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0] \\\\\nC = [0, 0, 1] \\\\\nD = [1, 1, 1]`}\n\nand let the same letters denote the faces opposite them.\n\nA billiard path in any tetrahedron can be unfolded by reflecting the tetrahedron across the faces it hits, so that the trajectory becomes a straight line ${tex`L`} through a chain of reflected copies. For a given face sequence ${tex`w`}, let ${tex`T_w`} be the affine isometry taking the first tetrahedron to the last in this chain. Then a closed billiard path with word ${tex`w`} exists exactly when there is a line ${tex`L`} with ${tex`T_w(L)=L`}, and whose segment through the chain intersects the corresponding faces in their interiors in the prescribed order.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":554,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":560,"value":"$35","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":256,"value":"## Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":127,"value":"THREE = require(\"three@0.150.1\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":139,"value":"OrbitControls = (await import(\n \"three@0.150.1/examples/jsm/controls/OrbitControls.js\"\n)).OrbitControls","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":508,"value":"TrackballControls = (await import( \"three@0.160/examples/jsm/controls/TrackballControls.js\"\n)).TrackballControls","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":476,"value":"Plot = require(\"@observablehq/plot\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W36","stdlib":"1","titling":"auto"}} 33:"$32:body"