Matpic
mathematics, in pictures
Theme

Doing infinitely many things

Sums that never end, subdivisions that never stop, and the care required to make either of them mean something.
131385322 × 131 × 81 × 51 × 31 × 22 × 1gcd(34, 13) = 1 Geometry

The oldest algorithm, drawn as a tiling

Euclid's method for finding a greatest common divisor is usually presented as a loop. It is also a way of tiling a rectangle with squares, and the tiling explains why it works.

 0.511.522.5301234xysum ≈ 5.790exact = 6.300 Analysis

Adding up rectangles until they stop being rectangles

The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.

 -111 term-113 terms-117 terms-1121 terms Analysis

A square wave built entirely out of round ones

Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.

 -2-1.5-1-0.50.511.52123456xyheight 0.37slope 0.37height 1.00slope 1.00height 2.72slope 2.72height 4.95slope 4.95 Analysis

The curve that is its own slope

There is exactly one shape of exponential curve whose steepness at every point equals its height at that point. The number that produces it is 2.71828…, and it was not chosen for elegance.

 area πr²base 2πr, height r — area ½ · 2πr · r Geometry

A circle unrolled into a triangle

Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.

 246810121416182022240123ntotal 3.776term 0.042 Analysis

A sum whose terms vanish and whose total does not

Add a half, a third, a quarter, and keep going. The terms shrink to nothing and the total passes every number there is — but so slowly that no computation will ever watch it happen.

 -8-6-4-22468-2-112xsin xdegree 9 is out by 4.7e+0 at x = 5.76 Analysis

One point's worth of information

A Taylor series claims that everything a function does, everywhere, is encoded in its behaviour at a single point. That claim is extraordinary, it is often true, and the cases where it fails are the interesting ones.

 0.511.522.512345xh = 1.2 slope 3.2000h = 0.8 slope 2.8000h = 0.5 slope 2.5000h = 0.28 slope 2.2800h = 0.12 slope 2.1200 Analysis

The slope of a single point

A slope needs two points. A derivative is the slope at one. The construction that bridges the gap is a sequence of secants, and the whole difficulty of calculus is in what "the limit of that sequence" is allowed to mean.

 246810121416182022240.60.70.80.91terms usedln 2 = 0.69315odd sums, from aboveeven sums, from below Analysis

The same terms, in a different order, adding to whatever is asked

Flip alternate signs in the harmonic series and it converges. Reorder the terms — add nothing, remove nothing — and it converges to any number chosen in advance. Addition stops being commutative, and the picture shows where it goes.

 050100150200250300350400-60-40-20204060steps takendistance from the start√n Probability

A walk that always comes home, until it does not

Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.

 2 × 3 × 5 × 7 + 1 = 211every listed prime tiles the product exactly, and then meets the extra unit÷ 21 left÷ 31 left÷ 51 left÷ 71 leftso no prime on the list divides 211211 is itself prime — but the argument never needed it to be Number

There is no last prime

Euclid's argument is often described as producing a new prime from any finite list. It does not, and the number it builds is frequently composite — which makes the proof more interesting rather than less.

 =φ1 +11 +11 +11 +11 +11[1; 1, 1, 1, 1, 1, …] — and it does not stop Number

A fraction that never closes

Euclid's algorithm throws away everything except the number of squares it peeled at each step. Those counts are a second name for the number it started from — one that terminates exactly when the ratio is a ratio.

 0/71/72/73/74/75/76/70φ5φthe fractional parts of φ, 2φ, 3φ, …5φ and 0φ landed in the same boxso 5φ is within 1/7 of the whole number 8 — an error of 0.0902which makes 8/5 accurate to 1.8e-2 Number

How close a fraction can get

Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.

 755the overlap is 7² = 49; the two corners are 2 × 5² = 50they differ by 1, which is exactly 17² − 2 × 12² = 1and 7, 5 is a smaller pair with the same discrepancy, the other way round Number

The square that cannot shrink

The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.

 every node has at most 3 children and the tree reaches level 416 nodes on the bottom row; the marked walk takes a surviving child at every step Logic

An infinite tree has an infinite path

A tree that goes on forever, in which every node has only finitely many children, must contain a single branch that goes on forever. The proof is a rule for walking, and the rule is the whole of why finite information can decide an infinite question.

 x10110101000x21101010000x30011111000x41010100000x50001001000x60111110000x71110011001x8010100000110011001neweach row is a number between 0 and 1, written in binary; the marked squares are thediagonalthe row underneath is the number that differs from the nth in its nth binary digit — so thelist is missing a number, and it was an arbitrary list Logic

The row that is not on the list

Write down a list of infinite sequences, any list at all, and there is a rule that builds a sequence missing from it. The rule reads one entry from each row, and it is the single most reused argument in this field.

 [0, 1](0, 1)01½0 goes to a half and 1/n goes to 1/(n+2); every other point of [0, 1] stays where it is8 moved points drawn, all distinct, all inside (0, 1) — and the two endpoints are gone Logic

Two injections make a bijection

If each of two collections fits inside the other without collisions, they are the same size. That sounds obvious and is not, because neither injection needs to be onto — and the proof is a rule for deciding which of the two to follow, one chain at a time.

All themes