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Concept

Projection

A map that drops a dimension, casting a shape onto a line, a plane or a surface along a chosen family of rays. It is what loses information deliberately, and what survives it is exactly what the projection was chosen to preserve.

Named by 15 essays across 6 fields — each of them below, with the objects they name alongside it.

Four conic sections from one cone. Circle, ellipse, parabola and hyperbola, produced by tilting a single cutting plane further and further.

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.

geometry · Conic sections One point on the sphere for every point of the plane. Lines from the north pole of a sphere through each of its points land on a plane below, matching the sphere minus one point with the whole plane.

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.

topology · Stereographic projection The dot product as a shadow. Two vectors and the shadow the first casts on the second. The shadow is 2.425 long and b is 4.123, so the dot product is 10.000.

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.

algebra · Inner product the trefoil. the trefoil, drawn as a closed curve with 3 crossings. At each crossing the strand passing underneath is broken, which is the only information the flat picture carries that the curve alone does not.

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.

topology · Knots Rational points on the unit circle. Lines of rational slope through the left-hand point of a circle, each meeting it again at a rational point.

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.

number · Pythagoras A whole line arrives at the origin. A linear map whose determinant is zero, drawn before and after. One line of the plane is sent to the origin and the whole plane is sent onto another line; the dimension lost and the dimension kept add to two.

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.

algebra · Linear maps The target, multiplied by one harmonic at a time. Four panels, each showing the square wave multiplied by a single sine. The areas cancel exactly except against the harmonics the wave actually contains.

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.

analysis · Fourier series The regular solids of four dimensions. 5-cell, tesseract, 16-cell, 24-cell, each turned in four dimensions and projected to the page; the edges are the pairs of vertices at the shortest distance apart.

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.

geometry · Regular polyhedra 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 A turn of the sphere, seen from the plane. A square grid in the plane and its image under the map obtained by lifting to the sphere, rotating by 62° about a tilted axis, and coming back down. The lines become arcs of circles and the crossings stay at right angles.

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.

topology · Stereographic projection 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.

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.

algebra · Inner product One subtraction clears a direction. Gram–Schmidt on two planar vectors, in 3 panels: the pair as given, the shadow of the second on the first, and the perpendicular pair that is left when the shadow is removed.

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.

algebra · Inner product The nearest point of the plane the columns span. A target vector in space, the plane spanned by two columns, the point of that plane nearest the target, and the residual joining them, which meets the plane at a right angle.

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.

algebra · Inner product Six points on an ellipse, and the line their opposite sides meet on. A hexagon with its six corners on an ellipse. Its three pairs of opposite sides are extended until they meet, and the three meeting points lie on one straight line. Moving one corner off the ellipse breaks the alignment.

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.

geometry · Conic sections "There is an x" is a shadow: x² + ax + 1 = 0 has a solution exactly when |a| ≥ 2. A grid with a horizontal and x vertical, marking the cells the curve x squared plus a x plus one equals zero passes through; beneath it, a strip marking the columns that contain a mark, which are exactly those with a at least two in size.

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.

logic · Quantifiers

Named alongside it

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

Inner productOrthogonalityDot productBasisConicDiscriminantStereographic projectionApproximationBijectionCauchy schwarzConformalDimension

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