Arc Length and Surface Area Calculus Techniques Meet History

1. Arc Length and Surface AreaCalculus Techniques Meet History David W. Stephens The Bryn Mawr School Baltimore, MD NCTM – Baltimore 2004 15 October 2004

2. Contact Information Email: stephensd@brynmawrschool.org The post office mailing address is: David W. Stephens 109 W. Melrose Avenue Baltimore, MD 21210 410-323-8800 The PowerPoint slides will be available on my school website: http://207.239.98.140/UpperSchool/math/stephensd/StephensFirstPage.htm

3. Why is Arclength a Fascinating Topic? • This is a late topic in BC Calculus. • The seniors are getting near the end of their high school years … and the AP exam is on the doorstep. • Calculus is a great capstone course in high school, because it brings together all of the mathematics that the students have previously learned.

4. How is Arclength a Fascinating Topic? • Calculus students already know about arclength on a circle from their geometry class • They understand radians (although perhaps they still struggle with the importance of radians) … and radians are crucial for calculus. • It is valuable to tie in new methods to ones they already know. Calculus topics often lend themselves to doing this.

5. Calculus Strategies: Integration • The definite integral is an accumulation of products … that is the sum of products of two quantities, so definite integrals can be thought of as measurements of areas.

6. Calculus Strategies: Integration • In any application of integration (such as areas under a curve, volumes, arclength, work, distances, or total costs), there is a three step strategy: • Cut the <area, volume, arclength, work, etc> into small pieces. • Code the quantity to be measured on a representative small piece, because we understand the geometry of the small parts. • Recombine the parts (with sums / definite integrals).

7. Calculus Strategies: Integration • Step 1 (Cut the desired result into small pieces.)

8. Calculus Strategies: Integration • Step 2 (Code the quantity to be measured on a representative small piece) • It looks like this: • dA = y dx • The width (x) is cut into infinitesimally small parts, and the height (y) depends on the function under which the area is to be measured.

9. Calculus Strategies: Integration • Step 3 (Recombine the parts with sums / definite integrals ) • dA = y dx • Adding up all of these simpler parts becomes 

10. A Whirlwind Histo-Mathematical TourHow Do We Calculate Length? 300 BC Euclidean Geometry Euclid(325 – 265 BC, probably at Alexandria, Egypt) The subject of plane geometry was known as far back as 2000 BC – 2500 BC. Perhaps the Chinese and other Asian cultures knew this information independently at about this same time as well. Distance is measured with a straightedge. History of mathematics available at: http://www-gap.dcs.st-and.ac.uk/~history/BiogIndex.html

11. A Whirlwind Histo-Mathematical TourHow Do We Calculate Length? 1629 to 1640’s Cartesian coordinates Rene Descartes (France 1596 – 1650) Points were located with numbers, “marrying” geometry and algebra. Fermat knew these results in about 1629 as well. Length is now calculated, rather than measured.

12. A Whirlwind Histo-Mathematical TourHow Do We Calculate Length? 1660-1670 Integral Calculus Isaac Newton (1643 – 1727, England) and Gottfried Leibniz (1646 – 1716, Germany) Ideas of cutting a length into small pieces and measuring the small pieces with plane geometry methods and then recombining the pieces was a new strategy. (Details to be shown later.)

13. A Whirlwind Histo-Mathematical TourHow Do We Calculate Length? 1920 – 1945 Measurement of Coastlines Lewis F. Richardson (England 1881 - 1953) Richardson investigated to find out that the reported length of coastlines in Europe (and he is known especially for a discussion of the coastline of England) varied by as much as 20%.

14. A Whirlwind Histo-Mathematical TourHow Do We Calculate Length? 1975 Fractals Benoit Mandelbrot (Poland 1924 - ) (His family was Lithuanian Jewish. He now resides in the USA.) Methods were developed to look at the similarity of small pieces of a line or surface to the whole line or surface. Measurements (and the accumulation of parts of the measurements) seemed to depend on the scale of the measurement tool.

15. A Whirlwind Histo-Mathematical TourHow Do We Calculate Length? 300 AD Theorems of Pappus Pappus ( 290-350 AD, Alexandria, Egypt) Pappus stated two useful theorems, long before the methods of calculus were in existence, which help to calculate volume and surface area. In uncanny ways, these ancient theorems are verified by the much newer methods of the integral calculus and the fractals.

16. A Whirlwind Histo-Mathematical TourHow Do We Calculate Length? 1980’s Gaussian Quadrature Texas Instruments Calculator Algorithm The method for performing the numerical integration fnInt is a fast, usually accurate, but complicated and fascinating algorithm. (This is a method used for any integration, not just for calculating length, but it has a connection to the other methods.)

17. Arclength Meets History • Here is how a class might proceed, building up the ideas for calculus in a historical-mathematical way. • This discussion will proceed as if all of you are not actually familiar with the calculus topic of arclength.

18. What is an “arc”? arc – Middle English word derived from Latin arcus meaning bow, as in bow-and-arrow, and, later, arch or curve. In his 1551 Pathway to Knowledge, Recorde used arche, arche lyne (also spelled archline), and bowe lyne (also spelled bowline) for the arc of a circle. Billingsley uses the word arke in his 1570 translation of Euclid’s Elements. This is from Historical Modules for the teaching and Learning of Secondary Mathematics (December 2002, Mathematical Association of America). This definition comes from “Lengths, Areas and Volumes” (page 193).

19. What do little pieces of most functions look like? • Most functions have a curve to them, so the question of the length of an arc amounts to calculating the length of a piece of a function. • Calculus students have been well trained to say that little pieces of most functions look like …. • line segments, because functions are usually locally linear.

20. Setting up Arclength • So calculating the length of a curve comes down to methods to measure the length of a line segment. • Cut the <area, volume, arclength, work, etc> into small pieces. • Code the quantity to be measured on a representative small piece, because we understand the geometry of the small parts. • Recombine the parts (with sums / definite integrals).

21. Setting up Arclength :Now we follow the history … • Pythagoras (569 – 475 BC, Samos, Ionia) • No coordinates available

22. Setting up Arclength:Use Pythagoras in a calculus class • We want to know the length of y = x2 on the interval [0 , 4]. • We do this in four pieces to begin.

23. Setting up Arclength:Use Pythagoras in a calculus class Here are the four triangles whose hypotenuses are straight, under the assumption that curves are locally linear. So s = about 16.747

24. Setting up Arclength:Use Pythagoras in a calculus class • Notice that we have • (1) cut the curve into small pieces even though Pythagoras would not have understood the idea of a function with coordinates, • (2) used the geometry of Pythagoras to calculate the lengths of the four pieces, and • (3) recombined with addition. No calculus was used, but the ideas of calculus were employed.

25. Setting up Arclength:Add Descartes to the question • For each of the triangles, coordinates are used to locate the points on the function, and the distance formula is that of Pythagoras with adaptations for the coordinates.

26. Setting up Arclength:Add Descartes to the question The coordinates of the points are (4, 16) , (3 , 9) , (2 , 4), ( 1, 1), and (0, 0)

27. Setting up Arclength:Add Descartes to the question • Using the distance formula on each of the triangles gives the same results as before.

28. Setting up Arclength:A Detour to the 20th Century • Calculus students accept the idea of local linearity fairly easily, even though it is a novel idea at first. • To challenge their acceptance of this idea (and recall it is late in the senior year at the end of a long and challenging AP course), let’s move to Richardson and Mandelbrot … and the coastline of England (and other places).

29. Length of a Coastline From Chaos by James Gleick (Penguin Books 1987, page 95)

30. Length of a Coastline • Some of the stories told about the measurement of coastlines include the importance of knowing the length of the coastlines of England and Norway during World War II, so that the navies knew how long a coastline they needed to defend. • Later it became a fascinating mathematical topic.

31. Length of a Coastline • We can actually do some measurements now to see how this paradox of Lewis Richardson goes. • We will simulate this with the maps of Jaggedland and Smootherland • We can measure with different “smallest” units available.

32. Length of a Coastline • Use a 3 inch straightedge. • Start at some point on the map. • Swing the 3 inch straightedge until it first hits another point on the map. • Move the end of the 3 inch straightedge until it is at the last endpoint • Count how many 3 inch measurements you can make, continuing until you are back at the starting point.

33. Length of a Coastline • Use a 1 inch straightedge. • Do the same process as above. • Use a ½ inch straightedge. • Do the same process as above. • Use the scale on the map to convert the total number of inches to miles.

34. Length of a Coastline Use actual maps of Florida, Norway, England, the Chesapeake Bay, and the Mississippi River in classes. A Student Worksheet Observations:

35. Length of a Coastline • Actual mileages …whatever “actual” means(since we are now skeptical about whether there is a real answer ????) • Florida ….1,350 miles • England … 5,581miles (6261 including islands) (11,072 miles for Great Britain, 19491 including islands) • Norway … • Chesapeake Bay … 11,864 miles of shoreline • Mississippi River … 2,350 to 2,552 miles • (depending on who you ask)

36. Length of a Coastline • What seems to be the results and connections? • As the measuring tool gets shorter, the total length gets longer, but not always! • What measurement tool does a geological survey use? Why? • Actual length seems to be the result of practical methods, but they are not definite answers.

37. Length of a Coastline • Small pieces on the maps are measured as the Greeks would have done it (!!), and the Pythagorean theorem could have been used to calculate from the vertical and horizontal. • Old meets new. • Mathematics is still evolving and new methods and ideas are still being added. • It is OKAY to combine new and old ideas!

38. Length of a Coastline Coastline Paradox Determining the length of a country's coastline is not as simple as it first appears, as first considered by L. F. Richardson (1881-1953). In fact, the answer depends on the length of the ruler you use for the measurements. A shorter ruler measures more of the sinuosity of bays and inlets than a larger one, so the estimated length continues to increase as the ruler length decreases. In fact, a coastline is an example of a fractal, and plotting the length of the ruler versus the measured length of the coastline on a log-log plot gives a straight line, the slope of which is the fractal dimension of the coastline (and will be a number between 1 and 2). from http://mathworld.wolfram.htm

39. Length of a Coastline How Long is the Coast of Great Britain? Figure 1: The coastline of Great Britain In 1967, Benoit Mandelbrot published [7] ``How Long is the Coastline of Great Britain'' in Nature. In it, he posed the simple question of how one measures the length of a coastline. As with any curve, the obvious answer for the mathematician is to approximate the curve with a polygonal path, each side of which is of length є . (See Figure 2.) Then by evaluating the length of these polygonal paths as є0 , we expect to see the length estimate approach a limit. Unfortunately, it appears that for coastlines, as є0 , the approximated length L(є) infinity as well. Figure 2: Approximating the coastline of Great Britain

40. Length of a Coastline • In a later book, [10, pp 28-33,] Mandelbrot discusses the extensive experimental work on this problem which was done by  Lewis Fry Richardson. Richardson discovered that for any given coastline, there were constants F and D such that to approximate the coastline with a polygonal path, one requires roughly Fe-D intervals of length . Thus, the length estimate can be given as L(є) = Fe 1-D • The reason has to do with the inherent ``roughness'' of a coastline. In general, a coastline is not the type of curve we are usually used to seeing in mathematics. Although it is a continuous curve, it is not smooth at any point. In fact, at any resolution, more inlets and peninsulas are visible that were not visible before. (See Figure 3.) Thus as we look at finer and finer resolutions, we reveal more and more lengths to be approximated, and our total estimate of length appears to increase without bound. http://www.math.vt.edu/people/hoggard/FracGeomReport/node2.html

41. Length of a Coastline • Contrast this idea with the foundations of calculus which assert that a limit is attained when we cut the length into smaller and smaller pieces. • We make the assumption …and conclusion… that there is a finite length and that our methods of the integral calculus will help calculate that length.

42. Length of a Coastline • What is the length of the coastline of Britain? Benoit Mandelbrot proposed this question to demonstrate the complexity of measurement and scale. There are a number of almanacs that provide this information. However, if one examines the measuring techniques used to determine the length of Britain's coastline, it becomes obvious that this measurement is only an estimate based on the accuracy of the measuring device.Smaller units mean greater accuracy. But we can continue that line of thinking indefinitely, just as we do with fractions. There are always smaller fractions, an infinite number. Therefore, the coastline of Britain is an infinite length, however, it is confined within a finite space. We can begin to understand then that perimeter can have an infinite length confined within a finite area. http://home.inreach.com/kfarrell/measure.html

43. Length of a Coastline • How Long Is Australia's Coastline? (an explanation) • At first blush the question seems eminently reasonable, but it is as open-ended as the classical "how long is a piece of string?" The answer to both is the same: it all depends. • Dr Robert Galloway of the CSIRO Division of Land Use Research in Canberra was recently confronted with the question when compiling an inventory of Australia's coastal lands. Looking up the published figures he found the following answers: • The great disparity has to do with the precision with which the measurement is made. The larger and more detailed the map, and the more finely the measurement is made, the longer will be the coastline. Ultimately one could walk around the coast itself with a measuring stick, but the answer still depends on whether you use seven-league boots or a metre rule. • (It's a philosophical point whether the coastline tends to any limit as precision improves. Some say it does, others not.) • To settle on a reliable, repeatable figure, Dr. Galloway got together 162 maps covering the Australian coast and enlisted the help of Ms Margo Bahr of the Division.

44. Length of a Coastline • A few points of methodology had to be agreed on before the exercise could begin: • How far up estuaries should the coastline be taken? It was decided that all inlets would be arbitrarily (but consistently) cut off whenever their mapped width was less than 1 km. Within Sydney Harbour, for example, Kirribilli Point was joined to Garden Island. Straits less than 1 km wide were ignored, treating the island as though it were part of the mainland. • Islands less than 12 ha. were ignored. Measuring the coastline of the 2600 islands larger than that would be tedious in the extreme. Instead, a 16% sample was taken and a graph of coast length against area drawn. • This plot gave a good correlation, allowing island coastlines to be derived simply from their area. However, the ten largest islands (including Tasmania) were, for accuracy, measured directly. (Macquarie Island and Lord Howe Island were not included.) • Mangroves were regarded as part of the land, with the coastline following their seaward fringe; channels between mangroves were treated as estuaries. All coral reefs were excluded. http://www.maths.mq.edu.au/numeracy/tutorial/cts2.htm

45. Length of a Coastline • Finally came the question, which tools to use: a pair of dividers, a map measuring wheel, or a length of string or fine wire? On a test run, dividers gave consistent results only if the same starting point was used; the wheel was rapid but inaccurate. Fine wire (not string) laid on the drawn coastline proved surprisingly consistent and accurate (as good as dividers set to a 0.7 km interval) and so was chosen for the task. Down to work! • When the 162nd map was put aside, the total length of the mainland coast plus Tasmania worked out to be 30 270 km. Adding on the length of the coast of all the islands greater than 12 ha, about 16 800 km, gave a grand total of 47 070 km. • As a matter of interest and undeterred by their prior efforts, the two workers examined the effect of different divider lengths on the measured coastline. As expected, the apparent length of the coast of the mainland diminished steadily as the divider length was increased; shrinking to 10 830 km at a 1000-km intercept. • A simple formula was derived that linked coast length to the measuring intercept. Using this formula to extrapolate a divider length of just 1 mm, gave a length of about 132 000 km for the mainland of Australian rather more than three times the circumference of the earth!

46. Length of a Coastline How Long Is Australia's Coastline? The correct answer is .... it depends! It depends on which source you read, apparently. Source Length Year Book of Australia (1978) 36 735 km Australian Encyclopedia 19 658 km Australian Handbook 19 320 km So who is correct? The answer is: all of them! Each source used a ruler with different sized increments on it. If you measure the coastline with a ruler that is just 1 mm long, you would get a length of 132 000 km!

47. Length of a Coastline • 4.2 Calculating coastline and population [in New Zealand for Maori tribes] • 4.2.1 Coastline calculations • Under the proposed allocation method, inshore fishstocks and 60% of deepwater fishstocks would be allocated according to the length of an Iwi’s coastline. Exactly how would coastline lengths be worked out? • It is proposed that a 1:50,000 scale map of New Zealand would be used. Iwi would have to reach agreement with neighbouring Iwi as to their respective coastline lengths. The exact coastline length for a quota management area would then be calculated as follows: • rivers would be cut off at the coast and the distance across the river mouth included in the coastline measurement; • the coastline length of harbours and bays whose natural entrance points are greater than 10 km apart would be included in the coastline measurement; • the juridical bay formula (see below) would be applied to harbours and bays whose natural entrance points are less than 10 km apart in order to determine whether those harbours and bays would be included in the coastline measurement; and • with the exception of the islands in the Chatham Islands group, coastline measurements would not include the coastline of islands claimed by Iwi to be part of their traditional takiwa.

48. Length of a Coastline • The juridical bay formula • The juridical bay formula is applied to bays where the natural entrance points are less than 10 km apart in order to determine whether the distance across the entrance of the bay or the actual coastline of the bay should be added to the coastline measurement. The formula works as follows (see Figure 3): • a straight line is drawn between the natural entrance points of the bay; • a semicircle is drawn on the straight line (using the straight line as the diameter of the circle) and the surface area of the semicircle is calculated; • the surface area of the bay enclosed by the straight line is also calculated using map information software; • if the surface area of the semicircle is smaller than the water surface area of the bay, then the distance between the natural entrance points is included in the coastline measurement, (see figure 3a, the shoreline of the bay is not included) ; or • if the surface area of the semicircle is bigger than the water surface area of the bay, then the shoreline of the bay is measured and included in the coastline measurement.( see figure 3b)

49. Length of a Coastline A drawing of the New Zealand juridical bay formula http://www.tokm.co.nz/allocation/1997/implementing.htm

50. Length of a Coastline The Florida Shoreline and its Measurement The FLDEP is responsible for monitoring and managing approximately 680 miles of Florida’s coastline. This includes the state’s entire coastline except for Monroe County (Florida Keys) and Federal sites. These management efforts are the result of two contributing factors. In addition, nature constantly changes the shoreline through normal coastal processes and occasional storm events. Other shoreline changes result from man’s engineering activities associated with ports and harbors and shoreline stabilization. The FLDEP must identify and quantify these changes and manage the coastline to preserve Florida’s most important natural resource – its beaches.