Plates on the curves that make a quantity as small as it can be, each one computed in the browser as you watch: light, which travels by the path of least time, off a mirror, into water, through a medium that slows it near the ground and through a raindrop, where the rays pile up at the angle of least deviation and make the rainbow; and the brachistochrone, the curve of quickest descent, which Johann Bernoulli found by bending light through layers of glass, and which is also the curve on which a bead takes the same time to reach the bottom from any height; and the hanging chain, the curve of its length with the lowest centre of mass, which turned over is the arch that stands in pure compression and laid in a row is a road for square wheels; and Dido's problem, the circle that encloses the most area for its length, reached through regular polygons and Steiner's symmetrization, with a semicircle against a coast and two bubbles whose walls meet at 120°; and soap films, which pull themselves into the shortest network they can reach between pins, meeting in threes at 120°, sometimes stuck in a longer shape than they need, and in a wire tetrahedron meeting along lines that come together at 109.47°. Every plate keeps in its corner the quantity being made least, measured along the paths on screen and set against the smallest value it can take.
图版
I–V
循环
每张 60 秒
分辨率
随播放区域自适应，最高 440 万像素
预先录好的帧
没有
短片
制作中
图版 I · 4 段 · 1:00 循环
Least time
Heron of Alexandria noticed that light reflected in a mirror takes the shortest of all the paths that touch the mirror. Pierre de Fermat, in 1662, made this a principle: light goes by the path of least time, and from it he derived the law of refraction that Snell had found by measuring. The plate draws the wavefronts from a lamp S as they spread, each point reached at the earliest moment light can get there, and a candidate path that slides until it finds the least; the curve at the right is the quantity along every candidate, flat at its minimum. The corner shows the length or the time along the path on screen, and the least it can be.
图版
说明
The quantity in the corner
Each plate is about a quantity that a curve makes as small, or as large, as it can: the length of a path, the time light or a bead takes along it, the height of a chain's centre of mass, the area a loop encloses. The corner measures that quantity on the path on screen, one straight piece at a time, each at the speed light or a falling bead has there, and sets it against the least value. Where there is a family of candidates, a small curve at the right plots the quantity for every one of them: at the best it is flat, and that flatness is where the calculus of variations begins. Some plates measure instead what only the best has: angles of 120° where films meet, a line of thrust inside the stone, an axle that neither rises nor falls.
Least, or only stationary
Fermat's principle is really about paths whose time does not change to first order when the path is nudged. Most of them are minima, but the principle needs only the flatness. The rainbow shows why it matters: the deviation of a ray through a drop is stationary at its least value, and it is the flatness there, not the smallness, that gathers the rays into a bright bow. The soap films of the last plate settle into whichever minimum they reach first, and it need not be the least.
Wavefronts and rays
The wavefronts are the points that light from the lamp reaches at the same moment, computed for every pixel as the least time over every route: straight, reflected in the mirror, bent at the surface of the water. The rays cross them at right angles, and the first front to reach the eye brings the ray of least time. The fronts of the falling bead, where it can be after the same time on the quickest route, cross every cycloid from A at right angles in the same way.