1 00:00:00,000 --> 00:00:07,720 Describe the girth of transportation since the early 1900s. 2 00:00:07,720 --> 00:00:10,040 What is mathematical about its girth? 3 00:00:10,040 --> 00:00:16,120 Hi, I'm Ardeth Williams, pilot and air traffic controller with the Federal Aviation Administration. 4 00:00:16,120 --> 00:00:18,960 Back in 1903, there was only one aircraft. 5 00:00:18,960 --> 00:00:21,400 Not much need for us to have an air traffic control system. 6 00:00:21,400 --> 00:00:27,120 However, by 1960, there were over 78,000 commercial and general aviation aircraft. 7 00:00:27,120 --> 00:00:32,600 And in 10 years, by the year 2010, we believe there will be almost 228,000. 8 00:00:32,600 --> 00:00:34,400 Air traffic is growing and growing. 9 00:00:34,400 --> 00:00:39,440 We anticipate by the year 2010, almost 1 billion people will be traveling by air. 10 00:00:39,440 --> 00:00:42,800 The year 2003 begins century number two of aviation. 11 00:00:42,800 --> 00:00:46,480 I hope in 10 years or so, you will be one of the visionaries that will ensure my safe 12 00:00:46,480 --> 00:00:52,400 and efficient flight by designing, building, maintaining, controlling, or flying the aircraft. 13 00:00:52,400 --> 00:00:55,200 The future of aviation is in your hands. 14 00:00:56,200 --> 00:00:58,520 You know, Ardeth is right. 15 00:00:58,520 --> 00:01:02,520 Mathematical concepts are everywhere and they help us explain the world we live in using 16 00:01:02,520 --> 00:01:03,520 a system of numbers. 17 00:01:03,520 --> 00:01:07,960 For example, remember when Ardeth used a bar graph to explain the growth in the number 18 00:01:07,960 --> 00:01:09,960 of airplanes since the Wright Brothers? 19 00:01:09,960 --> 00:01:11,520 Well, get this. 20 00:01:11,520 --> 00:01:16,520 We can also create a graph to show the growth of all types of transportation, from cars 21 00:01:16,520 --> 00:01:20,200 to planes to jets to future aircraft. 22 00:01:20,200 --> 00:01:21,880 Look closely at this graph. 23 00:01:21,880 --> 00:01:24,360 Can you see a pattern? 24 00:01:24,480 --> 00:01:27,880 It's like the growth of transportation are everywhere. 25 00:01:29,880 --> 00:01:31,880 You just have to look around. 26 00:01:31,880 --> 00:01:37,600 Speaking of patterns, a man by the name of Fibonacci discovered a very famous pattern 27 00:01:37,600 --> 00:01:41,160 of numbers a long time ago in Italy. 28 00:01:41,160 --> 00:01:47,640 This pattern of numbers is called the Fibonacci sequence and the ratio of certain numbers 29 00:01:47,640 --> 00:01:51,880 in this sequence is so special, it's called the golden ratio. 30 00:01:52,400 --> 00:01:55,400 Hey, how would you like to meet an expert on Fibonacci? 31 00:01:55,400 --> 00:01:57,400 He's also a poet. 32 00:01:57,400 --> 00:02:01,520 Hi everybody, this is Bud Brown talking to you from the Math Emporium at Virginia Tech 33 00:02:01,520 --> 00:02:03,120 in Blacksburg, Virginia. 34 00:02:03,120 --> 00:02:08,120 The Emporium is a large room with over 500 computers where students can come day or night 35 00:02:08,120 --> 00:02:09,800 to learn about math. 36 00:02:09,800 --> 00:02:14,720 And speaking of learning, here's a little verse I've written about a man called Fibonacci. 37 00:02:14,720 --> 00:02:16,760 How many ancestors do we have? 38 00:02:16,760 --> 00:02:22,080 That number is easily found, for we all have two parents, four grands, and eight greats. 39 00:02:22,080 --> 00:02:24,000 Just double the previous round. 40 00:02:24,000 --> 00:02:27,840 But the family tree of the honeybee is not like any other. 41 00:02:27,840 --> 00:02:30,840 The girls, good and bad, have a mom and a dad. 42 00:02:30,840 --> 00:02:33,360 But each boy has only a mother. 43 00:02:33,360 --> 00:02:38,240 It's true, each drone has a mom alone, but each female has parents too. 44 00:02:38,240 --> 00:02:43,960 In addition, you see, she has grandparents three, one fewer than me or you. 45 00:02:43,960 --> 00:02:49,280 Sakes alive, great grandparents five, that's even true for the queen. 46 00:02:49,280 --> 00:02:51,840 And next, twice great, that number is eight. 47 00:02:51,840 --> 00:02:54,880 And of thrice greats, she has 13. 48 00:02:54,880 --> 00:02:59,120 Now she's asking us, don't make a fuss, to do this calculation. 49 00:02:59,120 --> 00:03:02,920 How many ancestors does she have in every generation? 50 00:03:02,920 --> 00:03:08,280 So hop to it folks, let's crack no jokes, don't stop for meals or for slumber. 51 00:03:08,280 --> 00:03:12,680 Just work your mind, the answer you'll find is a Fibonacci number. 52 00:03:12,680 --> 00:03:17,400 Now, to help you learn more about Fibonacci numbers, here's Jennifer. 53 00:03:17,400 --> 00:03:19,560 Before we begin the student activity, let's 54 00:03:19,560 --> 00:03:23,080 learn a little more about the golden ratio and Fibonacci. 55 00:03:23,080 --> 00:03:25,360 Fib-a-who? 56 00:03:25,360 --> 00:03:29,160 Fibonacci was a 13th century Italian mathematician 57 00:03:29,160 --> 00:03:31,760 who was studying a rabbit problem. 58 00:03:31,760 --> 00:03:35,120 He wanted to know how many rabbits he would have at the end of the year 59 00:03:35,120 --> 00:03:39,800 if he started with only one pair of newborn rabbits. 60 00:03:39,800 --> 00:03:43,920 Fibonacci knew that newborns are able to breed after one month, 61 00:03:43,920 --> 00:03:47,880 then every month after, if the conditions were right. 62 00:03:47,880 --> 00:03:56,160 He found that the sequence 1, 1, 2, 3, 5, 8, 13, and so on, 63 00:03:56,160 --> 00:04:00,880 demonstrated the total number of rabbit pairs at the end of each month. 64 00:04:00,880 --> 00:04:02,680 So at the end of the first month, you have 65 00:04:02,680 --> 00:04:05,400 the original pair of newborn rabbits. 66 00:04:05,400 --> 00:04:08,200 At the end of the second month, you still have the original pair 67 00:04:08,200 --> 00:04:11,880 because it took a month for them to become old enough to breed. 68 00:04:11,880 --> 00:04:14,960 At the end of the third month, you will have two pairs of rabbits, 69 00:04:14,960 --> 00:04:17,400 the original pair and their newborn pair. 70 00:04:17,400 --> 00:04:20,400 At the end of the fourth month, you have the original pair, 71 00:04:20,400 --> 00:04:24,000 their first pair born the third month, and their newborn pair 72 00:04:24,000 --> 00:04:25,640 born the fourth month. 73 00:04:25,640 --> 00:04:31,920 Following this sequence, at the end of month 12, you'll have 144 pairs of rabbits. 74 00:04:31,920 --> 00:04:34,680 Fibonacci and others soon found this sequence 75 00:04:34,680 --> 00:04:37,560 occurring in many other things in nature. 76 00:04:37,560 --> 00:04:41,680 By counting the spirals of pine cones, pineapples, and sunflower seed heads, 77 00:04:41,680 --> 00:04:46,200 for example, you can find neighboring pairs of Fibonacci numbers. 78 00:04:46,200 --> 00:04:48,880 The way in which leaves are arranged on a stem 79 00:04:48,880 --> 00:04:51,520 also displays a Fibonacci relationship. 80 00:04:51,520 --> 00:04:54,800 So do the spirals found in seashells. 81 00:04:54,800 --> 00:04:58,640 Now, Fibonacci wasn't the only one who was fascinated with these numbers. 82 00:04:58,640 --> 00:05:02,680 The ratio obtained by successive terms in the sequence 83 00:05:02,680 --> 00:05:06,480 was thought by the ancient Egyptians and Greeks to be special. 84 00:05:06,480 --> 00:05:10,040 It was so pleasing that they used this special ratio 85 00:05:10,040 --> 00:05:14,000 to design their pyramids, their temples, and buildings. 86 00:05:14,000 --> 00:05:15,160 You know the Parthenon? 87 00:05:15,160 --> 00:05:17,640 That's a great example of what has come to be known 88 00:05:17,640 --> 00:05:20,920 as the golden ratio or golden proportion. 89 00:05:20,920 --> 00:05:22,480 Here's the Fibonacci sequence. 90 00:05:22,480 --> 00:05:27,960 Let's see if you can determine the operation used and find the next four terms. 91 00:05:27,960 --> 00:05:34,960 1, 1, 2, 3, 5, 8, 13. 92 00:05:37,920 --> 00:05:45,320 If you guessed 21, 34, 55, and 89 are the next four terms, you're right. 93 00:05:45,320 --> 00:05:46,800 How did you get it? 94 00:05:46,800 --> 00:05:49,960 The ratio of certain pairs of numbers in the Fibonacci sequence 95 00:05:49,960 --> 00:05:53,320 is used to describe things in nature. 96 00:05:53,320 --> 00:06:05,520 1 to 1, 1 to 2, 2 to 3, 3 to 5, 5 to 8, 8 to 13, 13 to 21. 97 00:06:05,520 --> 00:06:09,320 If you divide the denominator of each ratio by its numerator, 98 00:06:09,320 --> 00:06:11,760 the results look like this. 99 00:06:11,760 --> 00:06:17,120 The ratios begin to get close to the rounded number 1.62. 100 00:06:17,120 --> 00:06:20,280 What if you divide the small number in the pair by the large number? 101 00:06:20,280 --> 00:06:23,120 Well, you'll get 0.62 rounded. 102 00:06:23,120 --> 00:06:27,640 If something in nature can be described using the ratios in the Fibonacci sequence, 103 00:06:27,640 --> 00:06:29,880 well then, it's said to be golden.