show: "roots"`

The 5 5th roots of unity. 5 points spaced evenly around the unit circle, at the vertices of a regular 5-sided polygon.

`show: "inside"`

3 roots, and the circles that enclose them. The roots of the polynomial in the complex plane, with circles of several radii drawn round the origin and each labelled by how many times its image winds round zero.

`show: "euler"`

Euler's formula on the unit circle. Walking a distance around the unit circle; at half a turn the point sits exactly at minus one.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

the root at step 0 first returns to 1 after 1 multiplications ×26
root 0 raised to the 5 is 1 ×24
the coefficient of x^0 in the product is the one in xⁿ − 1 ×16
and that number is the Möbius function of 1 ×12
the primitive roots of order 1 add to a real number ×12
the factor belonging to the roots of order 1 has degree φ(1) ×8
the curve the image of the circle of radius 0.4 never passes exactly through the origin ×7
the image of radius 0.4 turns once for each root inside it ×7
the image of the circle of radius 0.4 closes up after a whole number of turns ×7
the walk along the image of the circle of radius 0.4 is fine enough to see which way it turned ×7
the circle of radius 0.4 has 0 roots inside and turns 0 times ×4
the curve the image of radius 0.4 never passes exactly through the origin ×4
the image of radius 0.4 closes up after a whole number of turns ×4
the walk along the image of radius 0.4 is fine enough to see which way it turned ×4
far out, the image turns once for each power of z, which is 3 times ×2
a polynomial of degree 3 has 3 roots ×1
all 12 roots add to zero ×1
and close in it does not go round the origin at all ×1
and has no imaginary part left ×1
and it crosses between them ×1
and multiply to the constant term, with a sign that follows the degree ×1
and the angles add ×1
and the degrees of the factors add to n, because every root has exactly one order ×1
and their imaginary parts cancel ×1
at a place where the discriminant is passing through zero ×1
each root found really is a root ×1
each step of the division gives a whole number ×1
every root found really is a root ×1
every root is on the unit circle ×1
in both coordinates ×1
the division leaves no remainder ×1
the equation is zⁿ = 1 for n between 4 and 20 ×1
the factors multiply out to a polynomial of the right degree ×1
the lengths multiply ×1
the number of powers drawn is between 2 and 24 ×1
the number of powers taken is between 1 and 24 ×1
the number of roots of unity is between 2 and 24 ×1
the polygon drawn has between 3 and 16 corners ×1
the polynomial has a degree ×1
the polynomial has a degree between 1 and 6 ×1
the radii drawn are between 2 and 5 positive numbers no larger than 4 ×1
the roots add to minus the next coefficient over the leading one ×1
the roots sum to zero ×1
the sign of the discriminant says how many roots are real ×1
the sums take all three of the values that function takes ×1
the sweep meets both cases ×1
the sweep takes between 20 and 800 samples ×1
the swept family is a depressed cubic with a negative linear term ×1
the table of sums runs to between 6 and 16 ×1
the table runs to between 6 and 16 ×1
the view is one the family draws ×1
with nothing imaginary left over ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Algebra

A loop that cannot miss the middle

Feed a circle into a polynomial and a closed loop comes out. A small circle gives a loop that does not enclose the origin; a large one gives a loop that goes round it as many times as the degree. Something has to happen in between, and that something is a root.

Algebra

A multiplication that remembers the order

Four characters — i² = j² = k² = ijk = −1 — define a multiplication in which ab and ba are different numbers. Everything follows from them, including the fact that a rotation of a solid body has two names, and that two full turns are needed to get one of them home.

Algebra

Multiplying is turning

Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.

Algebra

The lattice that runs the other way

The symmetries of a polynomial's roots form a group, and the fields between the bottom and the top form a lattice. The two are the same picture, one of them turned over — a bigger group of symmetries fixes less, so it names a smaller field.

Algebra

The polygon an equation forces

The n solutions of z to the n equals one are the corners of a regular polygon, and nearly everything about them — that they form a group, that they sum to zero, that the equation factors into pieces with whole-number coefficients — is that picture read carefully.

Algebra

What the coefficients already know

Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.

Analysis

When the period grows without bound

A repeating signal has a spectrum of separate lines. Stretch the gap between repeats and the lines crowd together while the curve they sit on stays exactly where it is — and at infinite period the lines are gone and the curve is the whole answer.

The whole library · What the figures prove