Question

Extra Credit (3 points): Try this with actual dough (playdoh, homemade dough, clay). Form a cube. Consider the cross sections that are formed when you slice into the cube as described. Draw each slice and describe the resulting cross section. A slice through the center, parallel to a side. A slice that cuts through the left top edge and the right bottom edge. A slice that cuts across a corner through three sides of the cube. Describe in a paragraph your observations and reflections.

Answer 1

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.
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.