calculus-2014-05-21

DRAFT c○ 2014 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse
Your answer to Investigation 6 illustrates our intuition about the relationship between the area
(or volume) of the whole and the area of the pieces that make up the whole. There is a sense that
when a region or a solid is cut up into pieces and the pieces are rearranged, nothing can be gained or
lost. The next section has several examples that will challenge our intuition about dissections.
2. Equidecompositions
The tangram examples in Section 1 illustrate one of the two important concepts we will be using
in this chapter. We say that two sets, X and Y are equidecomposable if we can cut both X and
Y into the same number of non-overlapping finite pieces such each piece of X is congruent to exactly
one piece of Y . The term congruent means that the two pieces are identical in shape and size and
that we can transform one piece into the other by only using some combinations of the following rigid
motions:
1. A translation; i.e shifting the entire piece a certain distance in a specific direction as shown
in Figure 7.4.
Figure 7.4. A Translation
2. A rotation; i.e rotating the entire piece through a specific angle as shown in Figure 7.5.
3. A reflection; i.e flipping the entire piece about a line or a point as shown in Figure 7.6.
7. We denote the natural numbers = {1, 2, 3, 4, . . . } by the symbol N. Let M = {1, 2, 3, 4, 6, 7, 8, 9, . . . }
(the natural numbers minis 5), S = {4, 5, 6, 7, 8, 9, . . . }, T = {2, 4, 6, 8, . . . }, U = {−2, −1, 0, 1, 2, 3, . . . }
and V = {. . . , −4, −3, −2, −1}. To which of the sets M, S, T, U and V (if any) is N congruent?
Explain your reasoning.
8. Do any of your answers to Investigation 7 surprise you? Explain.
9. The other important concept we will need in this chapter is called shifting to infinity. Use your
answers to Investigation 7 to explain what this means.
Include something about Hilbert’s Hotel here?
77