Reflection
Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
A bounce is a fold of the table
Reflect the room instead of the ball and every bounce disappears — the trajectory becomes a straight line through a tiled plane, and questions about what a ball does forever become questions about the slope of that line.
A table folded into a surface
Unfolding a square billiard gives a straight line on a torus. Unfolding any table whose angles are whole fractions of half a turn gives a straight line on some surface — and which surface it is decides how hard the dynamics will be.
A room that cannot be lit
Mirror the walls of a room and put a lamp inside it. Every point should be lit, since light bounces forever — and there are rooms with a dark spot no ray from the lamp ever reaches.
The triangle nobody can settle
Does every triangular billiard table have a path that closes on itself? Acute triangles do, right triangles do, triangles with rational angles do — and for the rest the question has been open since it was asked.
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.
Half a circle against a wall
Lay a fence of fixed length with both ends against a straight wall and the best shape is a half-circle, holding exactly twice what a full circle of the same fence holds. The proof is a mirror: doubled in the wall, any fence becomes a closed curve with twice the length and twice the area, and the closed-curve answer carries over. In a corner the same mirrors give a slice of a circle — until the corner's angle stops dividing a half-turn.
Three mirrors make every solid
Every symmetry of a regular solid, reflections included, is produced by just three mirrors meeting at its centre, reflected in one another over and over — a kaleidoscope. Put a single point between the three mirrors and its reflections are the corners of a solid: the regular solid itself if the point sits in a corner, and every one of its truncated and expanded relatives if it sits anywhere else.
Every point has a partner across the bisectors
Draw the three lines from the corners of a triangle through any point, reflect each in the bisector of its own angle, and the three reflections meet again. The pairing this makes swaps the centroid with the symmedian point and the orthocentre with the circumcentre, bends every straight line into a conic through the corners, and sends the circumcircle to infinity.
Named alongside it
The objects these essays reach for when they reach for this one.
ConicBilliardsFocusPeriodic orbitTangencyUnfoldingEllipseOptimalityCausticCounterexampleEccentricityEquidistribution