Homework 4

Notes

• In order to get full credit in the open ended questions, you need to spend some time reflecting upon and writing your answers. In most of these problems, the probability that a one-sentence answer will get full credit is close to 0 (and so will be what you learn from the exercise)
• Make sure you show all your work on the problems that require it so. Otherwise, even if you give a correct answer, if you do not explain how you obtained it, you'll get very little or no credit.
• It would be great if you discussed ideas with your classmates. The write-up, however, must be done individually.
• Recall that the slides of the lectures can be found here.

Suggestion: Play Euclidea!

Problems

1. In each of the next two questions below, determine the side length of the square, explain how you obtained the solution and provide a drawing with measurements marked.
   1. Divide an isosceles triangle with base of length 8 and height of length 9 into polygonal pieces from which a square can be assembled.
   2. Consider a right triangle T whose legs have length 2 and 25. Divide T into polygonal pieces from which a square can be assembled. (Hint: First break the triangle into pieces that can be reassembled into a rectangle)
   You can use cm or any other unit of our choice) "Make" your own solutions in paper or using Geogebra. Explain how you obtained the solution and provide photos of the pieces you made, on paper or in Geogebra. (If you complete this successfully, you will have produced quadratures of an isosceles and a right triangle!) Extra credit challenges: Consider two positive numbers a and b.
   1. Explain how to divide an isosceles triangle with base of length a and height of length b into polygonal pieces from which a square can be assembled.
   2. Consider a right triangle T whose legs have length a and b. Explain how to divide T into polygonal pieces from which a square can be assembled. (Hint: First break the triangle into pieces that can be reassembled into a rectangle)
   3. How would you proceed to divide any triangle into polygonal pieces from which a square can be assembled.
2. In the proof of "Proposition IX.20: Prime numbers are more than any assigned multitude of prime numbers", given three primes a new one is defined. Find four concrete examples of this procedure, assuming that one of the given primes is 41 and the other one is 2. (You need to choose the third prime).
3. You are given three points on a line, A, B and C, so that B is between A and C and the length of BC is 1 (in some unit). Describe how to construct a segment of length square root of (the length of) AB, using straightedge and compass. Implement your construction in Geogebra. and include a screenshot of your work in Geogebra. (Hint: the magic words here are "geometric mean")
4. Is it possible to construct a square with the same area as a given circle, only using straight-edge and compass? Why or why not?
5. Euclid's Elements is the earliest extant example of axiomatic mathematics. Describe its structure and how the axiomatic method depends on such a structure. (One or two paragraphs will suffice.)
6. Recall that in class we discussed the three impossible problems of antiquity.
   1. Explain which of these problems, in your opinion, is the hardest to tackle and why.
   2. One of these problems, the trisection of the angle is equivalent to constructing (with compass and straightedge) a segment whose length is the solution of a certain cubic equation. The other two problems are equivalent to constructing two numbers. Explain what these two numbers are and how they relate to the corresponding problem.
7. Explain in your own words the statement and proof of Proposition I.1 of Euclid's Elements. (You can start by "Given a segment, draw a circle with center one of its endpoints") Explain whether there is any gap in the proof, from a modern point of view.