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Geometric Dissections
By Solomon Golomb '51

In each of these problems, you are asked to cut the given figure into a specified number of parts congruent to each other (identical in size and shape, with turning over permitted), where the parts are also similar to the original figure (same shape, though smaller in size).

Here are two examples of figures that can be cut into four congruent parts similar to the original figure:

a) Any parallelogram:

b) Any triangle:

Cut each of the next seven figures into four congruent pieces that are similar to the original figure.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

(Figures 1, 3, and 4 are made up of six-, five-, and four-unit squares, respectively, while figure 2 is "a square and a half." Figures 5 and 7 are made up of three and six equilateral triangles, respectively, while figure 6 is "an equilateral triangle and a half.")

8. Cut this figure (half of an equilateral triangle) into three

Fig. 8
9. Cut this figure (a right triangle with legs in the ratio 2:1) into five congruent pieces similar to the original. congruent pieces similar to the original.

Fig. 9
10. Cut this figure (which can be formed from four-unit squares) into 16 congruent pieces similar to the original.

Fig. 10
Note: These are examples of figures that I studied many years ago and named "rep-tiles" because they can be dissected into replicas, and can also be used to tile the Euclidean plane.

Follow this link to solutions.

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