Projection
Named by 28 essays across 6 fields — each of them below, with the objects they name alongside it.
One cone, four curves
The circle, the ellipse, the parabola and the hyperbola look like four separate objects with four separate equations. They are one object, cut at four angles.
A sphere is a plane plus one point
Remove a single point from a sphere and what is left can be flattened out to cover an infinite plane exactly. The construction is one straight line, repeated.
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.
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.
Every triple, on one circle
Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.
What a map throws away
A linear map redraws the grid, and the determinant measures how much it stretches area. When that measurement comes out zero the map has flattened the plane onto a line — and the question worth asking is not how much was lost but how much survived, because the two always add to what there was.
Where the coefficients come from
The recipe for a square wave has a four over pi in front and a one over three on the second term, and the essay that built a square wave from sines used them without saying where they came from. They come from multiplying by one harmonic and taking the area.
Six in four dimensions, and three forever after
The count of regular solids goes five in three dimensions, six in four, and then three in every dimension above — for good. Four dimensions is the last place anything unusual happens, and it happens twice.
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.
The sphere that complex numbers live on
Add one point to the complex plane and it becomes a sphere. The rotations of that sphere are exactly the maps written as one linear expression divided by another, so a fact about turning a ball is a fact about dividing polynomials.
The square that cannot be negative
Cauchy–Schwarz is the load-bearing inequality of the whole subject and is nearly always stated without proof. 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.
One subtraction clears a direction
A basis is a set of directions to measure along, and most bases are awkward because the directions get in each other's way. Removing one shadow at a time turns any basis into one where every coordinate is a shadow and nothing interferes.
The nearest point of a flat thing
More equations than unknowns almost never have a solution. Asking instead for the point of a plane nearest to where the answer should have been turns an unanswerable question into a shadow, and the shadow is what a line of best fit is.
Six points on a conic, and the line they share
Put six points on an ellipse, join them into a hexagon, and extend each pair of opposite sides until they meet. The three meeting points always lie on one straight line. The statement uses no length, no angle and no focus — which is why it holds for every conic at once, and why a straightedge alone can draw the curve through any five points.
A quantifier is a shadow
'There is an x such that …' asks whether a column of a grid contains a mark — which is the same as asking whether a shape casts a shadow on the axis below it. Over the real numbers every such shadow can be described without the quantifier, by polynomial inequalities: 'x² + ax + 1 = 0 has a solution' is just a² ≥ 4. Over the whole numbers the same kind of shadow can carve out the primes, and any set a computer can list.
A band of a sphere is a band of its cylinder
Cut a sphere with two parallel planes and the band between them has exactly the area of the band the same planes cut from the cylinder wrapped round the sphere — whatever the band's latitude. Near the equator the sphere's band is wide and nearly upright; near a pole it is narrow and nearly flat; and the two effects cancel exactly. Archimedes proved it, wanted it on his tomb, and it gives the area of the sphere, the only honest way to pick a random point on it, and a map on which no country is the wrong size.
Six sticks tie a trefoil, and five cannot
Build a knot from straight sticks joined end to end and ask for the fewest. A trefoil takes six, and the six corners can be whole-number points in a box ten units wide. Five sticks can cross one another five times in a picture, as often as a cinquefoil needs, and still tie nothing — the five crossings always twist three one way and two the other. The fewest sticks is a measure of how knotted a knot is that no diagram shows directly.
Almost every long loop is knotted
Close a random walk into a loop and ask whether it is knotted. With ten steps almost never; with a hundred, more than one time in five it can be proved knotted by a single number; with two hundred and fifty, more than half. The chance of staying unknotted falls exponentially with length — Frisch, Wasserman and Delbrück guessed it for polymer rings around 1961, and it was proved in 1988 — because a knot needs only one small tangle somewhere, and a long loop has room for many.
A dust that almost every line misses
Keep the four corner squares of a square, then the four corners of each of those, and so on. What is left has a length, in the sense that measures length, and yet almost every straight line misses it entirely. Stage by stage the chance that a random line hits the dust falls — to 48% by the eighth stage — and how fast it falls to nothing is one of the few questions about a simple picture that is still open.
Regression is a square completed halfway
A bell in two variables, sliced at a fixed x, is a bell in y — and completing the square in y alone says where it is centred: at ρx, on a line flatter than the ellipse's own axis. That gap is regression to the mean, and the same completion says it runs backwards in time just as well as forwards.
An instrument that computes with circles
An astrolabe is a stereographic projection of the sky cut in brass. Because the projection sends circles to circles, every line on it can be drawn with compasses, and turning the star map over the plate is a rotation that answers questions in spherical astronomy without a single calculation. Of the lines medieval makers engraved, one family is not a circle on the sphere — and the arcs they drew for it miss the truth by a third of a millimetre.
A cube through a hole in a cube
Prince Rupert of the Rhine bet that a hole could be cut through a cube large enough for an equal cube to pass through it, and won. The trick is to look at the cube along a direction near its long diagonal, where its shadow is a hexagon wide enough to hold a square face with room to spare: a cube 6% larger can pass. All five Platonic solids turn out to have such a passage, found by searching over the directions from which a solid is seen — and in 2025 a solid was found that has none.
Holding one still turns the other over
A quantity can rise with an outcome and still predict it downwards once a second quantity is held fixed. Completing the square in two predictors says exactly when: the pair of correlations has to fall in a wedge whose angle is arcsin of the predictors' own correlation. Averaged over every possible correlation matrix the chance is exactly 1/4 − 1/π², about one coefficient in seven; two coefficients never reverse together, and with two dozen predictors almost two in five do.
The nearest point is not the best guess
Measure ten unrelated quantities once each, with noise, and report the measurements. Nothing seems better. Yet pulling all ten towards nought by an amount the data choose gives a smaller total error whatever the true values are — in three dimensions or more, and never in one or two. The reason is a right angle: in many dimensions noise is nearly perpendicular to the truth, so a measurement is always too long, and the best correction is the shadow of the truth on it.
Many crossings and a small linking number
Two random loops of a thousand steps, tangled round a common centre, cross each other about a thousand times in any picture of them. The linking number is half the signed count of those crossings — and it is typically about three. The signs cancel in nearby pairs, wherever one loop dips under the other and comes back, and what survives grows roughly as the fourth root of the length. Shut the same loops in a box and the cancellation stops: the crossings grow as n², and the linking number as n.
Which powers of x are enough
Every continuous function on an interval is nearly a polynomial, and it seems that all the powers of x should be needed for that. They are not. The even powers will do, and so will the primes, though the primes run out at a rate of one number in ln n. The squares will not: however many are combined, they stay 0.0905 away from x itself. The dividing line, found by Müntz in 1914, is whether the reciprocals of the powers add up to infinity.
The shortest wall that stops every line
A straight cable runs somewhere under a square field, and a trench has to find it for certain. Three sides of the field will do, at length 3. Two diagonals cost 2.83, and a Steiner tree 2.73. The best trench known has length 2.6390 and comes in two pieces that do not touch. Averaging over lines proves nothing shorter than 2 can work, and in over a century the gap between the two numbers has hardly moved.
Six lines all at one angle
Take the six long diagonals of an icosahedron: every pair meets at the same angle. Squaring each line into the matrix that projects onto it removes every sign and leaves a table no vector can kill, so at most d(d + 1)/2 lines can do this in d dimensions. The bound is reached at 2, 3, 7 and 23, and at no other dimension anyone can find.
Named alongside it
The objects these essays reach for when they reach for this one.
Inner productOrthogonalityCrossing numberDot productStereographic projectionBasisConformalConicDimensionDiscriminantInvariantKnot