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Proof without words — page 3

Arguments that are complete once they have been looked at properly. Not illustrations of proofs — the proofs themselves.
The midpoint of a segment, drawn with a compass and no straightedge. A segment with the arcs that step its length three times round one end to reach the point twice as far away, and the further arcs that send that point back to the midpoint, every one of them a circle. Computation

The straightedge buys nothing

Every point a compass and a straightedge can construct together can be constructed by the compass alone. The straightedge draws lines nobody needs; the compass does the work, and the proof that it does is an inversion performed with arcs.

 The nine-point grid, and the lines they force. 9 points with all 20 of their connecting lines drawn. The 12 carrying exactly two points are drawn solid and the rest faintly; the count is computed from the coordinates rather than read off the drawing. Geometry

The line with only two points on it

Scatter finitely many points on a page, not all in one line, and draw every line through two or more of them. However cunningly the points are placed, some line ends up carrying exactly two — and the proof is a minimisation with no algebra in it at all.

 A chord of x², and the curve under it. The curve x² with one chord drawn across it, the region between them shaded, and the midpoint heights of both marked. The comparison is computed at four hundred sample points. Analysis

The curve of the average, and the average of the curve

A curve that bends upwards keeps every one of its chords above it. That single fact, applied to a weighted average instead of a midpoint, turns into an inequality that produces the arithmetic–geometric mean inequality, Cauchy–Schwarz and the entropy bound as special cases.

 The level curve, and the axes the matrix chooses. The curve xᵀAx = 1 for the matrix [2, 0.8, 0.8, 1.4], drawn by solving for the radius at each angle, with the two eigen-directions marked; they cross at a right angle and are the axes of the curve. Algebra

Symmetry forces a right angle

A matrix equal to its own reflection across the diagonal always has real stretches and always has perpendicular directions to stretch along. Neither is true of matrices in general, and both follow from one line of algebra.

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

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.

 A sequence of 3² with no climb and no fall longer than 3. 10 terms plotted in order, each labelled with the longest climb and the longest fall ending at it; the first 9 keep both counters at 3 or below and the last one cannot. Discrete

The sequence that cannot avoid a staircase

Any ten numbers in a row contain four that climb or four that fall. The proof gives every term a pair of counters, notices that no two terms can share a pair, and is finished — with a bound that is exactly right.

 A parallelepiped of volume 2.94. The image of the unit cube under a three-by-three matrix, beside the six signed products whose sum is its volume. Algebra

The only function that behaves like a volume

Ask for a function of the columns of a matrix that scales when a column scales, vanishes when two columns agree, and gives one on the identity. Three conditions, and there is exactly one such function in every dimension.

 The product of (1 − qᵏ), and what survives at 12. The coefficients of the pentagonal product drawn as signed bars, with the partitions into distinct parts that Franklin's move leaves unpaired. Number

The terms that cancel almost everything

Multiply out the product of 1 − q, 1 − q², 1 − q³ and so on, and nearly every coefficient is zero. What survives is a single plus or minus one at 1, 2, 5, 7, 12, 15 — and the reason is a way of pairing partitions off so that each pair cancels.

 Every ear carried to a slice of a disc. The corridor's triangulation on the left and the same triangulation of a regular 20-gon on the right, with four points and their images marked; the map is affine on each triangle and agrees on every shared edge. Topology

Every loop is a circle in disguise

Separating the plane is the weak half of what the eye believes about a closed curve. The strong half is that the inside is a disc — that the whole plane can be bent until the curve is a round circle — and for a polygon that is a construction rather than an argument.

 Two inversions, and the number four points agree on. Four points, their images after one inversion and after a second in a different circle, with the cross-ratio computed at each stage; it is conjugated once and restored twice. Geometry

The number four points agree on

One inversion is a reflection and reverses orientation. Two of them compose to a motion, and what that motion leaves alone is a single number computed from any four points.

 The cube cut into 6 symmetric chains. The subsets of a set of 4 partitioned into 6 chains by the bracket rule, each chain running from size k to size 4 − k and passing once through the middle layer. Discrete

The cube cut into chains

Write a subset as a string of brackets, match them the ordinary way, and the unmatched ones say which chain it is on. Six chains cover all sixteen subsets of a four-element set, and the bound and the example arrive together.

 The triangulation as the underside of a hull. Eight points on a floor, lifted onto a paraboloid above them. The faces of the lifted set's lower convex hull are shaded, and their shadows on the floor are the Delaunay triangulation of the original points. Geometry

One dimension up, and the circles disappear

The Delaunay triangulation is defined by a condition about circles, which is awkward to compute and awkward to reason about. Lift every point onto a paraboloid and the circles turn into planes, the condition turns into convexity, and a two-dimensional problem is solved by looking at a three-dimensional shape from underneath.

 The two numbers that make up a width. A Reuleaux polygon with 3 sides, its centre marked as the origin, and the two supporting lines with normals 33° and 213°. The perpendicular distances from the origin to the two lines are marked; they add to the width. Geometry

The shape described from outside

A convex shape can be given by its boundary or by the family of lines that touch it, and the second description turns the constant-width condition into one line of arithmetic — after which the perimeter falls out, and curves with no corners at all can simply be written down.

 A line under every point of x². The curve x² with 4 tangent lines drawn, each extended across the whole interval and each staying below the curve throughout. Analysis

A line under every point

The chord above the curve is one definition of convexity. There is a second — a line under the curve at every point, staying under everywhere — and it is the one that turns a statement about a derivative at a point into a statement about the whole function.

 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. Geometry

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.

 The parabola that proves |a·b| ≤ |a||b|. The squared length of a − t b plotted against t. It is a parabola opening upward whose least value is 7.118; that this is never negative is exactly the Cauchy–Schwarz inequality. Algebra

The square that cannot be negative

Cauchy–Schwarz is the load-bearing inequality of the whole subject and is nearly always asserted. It is one line away from a fact nobody would argue with, and the line is a parabola with no room to cross the axis.

 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. Computation

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.

 A run down a diagonal, and the entry it adds to. 9 rows of Pascal's triangle with 5 entries shaded and the entry they add to marked. The claim is checked by adding the shaded entries: 1 + 3 + 6 + 10 + 15 = 35. Discrete

The run that lands one place along

Add up a run of entries down one of Pascal's diagonals and the total is another entry of the triangle — one row further down and one place along. The same triangle holds four more sums of that kind, and each is a different question answered by the same additive rule.

 Every ray aimed at one focus is turned towards the other. A hyperbola with its two foci and 13 rays aimed at the far one. Each strikes the near branch from outside and is turned towards the near focus — which is the property a Cassegrain telescope's secondary mirror uses. Geometry

Aimed at one focus, turned towards the other

An ellipse sends every ray leaving one focus through the other. A hyperbola does something that sounds like the same sentence and is not — it takes a ray aimed at the far focus and turns it towards the near one, which is what the second mirror of a telescope is for.

 7 steps and 3 withdrawn assumptions. A natural-deduction derivation of (p → q) → (¬q → ¬p). Each horizontal bar is one inference, named on its right; the bracketed formulas are assumptions, and each is withdrawn at the step that names it. Logic

The assumption a proof pays back

A tableau assumes the opposite once and takes it apart. Natural deduction assumes things freely, uses them, and then withdraws them — and the withdrawal is what turns a derivation of a consequence into a proof of an implication.

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