Chapter 5 Responses.

THE TRIO in the mix.

It says in the book there are no more regular objects in 3-space to be found. Is it possible to add and subtract regular shapes to and from each other in such a way that another regular 3-dimensional shape can be made? For instance, if you have a cube and it's sides are all, say, 5 units long, and in the exact center of each face you subtract out of the solid a smaller cube, say with sides of size 1. (It helps to draw this with grid lines on the cube faces) You have not violated rules of regularity, in that angles are still the same at all vertices, the shape is symmetrical, and there are a similar number of edges meeting at each vertex.

This leads to an interesting exercise that reminds us of fractals. Try to preserve rules of symmetry, angle similarity, and edge number at each vertex. You are allowed to add and subtract shapes, and change scale. You can get a 3-D fractal effect in the cube example if you subtract a cube out of each new face created by subtractions. The computer would lend itself well to this, but with care it can be done just as well in a drawing.

See if you can figure out a numerical pattern for the decrease in edge size at each recurrence of subtraction. For instance, the original cube in the example above was 5 units long per edge. The next cube was only 1 unit long. Why 1 ? Why not start with side 3 and subtract a cube of size 1 from each face? Look at the drawing of this, and see if subtracting different-size cubes makes the exercise work or not work.

To further the above idea of adding and subtracting, say you subtracted a shape's dual out from its inside, so you had a geode-like effect of the solid outside of the original shape, but inside there is a hollow area. Would the result be a regular?

There are a number of fold-out patterns given in the book to go from 2 to 3 dimensions. It would be fun to try to figure out how to go from a 3-D shape down to a 2-D cutout. A way to do that would be to start with a 3-D shape (use things like role-playing many-sided dice for hard-to-find shapes like tetra- and dodeca-hedrons. These can be gotten at hobby or game stores usually) and dip it in ink or paint, then carefully roll the dice side by side onto a piece of paper. You will find that some methods of rolling the dice may get all sides to the paper, but will overlap. Other methods may not work. Try to figure out if there is a method or order to exposing object sides to paper to get an accurate pattern. You don't have to dip the sides in paint, you can just mark them as visited, and trace their contour onto the paper, whatever is easiest.

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Prof. Banchoff's response