RJML Labs · Experiment
Nine circles
Every turn of a Rubik’s Cube is a permutation of its 54 stickers, and 9 circles are enough to draw all of them.
x layers L M Ry layers D E Uz layers B S F · pinch or scroll to zoom, drag to pan · drag the cube to turn your view
The cube has three axes, and each axis has three layers, so there are 9 layers in all. In the picture each layer is one circle, and each circle carries 12 dots. Every one of the 54 stickers is a dot where two circles from different axes cross. When the cube turns, the matching circle lights up and its dots slide round it.
A quarter turn moves the 12 dots on a circle 3 places. A turn of an outer layer also turns the 8 outer stickers on the face beside it. Together that moves 20 stickers in five cycles of length four, which is why four equal quarter turns put every sticker back. A turn of a middle layer moves only the 12 dots on its own circle. The scrambled cube is solved turn by turn, and each of the six clusters ends as one colour.
Each turn has a name in the standard notation. A letter names the layer, and a letter alone is a clockwise quarter turn seen from outside that face. A letter with a mark after it is the same turn the other way. The row under the picture collects the fifteen turns of this solve.
Group theory is the branch of algebra that studies this structure. The turns of the cube form a group, and each turn acts on the stickers as a permutation. The group has 43,252,003,274,489,856,000 positions. In 2010 Rokicki, Kociemba, Davidson and Dethridge showed that every one of them can be solved in 20 moves or fewer, when a half turn counts as one move.