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ひそむ秩序 · No.16 · リアルタイム図版
四次元の正多胞体
Plates on the regular solids of four-dimensional space, each one computed in the browser as you watch, and seen the only ways we can see them, by their shadows and by their slices: the tesseract, swept out of a cube pushed in a fourth direction, with its sixteen vertices, thirty-two edges, twenty-four squares and eight cubes, turning in a plane while a whole plane stays still, turning in two planes at once until no point but the centre stays still, and unfolding into the cross of eight cubes that Dalí painted; slices, a four-dimensional polytope passing through our space as a solid that grows and changes shape, the tesseract arriving corner first as a tetrahedron that becomes an octahedron, the 24-cell as an octahedron that becomes a cuboctahedron, and the 120-cell as a dodecahedron whose slices take on the patterns of a football and half a dozen other solids, the volumes of the slices adding up to the volume in four dimensions; only six, the reason there are five regular solids, that the angles of the faces around a corner must add up to less than 360°, moved up a dimension to the cells around an edge, where it leaves exactly six regular polytopes, shown side by side, and the 24-cell, raised out of the tesseract by a pyramid on each of its cubes; the 600-cell, six hundred tetrahedra on the sphere of four-dimensional space seen through stereographic projection, nine shells of vertices around one of them, twenty tetrahedra at a vertex and five around an edge closing up exactly, an isoclinic rotation that streams them out through infinity and lands them back on themselves after 36°, and its dual, the 120-cell; and the Coxeter plane, the one plane in which the product of a polytope's mirror reflections turns by 360°/h, where the shadow of the cube becomes a hexagon, the tesseract two octagons, the 24-cell two dodecagons and the 600-cell and the 120-cell rings of thirty. The corner keeps the polytope itself, what is true in four dimensions, and checks it against what is measured in the picture.
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Push a point along a line and it sweeps out a segment; push the segment at right angles to itself and it sweeps out a square, the square a cube, and the cube, pushed in a fourth direction at right angles to all three, a tesseract, with 16 vertices, 32 edges, 24 squares and 8 cubes. Nobody can see four dimensions, but anyone can see a shadow: the plate draws the tesseract in perspective from a point on the fourth axis, so that the cube farther off in that direction looks smaller and sits inside the nearer one. In every frame each vertex is moved in four dimensions and then projected; the corner keeps the tesseract itself, what is true in four dimensions, and checks it against what is measured in the picture.
A picture of a four-dimensional object is always a shadow or a slice, and shadows lie: a cube's shadow can be squashed flat, and an edge can look shorter than another of the same length. So the corner keeps the polytope itself, its coordinates and how many vertices, edges, faces and cells it has, all computed in four dimensions, and checks each against what is measured in the picture: the volume of a shadow, the faces of a slice, the speed of a point.
A slice of a polytope is the intersection of the half-spaces of its cells with the hyperplane, so a shader can draw it without being told any vertex. For every pixel it finds where the line of sight enters and leaves each half-space; the latest entry and the earliest exit bound the solid, and an edge is wherever the point it hits is close to a second face.
Nothing on screen is precomputed. The 600-cell is built from its 120 vertices, the unit quaternions of the binary icosahedral group, its 600 tetrahedra found as the sets of four vertices all at the same distance from each other, and the 120-cell from it by duality, every tetrahedron becoming a vertex and every vertex a dodecahedron. The volumes of the slices are added up exactly: between two levels of vertices the volume of a slice is a cubic polynomial in the height, and two-point Gaussian quadrature is exact for cubics. The Coxeter planes are not looked up either: the simple roots are picked out of each polytope's own reflections, their product is multiplied by itself until it returns to the identity, which gives h, and the plane is read off the eigenvectors of that product.