Beyond the Third Dimension
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Artist's Comments on Iced Cubes

In this image, we see a sequence of slices of a hypercube that is being cut by a hyperplane perpendicular to the hypercube's long diagonal. The dark blue ribbing is an orthographic view of the hypercube itself from a viewpoint that is parallel to the long diagonal, so the closest and farthest points on the hypercube are both projected to the same point in three-space, namely the center of the figure.

The slices at five different heights are shown in light blue. The symmetry of the view is reflected in the regularity of the slices, which are in the form of regular or semi-regular solids: the tetrahedron, truncated tetrahedron, and octahedron.

Below each hypercube is a plane containing an analogous view of a three-dimensional cube being sliced. The dark blue hexagon made from six equilateral triangles is actually an orthographic view of a cube when viewed from a direction parallel to its long diagonal (see the movies of orthographic projections). of a cube for more details). The light blue figures represent slices of the cube at five different heights.

Our goal is to try to use the slices of the standard cube together with how these correspond to the cube itself (which we understand) to try to use the slicing sequence for the hypercube to generate in our minds an analogous understanding of the hypercube in four dimensions.