Matpic
mathematics, in pictures
Concept

Area — where it appears

The amount of surface a shape covers, measured against a chosen unit square. It is additive over pieces that do not overlap, and it is what a dissection proof compares before and after the pieces are moved.

Named by 29 essays across 7 fields — each of them below, with the objects they name alongside it.

The Pythagorean theorem by dissection. Two squares of the same size. Each holds four copies of one right triangle. The space left over is a single tilted square on the left and two upright squares on the right.

Two squares, four triangles, and no algebra

The Pythagorean theorem is usually met as a formula to be memorised. It is much better met as a rearrangement that can be checked by eye.

geometry · Pythagoras 8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve.

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.

analysis · The integral A linear map redrawing the plane. The integer grid before and after a linear transformation; the shaded unit square becomes a parallelogram whose area is the determinant.

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.

algebra · Linear maps A disc unrolled into a triangle. A disc cut into 12 concentric rings, and the same rings straightened and stacked. The longest is the outer circumference; the shortest is nearly a point; the stack is a triangle.

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.

geometry · Circle area A Reuleaux triangle. A curve of constant width on 3 vertices, with 6 pairs of parallel supporting lines drawn across it. Every pair is 180.1 apart.

Round is not the only way to be the same width

A shape that measures the same in every direction sounds like a description of a circle. It is not — there are infinitely many others, one of them is on a coin in most people's pockets, and a drill built from one cuts a nearly square hole.

geometry · Constant width Bayes' theorem as two rectangles. A unit square split by how common the condition is (1.0%) and then by how the test behaves. Of everyone who tests positive, the fraction who have it is 16.7%.

Bayes' theorem is a picture of a square

A test that is 99% accurate returns a positive result. The chance it is right can easily be under one in five, and the reason is visible the moment the population is drawn as a square rather than described as a formula.

probability · Bayes Euclid's proof, without moving anything. The square on a leg and its share of the square on the hypotenuse are each exactly twice the same triangle, so they are equal. Nothing in the figure is cut or rearranged; the triangle is only looked at from the other side.

Euclid proves it without moving anything

The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.

geometry · Pythagoras Area is the undoing of slope. Above, a positive function with the area from 0 to 1.80 shaded. Below, that area plotted against where it stops. The lower curve's slope at 1.80 is 1.129, which is exactly the upper curve's height there.

Area is the undoing of slope

Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.

analysis · The integral Completing the square, as a square. An x by x square with the strip split in half and laid along two sides, leaving a square hole of side 1.5. Filling the hole costs 2.25 and buys a perfect square.

Completing the square, by completing a square

The step everybody is taught as an algebraic trick is a literal instruction about a literal square. There is a corner missing, its size is forced, and paying for it is the whole method.

algebra · Completing the square Three doors, as areas. Staying wins 33.3% of the time and switching wins 66.7%, because the host's choice is constrained by what the host can see, so opening a door rules a region out without moving any boundary.

The door that was not opened

Three doors, one prize, a host who opens a losing door and offers a swap. Switching wins two times in three, and the reason is not about doors — it is about what the host was allowed to do.

probability · Bayes The unit square, mapped: area × 5. The unit square and the parallelogram it becomes under a linear map, with the area of that parallelogram computed from its own corners and set against ad − bc.

The number that says how much room is left

A linear map takes the unit square to a parallelogram. The area of that parallelogram is one number, it is computable from the four entries of the matrix, and almost everything the determinant is used for is a restatement of that sentence.

algebra · Determinant A lattice polygon of area 22.5. A polygon with all its corners on the integer grid, with the 20 grid points strictly inside and the 7 on its boundary marked; its area is the first count plus half the second, less one.

Area by counting dots

Draw a polygon with every corner on a grid of dots. Count the dots strictly inside, add half the dots on the edge, subtract one — and the answer is the area, exactly, with no measuring anywhere.

discrete · Pick theorem One perimeter of 300, spent five ways. Regular polygons all of the same perimeter, drawn to scale beside the circle of that perimeter, with the area each encloses and the ratio 4πA/L².

The most area a fence can hold

One length of boundary, and the question of what shape to bend it into. The answer is a circle, everybody knows it, and the argument that convinced the nineteenth century turned out to prove something slightly different.

geometry · Isoperimetric A rectangle grown on two sides. A rectangle x by √x, with both sides grown by the change a step of h makes. The new area is the old one, two strips, and a small corner rectangle that has both increments in it.

A rectangle grown on two sides

A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.

analysis · The derivative A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

Equal area is enough, and equal volume is not

Any two polygons of the same area can be cut into each other with finitely many straight cuts. The same sentence with area replaced by volume and polygon by polyhedron is false, and what blocks it is an angle.

computation · Scissors congruence The area that names the number. The curve 1/x with the area under it from 1 to 2.7183 shaded, measuring 1.0000.

The area that names the number

The number e can be defined without mentioning slopes at all. Slide right along the curve 1/x until the area underneath reaches exactly one, and stop. That is where e is, and the reason logarithms turn multiplication into addition is visible in the same picture.

analysis · The exponential 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 measures a lattice, not the basis that happened to describe it — and that measurement is an exchange rate. Any symmetric convex region with more than four times that area has to swallow a lattice point.

algebra · Determinant A circle stays a circle, unless it meets the pole. 3 circles on a sphere beside their stereographic images in the plane, which are circles, together with one circle through the projection point whose image is a straight line.

Angles survive and areas do not

Stereographic projection takes every circle on the sphere to a circle or a line, and every crossing angle to itself. It does both exactly, with no approximation anywhere, and it destroys area so thoroughly that a patch near the pole can be a thousand times its neighbour's size.

topology · Stereographic projection Area at equal width: the triangle least, the circle most. A bar for each curve of constant width the family draws, all at the same width, with the bar's length its enclosed area and the extremes marked.

The least area a width can hold

Barbier's theorem says every curve of constant width has the same perimeter, which removes perimeter as a way of telling the family apart. Area is not like that — the circle holds the most and the Reuleaux triangle the least — and the reason the minimiser has corners is a constraint rather than a preference.

geometry · Constant width A dissection that never comes apart. The three pieces of the triangle-to-rectangle dissection drawn at 4 moments of the swing. Each top piece turns about a pin at the end of the slice it stands on, and the pieces stay joined throughout.

A dissection that never comes apart

The plane theorem lets the pieces be picked up and put down anywhere. Require instead that they stay joined at their corners and swing, and the theorem survives — which was open for a century and is a much stronger statement about the same cuts.

computation · Scissors congruence The quantity a cut cannot change and a turn can. 4 polygons, each with the spikes of its translation invariant drawn round a dial: the length of the edges facing each direction, less the length of those facing the opposite way. It vanishes everywhere for 3 of them.

Slid, but never turned

The classical dissections all turn their pieces. Forbid the turn — allow the pieces to be slid and nothing else — and equal area stops being enough, for a reason that is a single number attached to each direction and that a cut cannot change.

computation · Scissors congruence What the chain costs on a 6-gon: 39 pieces. A regular 6-gon fanned into 4 triangles, each with the three cuts that turn it into a rectangle, beside the running count of the pieces the whole chain produces — 39 of them.

Finitely many, and nobody says how many

The theorem promises a dissection exists and the proof produces one. Running the proof on a hexagon produces thirty-nine pieces, ingenuity produces five, and there is no method for proving that five cannot be four.

computation · Scissors congruence A spherical triangle becomes a quadrilateral with two right angles. A triangle on a sphere with the arc through the midpoints of two of its sides, the perpendiculars dropped from its three corners, and the quadrilateral with right angles at its base that the same area makes when the two corner pieces are moved.

Equal area on a sphere, without a rectangle

On a sphere, two polygons of the same area can still be cut into each other, exactly as in the plane. Almost nothing in the plane proof survives the move: a sphere has no rectangles, no parallel strips and no similar triangles of different sizes. What carries the theorem instead is a quadrilateral with two right angles, built from a triangle's midline.

computation · Scissors congruence Fences of 300 against a straight wall. Fences made of two, three and four straight pieces with both ends on a wall, beside a half-circle with the same length of fence, each labelled with the area it holds.

Half a circle against a wall

Lay a fence of fixed length with both ends against a straight wall and the best shape is a half-circle, holding exactly twice what a full circle of the same fence holds. The proof is a mirror: doubled in the wall, any fence becomes a closed curve with twice the length and twice the area, and the closed-curve answer carries over. In a corner the same mirrors give a slice of a circle — until the corner's angle stops dividing a half-turn.

geometry · Isoperimetric A hemisphere and a cylinder with a cone taken out, sliced at one height. Two solids drawn in profile — a hemisphere, and a cylinder with a cone removed — each cut at the same height, with the disc and the annulus the cut produces marked and their equal areas given.

The slice that has to match

The same slicing one dimension up gives the sphere's volume in a line, once one comparison is noticed: at every height a hemisphere's disc has exactly the area of a cylinder's slice with a cone's taken out of it. The principle that licenses that comparison also returns a false answer the moment the slices are not parallel.

geometry · Circle area A circular sector of area 1.10 and a hyperbolic sector of area 0.80. On the left, the unit circle with the sector from (1, 0) to (cos 2.2, sin 2.2), of area 1.100. On the right, the hyperbola x² − y² = 1 with the sector from (1, 0) to (cosh 1.6, sinh 1.6), of area 0.800. In both, the parameter is twice the shaded area.

The angle that is really an area

On a unit circle the angle t is the length of arc the point has walked, and it is also twice the area of the slice it has swept. The two readings agree on the circle and part company on the hyperbola — where arc length leads nowhere and area leads straight to cosh, sinh, and the exponential.

analysis · Circular functions Integration by parts is a rectangle. The increasing curve v = u²/4 between u = 1 and u = 3. The region under it is shaded one way and the region between it and the vertical axis another; together they fill the rectangle from the origin to (3, 2.25) minus the rectangle to (1, 0.25).

A rectangle cut by a curve

Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.

analysis · The integral Two children, and at least one is a boy. Four equally likely families drawn as quarters of a square: the question “is at least one a boy?” rules out only the girl–girl family, and leaves three equal quarters; the chance of two boys is 33.3%.

Two children and the sentence about one of them

A family has two children and at least one is a boy. The chance that both are boys is one in three — or one in two, or anything from one in three to certainty — and every one of those answers is right for some way the sentence could have come to be said. There is no host and no door, and the protocol is still the whole problem.

probability · Bayes One experiment, measured by runs and by awakenings. Two unit squares for the same coin and the same schedule: the same experiment weighed two ways: by runs, heads keeps half the square; by awakenings, heads is one of 3 equal slices.

One coin, counted by runs and by wakings

Beauty is put to sleep and a fair coin is tossed. Heads, she is woken once; tails, twice, with the first waking erased from her memory. Each time she wakes she is asked how likely heads is. One half, say some; one third, say others; and unlike every earlier puzzle of this kind, stating the protocol exactly does not end the argument.

probability · Bayes

Named alongside it

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

DissectionInvariantCongruenceConvexityCounting argumentLimitOperation setPolygonBayes' theoremConditional probabilityFundamental theoremBasis

All concepts