Each hexagon is the slice exactly halfway through one of the cubes in the boundary of the hypercube. Then it experiences truncation as the corners of the tetrahedron are cut off, and three-eighths of the way through, the slice is a solid figure having four equilateral triangles and four regular hexagons as faces. This tetrahedron expands until it reaches a level containing 4 vertices of the hypercube. Slices of the hypercube starting with an edge.įinally, if a hypercube comes through vertex first, we start with a single vertex, which expands to form a small triangular pyramid. Vertices of the hypercube are arranged in four sets of, in The triangular prism changes to a hexagonal prism, revertsīack to a triangular prism, and shrinks down to an edge. Triangular prism with height equal to the length of the originalĮdge. We slice a hypercube edge first, we see an edge expanding to a The 8 vertices of the cube areĪrranged in four sets of, in order, 1, 3, 3, and 1 vertices. Triangle, changing to a hexagon, reverting back to a triangle,Īnd shrinking down to a point. Slicing an ordinary cube corner first yields a point expanding to a Slices of the hypercube starting with a square. The 16 vertices of the hypercubeĪre thus arranged in three sets: 4 at the beginning, then 8 in They reach the length of a diagonal of the original square and
#Hypercube of suitable dimension series#
The comparable slicing sequence for the hypercube starts with a squareįace, which expands in a series of rectangular prisms all having The 8 vertices of the cube are partitioned into three sets: 2 in the beginning edge, 4 in the largest rectangular slice, and 2 in the final edge. Thus the rectangular slices of the cube will have one set of unchanging edges equal in length to an edge of the cube, and another set growing from a point to the length of a diagonal and shrinking back to zero. The slices of each of these squares are line segments: the segments start at a point, grow to a diagonal of the square, and shrink back to a point. The squares perpendicular to the leading edge are sliced by a line parallel to one of their diagonals. Thinking about the formation of the rectangular slices will guide us to the slices of a hypercube. Slices of the hypercube starting with a cube.Ī cube coming through edge first was sliced into a series of rectangles. Square for a while, and if the hypercube comes through our space If a cube comes through the horizontal plane square first, we see a But just as the slices of an ordinary cube depend on the orientation of the cube with respect to the slicing plane, so too do the slices of the hypercube. This is analogous to the appearance of the slices of an ordinary ball passing through Flatland. No matter which way the hyperball is rotated, we will get the same sequence. The slicing sequence of the hyperball has already been compared to the inflation and deflation of a spherical balloon.
How will the appearance of the slices change as the water level rises to cover the objects? We can imagine four-dimensional versions of Froebel's geometrical gifts, a hyperball and a hypercube suspended "above" our three-dimensional world, thought of as the analogue of the surface of the water.
We can exhibit the slices even though we cannot construct the objects themselves that are being sliced! What would happen if the hypercube came through our universe from another direction, for example square first or edge first or corner first? For such questions, the modern graphics computer is ideally suited. We have already seen how a hypercube would appear if it penetrated our universe cube first: at the first instant, we would see an ordinary cube with 8 bright vertices, then a dull cube for a while, and finally another cube with 8 bright vertices. The hypercube Q n Q_n Q n has 2 n 2^n 2 n vertices and each vertex is connected to n n n of the other vertices.Slicing the Hypercube Slicing the Hypercube
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