Matpic
mathematics, in pictures
Theme

Throwing things away

Progress made by deleting detail: the map that becomes a graph, the shape that becomes a number.
beforeafter · area × 2.50210.51.5 Algebra

A matrix is a picture of what happens to the grid

Four numbers in a box is not an object anyone has intuitions about. The same four numbers, shown as an instruction for redrawing the plane, are.

 Ndegree 3Idegree 5Edegree 3Sdegree 3 Discrete

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.

 Topology

The surface with one side, and what happens when it is cut

A strip of paper, half a twist, and a join. The result has one side and one edge, and cutting it down the middle does not produce two of anything.

 abshadowa · b = 10.00= 2.43 × 4.12 Algebra

The dot product is a shadow

Multiply the matching coordinates and add them up. That rule explains nothing, and it hides the fact that the answer is a length — how far one arrow reaches along another, times how long that other one is.

 λ = 3.00λ = 1.00[2, 1, 1, 2] Algebra

The directions a map leaves alone

Almost every arrow comes out of a transformation pointing somewhere else. A few come out pointing exactly where they went in, only longer or shorter. Those few decide nearly everything the map does.

 the mapwho touches whom Discrete

Four colours, and a proof nobody can read

Every map on a plane can be coloured with four colours so that no two neighbours match. The statement is understandable by a child, it resisted a century of attempts, and the proof that settled it cannot be checked by a human being.

 VEFV − E + Ftetrahedron4642cube81262octahedron61282dodecahedron2030122icosahedron1230202 Topology

Every corner pays for itself

Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself.

 Geometry

The plane, divided by whoever is nearest

Scatter some points and colour every other point of the plane by which one is closest. The result is a tiling nobody designed, and its dual triangulation has a property that no part of the construction mentions.

 012345678910118 + 9= 17= 5 (mod 12)1 lap of the dial,then the remainder Discrete

Numbers that wrap

A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.

 the trefoil Topology

Three moves, and what they cannot undo

A knot is a closed loop of string, and two knots are the same if one can be wiggled into the other. Reidemeister reduced all possible wiggling to three local pictures — which is what makes it possible to prove that a knot is knotted.

 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100every composite struck once, by its smallest prime factor25 squares are left standing, and they are the primes below 100 Number

The primes are what is left over

Eratosthenes' sieve does not find the primes. It removes everything else, one prime at a time, and whatever survives is prime by default — which is a strange way to reach the most studied objects in arithmetic.

 0612391017134511281401234012mod 3 ↓mod 5 →every one of the 15 pairs is reached, exactly onceso a remainder mod 3 and a remainder mod 5 together name one number mod 15 Number

Two dials at once

Watch one number on two clocks with different faces. If the faces share no factor, every pair of readings occurs exactly once — so two remainders name a number, and a hard calculation can be split into two easy ones.

 rspq00011110000111101110111011110010((p ∧ q) ∨ (r ∧ s)) ∨ (¬p ∧ ¬r) covered by 3 of its 6 primeimplicantsr∧s ∨ ¬p∧¬r ∨ p∧q — checked against the formula on all 16assignments Logic

The map that puts neighbours side by side

Reorder the rows of a truth table so that neighbouring squares differ in one letter, and finding a short formula stops being algebra and becomes the problem of covering a shape with rectangles.

 ℚdim 12ℚ(√2)dim 22ℚ(√2, √3)dim 41√2√3√61√2√3√61√2√3√6√22√62√3√3√633√2√62√33√262 square roots taken, one at a time, and the degree doubles at each: 1 → 2 → 4the 4×4 table is the closure check — every product of basis elements landed on a whole-numbermultiple of another Computation

Every step is a square root

A line meets a line by solving a linear equation and a circle by solving a quadratic one. There is no third case, so the numbers a construction reaches can only ever double in complexity — and a doubling is a thing that can be counted.

 received1011010H111010001110101101001010synsyndromethe bit it names101bit 1111bit 2110bit 3011bit 4100bit 5010bit 6001bit 7000no error1011010 gives syndrome 010, which is column 6 — so bit 6 is wrong and 1011000 is thecodewordthe syndrome is three bits and the message is four: the check finds the error without everrecovering what was sent Computation

Finding the error without reading the message

Three parity checks on a seven-bit word produce three bits. If they are all zero nothing is wrong; otherwise they are the number of the position that broke. The message is never consulted, because the answer does not depend on it.

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