Homework II (Geometric Structures)

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451



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Suppose one is given a plane convex quadrilateral.

Such a polygon has 4 sides and 4 angles.

A partition of a positive integer n (2 or more) is a way writing n as a sum of positive integers. For example, partitions of 5 include: {4,1}, {5}, {2, 1, 1, 1}. Note that we do not care about the order in which we write n as the sum of these integers.

1. Write down all of the partitions of 4. How many partitions are there?

2. We can classify convex quadrilaterals, based on the partitions that exist for the number of angles and sides of the quadrilateral.

For example for sides we might have the partition {2, 2} and for angles the partition {4}. This can be interpreted as having a quadrilateral with two pairs of sides with equal lengths, and all four angles being of equal measure. We can now ask if there is a convex quadrilateral with this pair of partitions.

3. Based on the partitions of 4 for the angles and sides of convex quadrilateral how many different types of such quadrilaterals might there be?

4. How many of the types of quadrilaterals that might exist in Question 3 actually exist?

5. Is this classification fine enough to actually enumerate all the kinds of convex quadrilaterals? Explain.

6. What properties of quadrilaterals does the classification system which we just looked at not address? How might one try to extend the partition approach to these quadrilaterals?

7. How many edges does a tetrahedron have? Write down the partitions of this number.

8. How many different types of tetrahedra are there based on the partitions of the number edges of the tetrahedron and which address the fact that there might be "inequivalent" tetrahedra with the same partition for the number of edges? (Again, the interpretation of a partition is that there are edges of equal length corresponding to the numbers in the partition. For example, a "combinatorial" cube (a truncated pyramid is an example) has 12 edges, and the partition {10, 2} would refer to having 10 edges of one length and two ends of a different length. However, these two edges might be in different places on the combinatorial cube: two adjacent edges on one face, two parallel edges , etc. There might or might not be an actual combinatorial cube which would be consistent with these requirements.)

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