Generator

The Pythagorean theorem by dissection

A generator in the geometry library, called 58 times across 12 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

pythagoras is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

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.

A box with whole diagonals on every face

The smallest Euler bricks

A brick from every right triangle

How many bricks there are

Factors every brick must have

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

157 is among the congruent numbers whose triangles are too large for the search ×1
361 congruent square-free numbers up to 1,000 by Tunnell's criterion ×1
44, 117, 240 has three whole face diagonals and no whole space diagonal ×1
a larger exponent gives a ball that contains the smaller one ×1
a marked third side is inside the range drawn ×1
a triple whose squares add gives a right angle ×1
a triple whose squares do not add gives an angle that is not right ×1
and below one it is not, which is why p < 1 gives no distance at all ×1
and every one the search found is in the tree ×1
and half the rectangle — same base, apex on the parallel through the altitude ×1
and has not appeared before in the tree ×1
and is primitive ×1
at nine dimensions the inner sphere reaches the cube's faces ×1
at ten it is outside them ×1
consecutive knots are one unit apart ×1
doubling (12, 36) gives (25/4, −35/8) on the curve ×1
each fixed ball drawn behind is between 1 and 12 ×1
each is a right triangle with the stated area ×1
each leg at the corner is a positive length ×1
each leg is between nothing and a quarter turn ×1
each leg of the triangle is a positive length ×1
each side is a positive length ×1
each side of the box is a positive length ×1
every box from Saunderson's formula is an Euler brick ×1
every node of the tree is a right triangle in whole numbers ×1
every number the search finds a triangle for passes Tunnell's test ×1
every point drawn is at distance one in this norm ×1
every primitive brick has an edge divisible by this number ×1
every square-free number of the form 8k + 5, 6 or 7 up to 1,000 passes Tunnell's test ×1
four triangles and the tilted square fill the left square ×1
four triangles and the two upright squares fill the right square ×1
no perfect cuboid has every edge below 300,000 ×1
no primitive triple has area a square, twice a square or three times a square ×1
so the square and the rectangle are equal ×1
the altitude cuts the big square into exactly these two rectangles ×1
the angle at C is a right angle ×1
the angle at the corner is a right angle ×1
the ball meets the diagonal at 2^(-1/p) in each coordinate ×1
the box has three positive sides ×1
the constructed triangle's hypotenuse is the root of the sum of the squares ×1
the corner is square exactly when the squares add ×1
the dimensions drawn run from 1 to between 4 and 40 ×1
the doubled point gives the triangle (7/10, 120/7, 1201/70) of area 6 ×1
the drawn third side is c ×1
the eight triangles are all the same ×1
the exponent is between 0.4 and 12 — below 1 it is drawn as a warning ×1
the exponent of the norm is between 0.4 and 12 ×1
the first right face is half the product of its legs ×1
the fixed balls behind it are between 1 and 12 ×1
the floor diagonal meets the vertical edge at a right angle ×1
the given triangle and the constructed right triangle are congruent ×1
the hyperbolic excess in c² is a²b²/3 at small size ×1
the inner circle touches each corner circle ×1
the largest triangle drawn has legs under a quarter turn ×1
the legs are positive lengths ×1
the long leg is b ×1
the point (12, 36) comes from the triangle (3, 4, 5) ×1
the proof is one the family draws ×1
the rope carries one knot per unit of its length ×1
the rope's three sides are whole numbers of knots, each under sixty ×1
the second right triangle has the floor diagonal and the height as its legs ×1
the sensitivity at the right angle is c/ab radians per unit ×1
the short leg is a ×1
the slanted face's area squared is the sum of the three right faces' areas squared ×1
the smallest box with a whole space diagonal and two whole face diagonals is 104, 153, 672 ×1
the smallest primitive brick by its longest edge is 44, 117, 240 ×1
the space diagonal squared is the sum of the three squares ×1
the sphere shortens the hypotenuse and the hyperbolic plane lengthens it ×1
the spherical hypotenuse is shorter than the flat one ×1
the spherical hypotenuse satisfies cos c = cos a cos b ×1
the spherical shortfall in c² is a²b²/3 at small size ×1
the square on the hypotenuse is c² ×1
the square on the leg has the leg's area squared ×1
the three faces at the corner meet at right angles ×1
the three lengths close into a triangle ×1
the tilted square equals the two upright squares ×1
the tilted square is c² ×1
the tree finds exactly as many primitive triples under 200 as the search does ×1
the tree is checked against a search up to a hypotenuse between 50 and 2000 ×1
the tree is drawn to between 1 and 3 generations ×1
the tree of triples is drawn to between 1 and 3 generations ×1
the triangle is half the square — same base, apex on the opposite side ×1
the triple 3, 4, 5 gives Halcke's brick ×1
the two floor edges meet at a right angle ×1
the two triangles are the same triangle turned ×1
with p at least one, the midpoint of any two points of the ball is inside it ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Geometry

A box with every face diagonal whole

Pythagoras gives a rectangle with whole sides and a whole diagonal — 3, 4, 5. A box with whole edges and whole diagonals on all three faces is far rarer: the smallest is 44 × 117 × 240, found in 1719, and there are only 113 essentially different ones with every edge below 300,000. Ask for the long diagonal through the box to be whole as well and the answer is unknown. No such perfect box has ever been found, and nobody has proved there is none.

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.

Number

A tree that holds every triple

Three fixed matrices, applied to 3-4-5 over and over, produce every primitive Pythagorean triple there is — each of them once, none of them twice, and with no test for common factors anywhere in the procedure.

Geometry

Circles that are diamonds and squares

The theorem hands over a formula for distance. Take the formula as a definition, change the exponent in it, and the set of points one unit from the origin stops being round — while remaining, in every sense that matters, a circle.

Computation

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.

Geometry

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

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.

Topology

Seven hundred and twenty degrees of gap

Unfold the faces around any corner of a solid and they do not close up. The gap left over is different at every corner and on every solid, and the gaps always add to two full turns.

Geometry

The areas a right triangle can have

A right triangle with sides 3, 4 and 5 has area 6; one with sides 3/2, 20/3 and 41/6 has area 5. No right triangle with rational sides has area 1, 2 or 3. Which whole numbers are areas is a thousand-year-old question, and its best answer — two counts of solutions to quadratic equations — rests on one of the great unproved conjectures.

Geometry

The rope that squares a corner

The theorem turns two sides into a third. Run it backwards and it turns three lengths into a right angle — which is a different statement, needs its own proof, and is the only one of the two that has ever been used to build anything.

Geometry

The triangle that a globe gets wrong

On a sphere, a right triangle with legs of fifty and sixty degrees has a hypotenuse of seventy-two, not seventy-eight. The theorem is not approximately true there — it is false, and what replaces it says exactly how much room the surface has.

Geometry

Two right angles and the diagonal of a box

The theorem applied once gives the diagonal of a floor. Applied again, standing on the first result, it gives the diagonal of the room — and the pattern does not stop at three, which is where a fact about triangles quietly becomes the definition of distance.

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