Mathematics 320: Project 1 Due: In class February 14.

Directions. Each group will submit answers to the following set of questions. Each question should be answered in complete sentences unless otherwise indicated. For example, when the answer is expected to be a graph. Projects that are mathematically accurate, clearly articulated, and most importantly contain insights that lead to increased understanding of the concept being investigated will be awarded the most points.

(1) The Fibonacci Sequence. How many pairs of rabbits can be bred from one pair in a year? A man has one pair of rabbits at a certain place entirely surrounded by a wall. We wish to know how many pairs will be bred from it in one year, if the nature of these rabbits is such that they breed every month one other pair and begin to breed in the second month after their birth. ... Liber abaci (1202) Leonardo Pisano, better known by his nickname Fibonacci, proposed the above problem in his book Liber adaci in 1202. Leonardo’s result is the well known Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34,.... In addition to counting the offspring of rabbits, the Fibonacci sequence arises in many other places in nature such as in the number of petals on a flower, and the number of rows on a pine cone, pineapple, or sunflower. (a) Find the next four terms of the Fibonacci sequence. What is the rule or formula for obtaining the successive terms of the Fibonacci sequence? (b) Calculate the sum of the first n Fibonacci numbers, 1+1=2, 1+1+2=4, 1+1+2+3=7,... up to 1+1+2+...+34. Discover a pattern by relating each sum you obtain with one of the Fibonacci numbers. Then complete the following statement The sum of the first n Fibonacci numbers is equal to ...... (2) The Golden Ratio. If you were going to design a rectangular TV screen or swimming pool, would one shape be more pleasing to the eye than another? Since the early Greeks a rectangle with a ratio of length to width that satisfies the equation l l + w = , w l has been considered the most visually appealing. This ratio, called the Golden Ratio, not only appears in art and architecture, but also in natural structures. The Golden ratio is usually denoted by the Greek letter Φ, and a rectangle with a ratio of length to width of Φ is called a Golden Rectangle. (a) Compute the exact value of Φ. Try setting the width equal to one in the equation above so that the length is equal to Φ. You will need the quadratic equation to solve for l, that is, √ −b ± b2 − 4ac l = . 2a This will yield two answers for l, and you will have to figure out which one to use. (b) Measure the two sides of a computer screen, a legal pad, a door, and the hood of a car. Calculate the ratio of the longer side to the shorter side to 3 decimal places. How close is your average to Φ? Your answer should be in terms of a percentage of Φ. (c) Draw a Golden rectangle with height one inch and length Φ inches. Now, divide your rectangle into two pieces with the first a square of length one inch and the second, a smaller triangle, with length one and width denoted by R2. Computer the ratio of the length to width of the smaller triangle. Repeat the process by dividing R2 into a square and smaller rectangle R3. Again compute the ratio of the length to width of R3. What do you discover? Continue dividing the smaller rectangle into a square and an even smaller rectangle until it becomes to difficult to draw. Now connect the opposite corners of the successive squares with an arc to form a spiral (you will need to orient the squares in a particular manner). This is a spiral of the nautilus shell, which is sometimes called a Fibonacci spiral. The three oldest Giza pyramids in Egypt lie on a Fibonacci spiral. The next question will address why this spiral is denoted by the name Fibonacci.

1 (3) Connecting the Fibonacci Sequence and the Golden Ratio. (a) Calculate to three decimal places the ratios 1/1, 2/1, 3/2, 5/3, 8/5. Do the same thing for the next six fractions in this sequence (using a calculator). What decimal value do these fractions appear to be approaching? Draw a graph on the xy- where x denotes the order in which the ratio appears, and y is the value of the ratio. For example, when y = 5/3, x = 4. Explain what is happening your graph. (b) How can you use this discovery to draw a rectangle that, for practical purposes, is very close to a golden rectangle? (4) Taxicab . In , the geometry in our every day experience, the between two points (x1, y1) and (x2, y2) is given by the well-know formula p 2 2 dE(x, y) = (x1 − x2) + (y1 − y2) . Taxicab geometry differs from Euclidean geometry by how we compute the distance between two points. The distance formula for the taxicab geometry between points (x1, y1) and (x2, y2) and is given by:

dT (x, y) = |x1 − x2| + |y1 − y2|. (a) Let A = (−2, −1). Plot A on a graph. For each point P below calculate the taxicab distance between A and P , and plot P for P = (1, −1), P = (−2, −4), P = (−1, −3), P = (0, −2), P = (1, −1), P = (0, 0), P = (−3, 1), P = (−4, 0), P = (−5, −1), P = (−4, −2), P = (−3, −3) and P = (−2, 2). (b) Connect the points P in your previous graph. This is a taxicab centered at A with 3. Convince yourself that each point on the outline you drew is a taxicab distance 3 away from point A. Does is make sense to call this a circle? How does a taxicab circle compare with a circle in Euclidean geometry? Would the definitions be the same? Explain. (c) The Taxicab geometry is a better mathematical model for urban geography than Euclidean geometry. Assume all the streets run straight north and south or straight east and west, streets have no width, and buildings are point size; etc. We denote this city, with these assumptions, Ideal City. Indicate the answers for each of the following questions on its own appropriately sized graph. (i) Alice and Bruno are looking for an apartment in Ideal City. Alice works as an acrobat at amusement park A = (−3, −1). Bruno works as a bread taster in bakery B = (3, 3). Being ecologically aware, they walk wherever they go. They have decided their apartment should be located so that the distance Alice has to walk to work plus the distance Bruno has to walk to work is as small as possible. Where could they look for an apartment? (ii) The dispatcher for the Ideal City police department receives a report of a car accident at X = (−1, 4). There are two police cars in the area, C = (2, 1) and D = (−1, 1). Which car should she send to the scene of the accident? (iii) There are three high schools in Ideal City: Fillmore at (−4, 3), Grant at (2, 1), and Harding at (1, 5). Draw in school district boundary lines so that each student in Ideal City attends the high school nearest his home. (iv) If Burger Baron wants to open a hamburger stand equally distant from each of the three high schools, where should it be located? (v) A fourth high school, Polk High, has just been built at (2, 5). Redraw the school district boundary lines (d) Give another application of taxicab geometry to a real world situation that is different than modelling a city. You must reasonably justify your answer. (5) What is Mathematics? A 5th grade student asks you to explain the following definition of mathematics given in the Concise Oxford Dictionary: The abstract science of number, quantity, and space studied in its own right (pure math- ematics), or as applied to other disciplines such as physics, engineering, etc. (applied mathematics). Answer this question by explaining what the terms “abstract science,” “number,” “quantity,” and “space” mean in the context of mathematics. Furthermore, explain the difference between pure and applied mathematics. Your explanation should contain examples a 5th grader would understand. Would your answer change if a NASA physicist asked the same question? What if a musician asked? Explain. Department of Mathematics, Kansas State University E-mail address: [email protected]