Plates on the motions that leave a pattern unchanged, each one computed in the browser as you watch: a band of ornament with exactly seven kinds of symmetry, slid, glided, flipped and turned onto itself, then bent into a ring where only its turns and mirrors survive; and the seventeen ways a pattern can repeat across the plane, on five lattices, with turns of order 2, 3, 4 and 6 and never 5; and the kaleidoscope, two mirrors that repeat what lies between them without a seam only at angles of π/n, and three that enclose a triangle whose angles decide whether its images close up on a sphere, fill the plane or fill the hyperbolic plane; and Wythoff's construction, a point in a kaleidoscope's triangle whose images are the corners of the Platonic and Archimedean solids; and the belt trick, a cube turned once that leaves a twist no moving of the belt can remove, and turned twice that can be untwisted, because the unit quaternions cover the turns of space twice over. The friezes and wallpapers are functions built from a few waves averaged over their groups, and every plate keeps in its corner a measurement taken on the figure on screen: a pattern compared with its own image under a motion, point by point, or the angles of a kaleidoscope, or the vertices, edges and faces of a polyhedron counted one by one, or the quaternion of a belt.
図版
I–V
ループ
各 60 秒
解像度
再生エリアに合わせて自動調整（最大 440 万画素）
あらかじめ録画したコマ
なし
映画
制作中
図版 I · 全 4 楽章 · 1:00 のループ
Seven friezes
Friezes run round the walls of temples, pots and buildings in every culture, and all of them fall into seven classes, told apart by which motions carry the band onto itself: a slide along it, a glide that slides by half a period and flips, a mirror across it or along it, a half-turn about a point on its centre line. John Conway named the seven classes after the tracks that feet leave, from hop to spinning jump, and wrote each as an orbifold, whose ∞ is a rotation that never comes back. In the corner the band is compared with the moving copy at scattered points on screen, and the largest difference is shown.
図版
解説
The test in the corner
A symmetry of a pattern is a motion that leaves it unchanged. The friezes and the wallpapers measure that directly: the pattern is evaluated at a few hundred scattered points on screen and again at the same points moved by the motion, and the corner shows the largest difference between the two, divided by the largest value. For a true symmetry it is zero to every decimal shown; for a motion the pattern lacks, it is plainly not. The kaleidoscope reads the angle of its mirrors, the seam between them, and the number of triangles its reflections make.
Patterns from waves
Each pattern is a sum of a few plane waves, averaged over its group: a wave is moved by every motion of the group, one for each class modulo the translations, and the results are added. The sum is then unchanged by every motion of the group, whichever waves were chosen, and choosing them generically gives it no other symmetry. It is the method of Frank Farris, drawn here as fine contour lines of the modulus over a faint colouring by phase.
Orbifold names
Conway's orbifold notation names a group by what is left when the plane is folded up along its symmetry: a digit for each kind of centre of rotation, a star for a mirror, with the digits after it counting the corners where mirrors meet, a cross for a glide that crosses no mirror, and an o for a pattern with translations only. For friezes, the ∞ is the rotation of infinite order that a slide along the band becomes.