Matpic
mathematics, in pictures
Concept

Degree

The highest power appearing in a polynomial, or the number of edges meeting a vertex in a graph. In both senses it bounds what the object can do: a polynomial has at most that many roots, and a graph's degrees sum to twice its edge count.

Named by 16 essays across 5 fields — each of them below, with the objects they name alongside it.

Königsberg as a graph. The four landmasses as circles and the seven bridges as edges; every circle has an odd number of edges.

Seven bridges, and the invention of throwing things away

Euler solved a puzzle about a Prussian city by deleting the city. What survived the deletion was a new branch of mathematics.

discrete · Eulerian paths 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.

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 · Polynomial roots One line, and both shapes halved. Two shapes and the single straight cut that divides each of them into two equal areas. The direction was found by sweeping every angle and watching the imbalance change sign.

One line that halves them both

Two shapes lying anywhere on a page, of any sizes and any shapes at all. There is always a single straight line that cuts both of them into two equal halves at once — and finding it needs no cleverness, only the observation that a quantity which reverses sign has to pass through zero.

topology · Borsuk ulam 3 roots, and the two numbers the coefficients already knew. The roots of a degree-3 polynomial, found numerically, with the point they average to. That average, and their product, are readable straight off the coefficients without finding the roots at all.

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.

algebra · Polynomial roots The subgroups of a polynomial's symmetries, against the fields they name. Two lattices side by side, one of the subgroups of the symmetry group of the cube roots of two and the other of the fields between the rationals and the splitting field, drawn so that one is the other turned upside down.

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 · Galois correspondence An angle of 60° cut in three with one mark. A circle with a marked point on it, a straightedge laid through that point so the segment between the extended diameter and the circle equals the radius, and the third-angle it makes.

The mark that changes what is reachable

Two thousand years of failure to trisect an angle with compass and straightedge was failure at a stated set of operations. Scratch two marks on the straightedge and Archimedes trisects any angle in four steps — because the new operation solves a cubic, and the old ones could only ever solve quadratics.

computation · Neusis How closely a fraction can come, and the barrier that says no closer. Two panels at very different scales: the approximations to √2, which stay above the barrier a degree-two number obeys, and the truncations of a constructed number, which fall below every barrier drawn.

Approached too fast to be algebraic

An algebraic number of degree d cannot be approached by fractions faster than the denominator's dth power. So a number that is approached faster than that is the root of no polynomial at all — and one can be built by choosing where its decimal digits go.

number · Irrationality A ray through a knotted tube, crossing it 5 times. A closed surface in space — a tube round a trefoil knot — with a point, a ray from it and every crossing of the surface marked; the parity of the count says which side of the surface the point is on.

Two pieces, in every dimension

A closed curve cuts the plane in two. A closed curve in space cuts nothing at all, and it takes a closed surface to do the job — which is the shape of the general theorem, and the reason the word "dimension" means anything.

topology · Jordan curve The zeros of x² + y² + z² over GF(5), and of x² + y² over GF(7). Grids of every point over a small prime field with the solutions of a quadratic equation filled in: the three-variable equation drawn as one slice per value of z, beside a two-variable equation with far fewer solutions.

Solutions that come in multiples of p

Count the solutions of x² + y² + z² = 0 in the field with five elements and there are 25; with seven, there are 49. Whenever a system of equations has more unknowns than its total degree, its number of solutions is a multiple of the characteristic — which forces a solution besides zero, and the reason is a sum over the field that vanishes because its non-zero elements form one cycle.

computation · Finite fields Two opposite points on a globe with the same temperature and the same pressure. A world map in longitude and latitude with one curve where each point's temperature matches its opposite point's and another where the pressures match, crossing at a pair of opposite points that are marked.

Two opposite points that agree twice

At any moment there are two points on opposite sides of the Earth with the same temperature and the same pressure. On a seeded globe they sit at 11.9°N 44.6°E and 11.9°S 135.4°W. The reason is the circle argument that halved two shapes, run one dimension up: the differences between opposite readings, walked round the equator, wind round zero an odd number of times — and an odd number cannot be zero.

topology · Borsuk ulam Which polygons two instrument sets reach, up to 24. A strip of the polygons from 3 to 24 sides, each marked according to whether compass and straightedge reach it and whether a conic or a trisector does, with the degree of its cosine beneath.

Two instruments with one reach

Allow every conic to be drawn at will, or allow an angle to be cut in three. The two permissions look nothing alike and reach exactly the same numbers — because what an operation buys is a degree, and both of these buy three.

computation · Neusis The polynomial the 11-sided polygon needs. The minimal polynomial of twice the cosine of the central angle of an 11-sided polygon, with its degree, the rational root test applied to it, and whether that degree is reachable by cubic steps.

A quintic a sliding mark reaches

The eleven-sided polygon needs a number of degree five, and five is not a product of twos and threes — so no conic and no angle trisector reaches it. A ruler with two scratches does, which places the marked ruler strictly above the conics and leaves its exact reach unknown.

computation · Neusis Hierholzer's construction on 6 vertices and 9 edges. Three views of one graph whose vertices all have even degree: a first closed walk that stops back at its start, the loops walked from vertices on it with edges left over, and the single circuit made by splicing them, with every edge numbered in order.

A walk that splices in its own detours

Euler proved that a walk crossing every bridge once needs every landmass to have an even number of bridges, and then stated, without proof, that this was enough. The missing half took 137 years, and it is not an argument but a procedure: walk until stuck, notice that stuck can only mean home, and splice in a detour from anywhere with edges left. The procedure never fails, and the reason fits in one sentence about arriving and leaving.

discrete · Eulerian paths The postman's route: 24 blocks of street, walked in 28. A street network with its odd-degree vertices marked and the streets a shortest closed route must walk twice drawn doubled, dashed in a second colour, pairing up the odd vertices.

The streets a postman walks twice

A postman must walk every street of a district and come back. If every corner has an even number of streets, no street needs walking twice. If not, some must — and the ones repeated always join the odd corners in pairs. Pricing every way of pairing them finds the shortest round; pairing the nearest corners first does not.

discrete · Eulerian paths A line in every direction in the plane over GF(7), in 31 points. A square grid of the points of a small finite plane with the points of a Kakeya set filled, beside a list of the lines it contains, one for each direction.

No set with a line in every direction is small

In the plane over the integers modulo 7 there are 49 points and lines in 8 directions. A set holding a whole line in every direction needs 31 of the points — more than half — and in any dimension such a set fills a fixed share of the space. In the real plane the same sets can have area zero. Over a finite field one polynomial of low degree shows they cannot be small.

computation · Finite fields Greedy colouring of the crown graph in two orders: two colours, and four. The crown graph on eight vertices coloured greedily twice: in the order top row then bottom row it uses 2 colours, alternating between the rows it uses 4. Numbers on the vertices give the order.

The order decides the colours

The simplest way to colour a graph is to take the vertices one at a time and give each the first colour its neighbours are not already using. It never needs more than one colour beyond the largest degree — and on a graph that needs only two colours it can be made to use as many as there are vertices on a side, depending on nothing but the order it is handed.

discrete · Graph colouring

Named alongside it

The objects these essays reach for when they reach for this one.

Existence proofField extensionParityPolynomialConstructible numberContinuityCubicGraphMarked straightedgeNeusisNonconstructiveOperation set

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