1. Exercise 1.4.2.

2. Use the methods of Section 1.3 to find a parametrization by rational functions for the hyperbola defined by
x²-2y²=1.

3. True or False? Every integer N>2 occurs in some Pythagorean Triple. (Justify your answer.)

1. Use reductio ad absurdum to prove that √3 is irrational.

2. Give a geometric proof that √3 is irrational. (Hint: It might be easier to show that 1+√3 is irrational, then deduce that √3 is also irrational.)

3. Use the Euclidean algorithm to show that gcd(270,168) = 6.

1. What is the height of a regular tetrahedron of side length 1?

2. Exercise 2.2.2 from the text.

3. Let P be a polyhedron. Suppose F₁, …, F_k are the faces of P that meet at vertex V, and that A₁, …, A_k are the angles of these faces at V. Define the defect at vertex V to be (360-(sum of the angles A_i)). (For example, in a cube there are three squares meeting at any vertex, so the defect at any vertex is (360-(90+90+90)) = 90 degrees.) The total defect if P is the sum of the defects at all of the vertices of P. Exercise: find the total defect of each of the Platonic solids.

1. Verify the correctness of the construction of the regular pentagon indicated above.

2. Using the result of Exercise 2.3.4 of the text, briefly outline a second construction of the regular pentagon.

3. Briefly explain how to construct a regular 15-gon.

2. Show that if a polygon is constructible, then its area is a constructible number. (Hint: start with triangles.)

3. Show that if a regular polygon of circumradius 1 has constructible area, then it is possible to construct a copy of the polygon. (The circumradius is the radius of the circumscribing circle.)

1. Above is an animated gif I found on the web page of Takaya Iwamoto, which indicates Descartes' method of trisecting angles using the parabola y=x². You are to show that the method is correct.

You start with two curves drawn in the plane: the parabola y=x² (yellow) and the circle of radius 2 centered at the origin (white). You draw the angle you want to trisect (blue) and find the point A where it intersects the white circle. You construct the point C whose x-coordinate is half the x-coordinate of A and whose y-coordinate is the same as the top point of the white circle. Draw a circle (green) centered at C and passing through the origin. If P is an intersection point of the parabola and the green circle, then the vertical line through P intersects the white circle at an angle 1/3 of your original. (Show this!)

It is easy to explain geometrically why an arbitrary angle can be bisected with straightedge and compass. In the next two problems you will be asked to explain algebraically why an arbitrary angle can be bisected (and 4-sected) with straightedge and compass.

2. Derive a monic quadratic equation q(x)=0 whose coefficients are rational combinations of 2cos(α) and rational numbers, which has x=2cos(α/2) as a root.

3. Derive a monic quartic equation whose coefficients are rational combinations of 2cos(α) and rational numbers, which has x=2cos(α/4) as a root. Show that the real roots of this quartic can be expressed using only +, -, 0, *, ^-1, 1 and square roots of positive real numbers.

1. We work in the Poincare unit disk model of the hyperbolic plane. Let P be the point with coordinates (0,1/2). Your task is to find two distinct hyperbolic lines through P that are both parallel to the hyperbolic line lying on the x-axis. Your hyperbolic lines will have to be circular arcs, so express your answer by describing the circles involved.
2. We work in the Poincare unit disk model of the hyperbolic plane. Suppose that A, B and C lie on a line with B between A and C. Show that d(A,C)=d(A,B)+d(B,C).
3. Find the coordinates of the point in the Poincare unit disk model that is halfway between the Euclidean points (0,0) and (c,0) in the sense of the hyperbolic metric.