MATH 350. Assigned readings and homework All numbered readings are from Stillwell s The four pillars of geometry. Reading Jan 30, Feb 1: Chapters 1.1 1.4 Feb 6, 8: Chapters 1.5 2.2 Feb 13, 15: Chapters 2.3 2.6 Feb 20, 22: Chapters 2.7 2.9 Feb 22: Midterm 1 Feb 27, Mar 1: Chapters 3.1 3.4 Mar 6, 8: Chapters 3.5 3.6 Mar 13: Chapters 3.7 Mar 20, 22: Spring Break Mar 27, 29: Chapters 3.7 4.2 Apr 3, 5: Chapters 4.3 4.5 Apr 10: Chapter 4.6 Apr 12: Midterm 2 Apr 17, 19: Chapters 5.1 5.5 Apr 24, 26: Chapters 5.6 5.9 May 1-15: No reading we ll be covering material not in the book. 1
Homework HW 1 (due Thursday, Feb. 1) Group homework. (In the groups formed in class. For this part, turn in only one writeup per group.) 1. On the first day of class, how did you construct a square with the tools available to you? 2. Is it a perfect square? Why or why not? 3. What advantages or drawbacks does your construction have? Individual homework. In each problem: Actually carry out the construction using a straightedge and a compass. Explain step by step how you are carrying out the construction. Explain why your construction is correct. I am looking for a convincing explanation that a(n interested and critical) random passerby on the street will agree with, not a formal proof. 1. Construct the center of a given equilateral triangle. 2. Construct a perfect square. 3. Construct a regular octagon. 4. Given three segments of lengths a, b, c, construct a triangle whose sidelengths are a, b, c. When is this possible? 2
HW 2 (due Thursday, Feb. 8) Group homework. 1. To be announced. Individual homework. In each problem, explain and justify your answer clearly. 1. Construct an isosceles triangle ABC with AB = BC such that angle ABC equals a given angle and AC equals a given length. 2. Draw a segment that is 3/5 of the the length of your left thumb. 3. Let l and m be two line segments that do not intersect. Let A, B, C be points on line l (with B between A and C) and D, E, F be points on line m (with E between D and F ) such that the lines AD, BE, and CF are parallel. Prove that AB BC = DE EF. 4. If ABCDEF is a regular hexagon of side length 1, find the length of AC. 5. If a quadrilateral has all sides of equal length and one angle equal to 90, prove that it is a square. 6. (Bonus.) If a regular heptagon has sidelengths equal to s and diagonals of lengths p < q, prove that s = pq p + q. 7. (Bonus.) Prove that sin 9 = 1 8 2 10 + 2 5. 4 3
HW 3 (due Thursday, Feb. 15) Group homework. (In the groups formed in class. For this part, turn in only one writeup per group.) 1. Estimate the area of California. 2. Go to the main entrance of the Business building that faces the Quad. If you stand there and look towards the Quad, there is a tall tree almost directly in front of you, slightly to your right. How tall is it? Individual homework. In each problem, justify your answer clearly. 1. Find the sum of the angles of a convex polygon with n sides. 2. Let A, B be points on the same side of line l, and let X, Y be the feet of the perpendiculars from A and B to l, respectively Let C be the intersection point of AY and BX, and let Z be the foot of the perpendicular from C to l. Prove that 1 CZ = 1 AX + 1 BY. 3. Give a (three-dimensional) geometric proof of the identity a 3 b 3 = (a b)(a 2 + ab + b 2 ). 4. Let ABC be an equilateral triangle of area 1. Let P, Q, R be points on the segments BC, CA, AB, respectively, such that BP P C = CQ QA = AR RB = 2. (a) Prove that triangles AQR, BRP, and CP Q have area 2/9. (b) Conclude that triangle P QR has area 1/3. 5. Let ABC be a right triangle with side lengths a, b, c, respectively, and C = 90. Consider the points P and Q on line AB such that BP = BQ = a, where P and Q are, respectively, inside and outside of the segment AB. (a) Prove that P CQ = 90. 4
(b) Prove that AP C ACQ. (c) Use (b) to prove that a 2 + b 2 = c 2. Bonus. Let ABCD be a square of area 1 and let P, Q, R, S be the midpoints of segments AB, BC, CD, DA, respectively. Find the area of the square cut out by the lines AR, BS, CP, DQ. 5
HW 4 (due Thursday, Feb. 22) 1. Construct a triangle whose angles are 90, 45 and 45. 2. Solve Exercise 1.4.3 in the book. 3. If ABCD is a rectangle, prove that AB = CD. 4. If ABC be a triangle and let D, E, F be the midpoints of sides BC, CA, AB, respectively. Prove that area( DEF ) = 1 4 area( ABC). 5. Let A, B, C be points on a circle with center O, such that A and O are on the same side of line BC, and O lies outside of triangle BAC. Prove that BOC = 2 BAC. 6
HW 5 (due Thursday, Mar. 8) Group homework. (In the groups formed in class. For this part, turn in only one writeup per group.) Answer the question you were assigned in class: 1. Compute the intersection of two given lines. When do they intersect? 2. Compute the intersection of a given line and a given circle. When do they intersect in 2 points? 1 point? 0 points? 3. Compute the intersection of two given circles. When do they intersect in 2 points? 1 point? 0 points? The answer to the second part of your question should be given in terms of the parameters (a, b, c, r,...) defining the lines and circles. Individual homework. In each problem, justify your answer clearly. 1. Find the coordinates of the midpoint of the segment connecting points (x 1, y 1 ) and (x 2, y 2 ). 2. Find the coordinates of the center of the circle passing through the points (p 1, q 1 ), (p 2, q 2 ), and (p 3, q 3 ). What happens if these three points are collinear? 3. Find the points of intersection of the circles (x 2) 2 + (y + 1) 2 = 4 and x 2 + y 2 = 1. 4. Let A = (0, 0), B = (3, 0), and C = (2, 8). Find a point P such that AP is perpendicular to BC, BP is perpendicular to AC, and CP is perpendicular to AB. Does it surprise you that such a point exists? Bonus. Let ABC be a triangle and let D, E, and F be the midpoints of sides BC, CA, and AB, respectively. Prove that 4( AD 2 + BE 2 + CF 2 ) = 3( AB 2 + BC 2 + CA 2 ). 7
HW 6 (due Thursday, Apr. 5) 1. Draw a nice, large, general picture of two congruent triangles S And T. (To make them general, please make sure they are not rightangled, not equilateral, not isosceles). Show one, two, or three reflections taking S to T. 2. Consider the function f : R R given by f(x, y) = (2 y, 2 x). (a) Prove that f is an isometry. (b) Draw the triangle with vertices A = (1, 2), B = (3, 1), C = (3, 2), and the triangle with vertices f(a), f(b), f(c). (c) Is f a rotation, a translation, or a glide reflection? Explain your answer. 3. If l and m are parallel lines in the plane, and r l and r m are the reflections across l and m, prove that the composition r l r m is a translation. 4. If l and m are lines in the plane that are not parallel, prove that the composition r l r m is a rotation. Where is the center of rotation? What is the angle of rotation? Bonus 1. If l, m, n are lines in the plane, prove that r l r m r n is a glide reflection. 5. This problem concerns the isometries of the plane. (a) Prove that the composition of two isometries is an isometry. (b) Prove that every isometry is bijective, and hence has an inverse function. (c) Prove that the inverse of an isometry is an isometry. (d) (Bonus 2.) Prove that the set of isometries of the plane form a group under composition. Bonus 3. Prove that any isometry maps a dinosaur to a dinosaur. 8
HW 7 (due Tuesday, Apr. 10) 1. If the vertices of a square are a, b, c, d, prove that the center of the square is 1 4 (a + b + c + d), 2. Consider the triangle formed by vectors 0, v and w. Let x = 1 3 v and y = 1 3w. Let m be the midpoint of the segment from v to w. Let p = 1 2 m. (a) Draw a diagram illustrating these vectors. (b) Prove that p, v and y are collinear. (c) Prove that p, w and x are collinear. 9
MiniHW 8 (due Thursday, Apr. 19) 1. Draw your initials in one-point perspective and in two-point perspective. Feel free to get creative! 2. (Bonus.) In the square shown below, let A, B, C, and D be the midpoints of the sides. Find the coordiates (a) Prove that the shaded region is a square. (b) Find the coordinates of E and F. (c) Find the area of the shaded region. W=(0,1) A X=(1,1) E D F B Z=(0,0) C Y=(1,0) 3. (Bonus.) Is it true that, in a 2-point perspective drawing of a floor with a square tiling, the tiles closer to the horizon look more square? 10
HW 9 (due Thursday, Apr. 26) 1. (a) Make a Möbius band by (taping/gluing/sewing/knitting/baking/...) together the top and bottom side of a rectangle in opposite orientations. (b) Make a torus by (taping/gluing/sewing/knitting/baking/...) together each pair of opposite sides of a rectangle in the same orientation. (c) (Bonus.) Draw/sew/bake/... the Pac-Man board on your torus. 2. (a) Draw two intersecting planes in R 3, which you will consider screen 1 and screen 2. Draw a point outside of these planes, which you will consider the projector. Draw your initial on screen 1, and show how it projects onto screen 2. (b) (Bonus). Do the same experiment in three dimensions: cut your initial out of a piece of cardboard ( screen 1 ), and use a flashlight to project it onto the wall ( screen 2 ). Move the light and the cardboard around to see how the image changes. E-mail me a video or some photos documenting your experiment. 3. Prove that the composition of two linear fractional functions is also a linear fractional function. 4. Prove that the inverse function of a linear fractional function is also a linear fractional function. 5. Express the function f(x) = 2x 3 3x 2 as a composition of functions of the form a k (x) = kx (k 0), b(x) = 1 x, and c l (x) = x + l. Bonus 1. Find and discuss interesting examples of drawing in perspective. As we already discussed in class, and is often discussed in this context, there are many examples throughout the Western European tradition. Those are welcome, but I invite you to dig deeper and look for other examples. (Murals? Street signs? Computer games? Record covers?) Explore whether your examples are accurately or inaccurately drawn in perspective. 11
Bonus 2. (This problem is highly recommended.) Draw a big picture of two lines l 1 and l 2 ( screens ) intersecting at a point P ( projector ), and 5 evenly spaced points on the first line l 1, called 2, 1, 0, 1, 2. Let P project their images to the points f( 2), f( 1), f(0), f(1), f(2) on the second line l 2. Choose a point 0 and a positive direction for a number line on l 2, using the same units that you chose for l 1. Measure the points f( 2), f( 1), f(0), f(1), f(2) on that number line. What is the linear fractional transformation f(t) taking i to f(i) in your picture? How many measurements f(i) (for i = 2, 1, 0, 1, 2) do you need to compute f(t)? Make sure that the function you compute (at least approximately) matches all 5 measured values of f(t). 12
HW 10 (due Thursday, May 3) 1. Carefully draw a 3 3 square grid in two-point perspective. By measuring them accurately, verify that the cross-ratio of the four points on each side of the square is 4/3. 2. To draw a tiled floor in one-point perspective, people sometimes make the width of each row of tiles a constant fraction c of the one before. Is this accurate? 3. Consider a linear fractional function f such that f(0) = 1, f(1) = 0, f(2) = 2. Find f( 1). Find f( ). Find the point x such that f(x) =. 4. (Worth 2 problems.) Using the photo on the next page, estimate: How far from the end line was the ball when Roberto Carlos scored his legendary 1 free kick against France on June 3, 1997? 5. (Bonus - Worth 2 problems.) Do you think Leonardo da Vinci painted The Last Supper using accurate perspective? Please provide a thorough and clear explanation. 1 Que why legendary, you ask? Watch: https://www.youtube.com/watch?v=itzwynwuonw 13
HW 11 (due Tuesday, May 15) 1. Using the information we learned about them in class the values of V, E, F, n, k draw the five platonic solids. You may want to draw the squished version where the whole polyhedron is projected inside one of its faces. (Of course these pictures are on the internet, books, etc. Please don t look them up.) 2. (a) Does there exist a polyhedron with 6 vertices, 9 edges, and 5 faces? If so, draw one. If not, prove it. (b) Does there exist a polyhedron with 7 vertices, 13 edges, and 7 faces? If so, draw one. If not, prove it. [Bonus.] Prove that any polyhedron must have a face having fewer than 6 sides (a triangle, a quadrilateral, or a pentagon). 14
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