Tangency
Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.
Every fraction, exactly once
Take two fractions, add the tops and add the bottoms. That is not how fractions are added, it is not an average, and repeating it produces every positive rational exactly once, already in lowest terms.
Every ray comes back to the other focus
An ellipse has two foci and one property everybody remembers: the distances to them add to a constant. What that property forces is stranger and more useful — a mirror shaped like an ellipse sends every ray leaving one focus, in every direction, through the other.
The slope of the mirror image
Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.
The map that trades circles for lines
Send every point to the one on the same ray whose distance multiplies with it to a fixed number, and circles become lines, lines become circles, angles survive untouched, and a ring of tangent circles falls out of a ring of equal ones.
Eight circles touching three
Draw three circles. How many circles touch all three? The answer is eight, the count is a fact about signs rather than about geometry, and the classical way to find them is to move the problem somewhere it becomes easy.
Curvatures that stay whole
Four circles touching one another satisfy an equation in their curvatures. Read it as a quadratic and the second solution is the first subtracted from something — so a packing that starts with whole numbers stays whole forever.
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.
Two families that cross at right angles
Fix two points and draw every ellipse with those foci, then every hyperbola with the same two. Each curve of the first family meets each of the second at a right angle, so between them they are a coordinate system — and the reason is one sentence about angle bisectors.
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.
One circle touching four
The nine-point circle touches the inscribed circle and each of the three escribed ones. Nothing in its construction mentions them, the two families of centres are built from different kinds of number, and the tangency is four exact equalities between distances and radii.
Where two roots run into each other
Put every quadratic equation at a point of a plane, one coordinate per coefficient. Each possible root becomes a straight line there, every one of those lines touches the same parabola, and that parabola is the discriminant — the crease where the plane of roots is folded onto the plane of equations.
The fewest ordinary lines a polygon allows
Take the corners of a regular polygon and add the points at infinity where its parallel chords meet. Every chord then carries three points, the line at infinity carries all the new ones, and the only lines left with exactly two points are the tangents at the corners — half as many as there are points. Dirac guessed in 1951 that nothing does better, and Green and Tao proved it in 2013.
A slope for a curve that is no function
The folium x³ + y³ = 3xy loops back over itself, so no formula y = f(x) describes it, and yet at almost every point it has a perfectly good tangent. Differentiating the equation as it stands gives the slope, −(∂F/∂x) ÷ (∂F/∂y), and the only points where that fails are the ones where the curve turns vertical or crosses itself — which are exactly the points where it stops being a graph.
Three circles that touch and are not the largest
In 1803 Gian Francesco Malfatti asked how to cut three round columns from a triangular block of marble with as little waste as possible, and answered with three circles each touching the other two and two sides. The circles exist in every triangle and are found by solving three equations. They are never the answer to his question: the plain greedy rule — the inscribed circle first, then the largest circle that still fits, twice — always does better, by 1.4 per cent on the equilateral triangle and by almost double on a thin one.
Named alongside it
The objects these essays reach for when they reach for this one.
CircleConicEllipseReflectionFocusHyperbolaInversionQuadratic polynomialsApproximationConformal mapConstructionCounterexample