Corner View of Cube Slices
This movie shows stereographic and orthographic views of a cube being sliced face first. The first direction shows the slice from the back left to the front right, while the second slice goes from bottom to top. Note that in the orthographic view, every slice appears the same shape. Each square slice is projected as a rhombus. As the slice progresses across the cube, the rhombus seems to move across the hexagon to the other side.
Here we slice the cube edge first. Slices are taken in two directions. The first is from the left vertical edge. In the orthographic view, this appears as a line since the slicing plane is parallel to our viewing direction. The second slicing sequence starts at the lower, back, left-hand edge. This one appears as a thin rectangle which thickens, achieving its widest point half-way through, then shrinks back to an edge. Every edge in the orthographic view is symmetric to one of these two edges, so we have seen every important slicing sequence.
This time we slice the cube corner first. The sequence begins with a triangle, and as the slice passes three corners of the cube, the triangle becomes cut off. The triangle becomes more cut off and eventually becomes a perfect hexagon. The sequence continues by becoming a triangle again as it passes the next three vertices, then shrinks to a point. This is seen most clearly in the second slicing sequence which shows a symmetric set of slices starting at the back-most vertex. The first set of slices start at one of the side vertices, and in this sequence, the triaganles and hexagons do not seem to be regular due to the fact that they are tilted with respect to our line of sight. These two sequences represent all the possibilities for the orthographic view.