December 2010 - mathfun@mailman.xmission.com

[math-fun] racing around a triangle
by Henry Baker 08 Dec '10

08 Dec '10

I'm still trying to solve for the optimum path around a triangle, subject to the acceleration constraint |a|<=1. On a line segment of length 1 (one kind of degenerate triangle) on the x-axis centered at the origin, one guess at an optimal path would be to start from the origin towards 1/2 with a velocity of 1 and decelerate at the constant rate of -1, achieving 1/2 after t=1 with velocity 0, and continuing to accelerate back towards the origin, reaching it at time 2 where we achieve velocity -1 & immediately start decelerating with acceleration of +1, achieving -1/2 at time 3, and continue accelerating at +1 to reach the origin again at time 4, for a complete cycle. x = x0+v0*t+a0*t^2/2 v = v0+a0*t v1 = v0-1*t=0 => t=v0 x0 = 0 x1 = 0+v0*t+a0*t^2/2 = 0+v0*v0-1*v0^2/2 = v0^2-v0^2/2 = v0^2/2 = 1/2 => v0=1 => t=1 However, if we were to simply go in a circle around this segment as a diameter with velocity |v|=1, then it would take time 2pi=6.28 to complete a cycle. Although the direction of the acceleration is constantly changing, its absolute value is still always |a|=1. So it would seem that we can go around some circles faster than simply following the circumcircle. Is there any way to see how to compute an optimum trajectory around an arbitrary triangle?

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[math-fun] Math rankings of various countries
by Dan Asimov 07 Dec '10

07 Dec '10

This graphic: < http://www.nytimes.com/imagepages/2010/12/07/education/07education_graph.ht… >, third column, shows the U.S. at 32nd out of 65 coutries whose 15-year-olds took a standardized math test. Why is it so hard for the U.S. to improve its math education? --Dan Those who sleep faster get more rest.

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Re: [math-fun] Math PuzzlesX Oldest Ancestors Took Form on Egyptian Papyrus
by Dan Asimov 07 Dec '10

07 Dec '10

I always thought the St. Ives problem was a trick question: "As I was going to St. Ives, I met a man with seven wives . . .." While walking, someone one meets is usually going the opposite direction. So only "I" -- one person -- was going to St. Ives. (At least that's how it was once explained to me.) --Dan << . . . Scholars have found a few errors in the problems, and Ahmes even wrote an incorrect number in his St. Ives problem. But over all, the equations are considered remarkably accurate. >> Those who sleep faster get more rest.

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Math Puzzles Oldest Ancestors Took Form on Egyptian Papyrus
by Henry Baker 07 Dec '10

07 Dec '10

FYI -- Computing the volume of truncated pyramids? Slopes of pyramids? Hmmm... I guess these issues became important after some early pyramid failures due to too-steep pyramid design. http://www.nytimes.com/2010/12/07/science/07first.html December 6, 2010 Math Puzzles Oldest Ancestors Took Form on Egyptian Papyrus By PAM BELLUCK As I was going to St. Ives I met a man with seven wives. ... You may know this singsong quiz, But what you might not know is this: That it began with ancient Egypts Early math-filled manuscripts. Its true. That very British-sounding St. Ives conundrum (the one where the seven wives each have seven sacks containing seven cats who each have seven kits, and you have to figure out how many are going to St. Ives) has a decidedly archaic antecedent. An Egyptian document more than 3,600 years old, the Rhind Mathematical Papyrus, contains a puzzle of sevens that bears an uncanny likeness to the St. Ives riddle. It has mice and barley, not wives and sacks, but the gist is similar. Seven houses have seven cats that each eat seven mice that each eat seven grains of barley. Each barley grain would have produced seven hekat of grain. (A hekat was a unit of volume, roughly 1.3 gallons.) The goal: to determine how many things are described. The answer: 19,607. The Rhind papyrus, which dates to 1650 B.C., is one of several precocious papyri and other artifacts displaying Egyptian mathematical ingenuity. There is the Moscow Mathematical Papyrus (held at the Pushkin State Museum of Fine Arts in Moscow), the Egyptian Mathematical Leather Roll (which along with the Rhind papyrus is housed at the British Museum) and the Akhmim Wooden Tablets (at the Museum of Egyptian Antiquities in Cairo). They include methods of measuring a ships mast and rudder, calculating the volume of cylinders and truncated pyramids, dividing grain quantities into fractions and verifying how much bread to exchange for beer. They even compute a circles area using an early approximation of pi. (They use 256/81, about 3.16, instead of pis value of 3.14159....) It all goes to show that making puzzles is the most ancient of all instincts, said Marcel Danesi, a puzzle expert and anthropology professor at the University of Toronto, who calls documents like the Rhind papyrus the first puzzle books in history. Dr. Danesi says people of all eras and cultures gravitate toward puzzles because puzzles have solutions. Other philosophical puzzles of life do not, he continued. When you do get it you go, Aha, there it is, damn it, and it gives you some relief. But the Egyptian puzzles were not just recreational diversions seeking the comforting illusion of competence. They were serious about their mission. In the Rhind papyrus, its scribe, known as Ahmes, introduces the roughly 85 problems by saying that he is presenting the correct method of reckoning, for grasping the meaning of things and knowing everything that is, obscurities and all secrets. And the documents were practical guides to navigating a maturing civilization and an expanding economy. Egypt was going from a centralized, structured world to partially being decentralized, said Milo Gardner, an amateur decoder of Egyptian mathematical texts who has written extensively about them. They had an economic system that was run by absentee landowners and paid people in units of grain, and in order to make it fair had to have exact weights and measures. They were trying to figure out a way to evenly divide the hekat so they could use it as a unit of currency. So the Akhmim tablets, nearly 4,000 years old, contain lists of servants names, along with a series of computations concerning how a hekat of grain can be divided by 3, 7, 10, 11 and 13. The Egyptian Mathematical Leather Roll, also from about 1650 B.C., is generally considered a kind of practice test for students to learn how to convert fractions into sums of other fractions. The Rhind papyrus contains geometry problems that compute the slopes of pyramids and the volume of various-shaped granaries. And the Moscow papyrus, from about 1850 B.C., has about 25 problems, including ways to measure ships parts and find the surface area of a hemisphere and the area of triangles. Especially interesting are problems that calculate how efficient a laborer was by how many logs he carried or how many sandals he could make and decorate. Or the problems that involve a pefsu, a unit measuring the strength or weakness of beer or bread based on how much grain is used to make it. One problem calculates whether its right to exchange 100 loaves of 20-pefsu bread for 10 jugs of 4-pefsu malt-date beer. After a series of steps, the papyrus proclaims, according to one translation: Behold! The beer quantity is found to be correct. The problems in these ancient texts are not difficult by modern mathematical standards. The challenge for scholars has come in deciphering what the problems are saying and checking their accuracy. Some of the numerical equivalents are written in a symbolic system called the Eye of Horus, based on a drawing representing the eye of the sky god Horus, depicted as a falcon. Sections of the falcons eye are used to represent fractions: one-half, one-quarter and so on, up to one sixty-fourth. Scholars have found a few errors in the problems, and Ahmes even wrote an incorrect number in his St. Ives problem. But over all, the equations are considered remarkably accurate. The practical answers are solved, Mr. Gardner said. What is unsolved about them is the actual thinking in the scribes head. We dont know exactly how he thought of it.

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Re: [math-fun] Exponential map of the Mandelbrot set, and the "logarithm of a circle"
by Henry Baker 06 Dec '10

06 Dec '10

I forgot to emphasize that the "logarithmic mapping" in the image processing application used the _complex logarithm_, so that a circle around the origin in "unwrapped" into a periodic function (in the imaginary coordinate), while circles not enclosing the origin are extremely distorted. Objects not enclosing the origin change size by simple translation in the ln (horizontal) coordinate, while rotations about the origin become translations in the imaginary (vertical) coordinate. At 10:30 AM 12/6/2010, Robert Munafo wrote: >The connection with human vision sensitivity is also related to the use of the logarithmic map to optimize the generation of "fractal zoom" animation videos. > >Another Mandelbrot artist with whom I am corresponding separately just suggested to me that the image can be produced at lower precision than needed to resolve every pixel in every frame of a zoom animation. The pixels near the edge will be a bit fuzzy, but this typically happens in compressed video because typical motion compensation algorithms smooth out pixel details in areas that are moving quickly. Since the frame rate naturally blurs high-speed motion, and since the viewer typically focuses on the center (at least when the zoom is an "inward" zoom) this fuzziness isn't noticed. > >- Robert > >On Mon, Dec 6, 2010 at 13:16, Henry Baker <hbaker1(a)pipeline.com> wrote: >In computer vision/image processing, a logarithmic map was suggested ~30 years ago as a way to focus attention/computational cycles on areas of greatest interest. For example, the fovea of the human eye has the highest density of pixels (as well as all of the color pixels), and the density of pixels falls off pretty quickly the further from the fovea that one goes. By utilizing a logarithmic map, one can theoretically achieve any desired resolution, at the cost of moving the center point -- either mechanically or digitally. > >I believe that a number of common geometric objects (circles, lines, etc.) were analyzed by these researchers to understand how they mapped under this logarithmic transformation. > >At 02:24 PM 12/5/2010, Robert Munafo wrote: >>I was working on a page about the Mandelbrot set as seen through an >>exponential (or logarithmic) coordinate transformation: >> >> http://mrob.com/pub/muency/exponentialmap.html >> >>[...] > >-- > Robert Munafo -- mrob.com > Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com

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Re: [math-fun] Exponential map of the Mandelbrot set, and the "logarithm of a circle"
by Henry Baker 06 Dec '10

06 Dec '10

In computer vision/image processing, a logarithmic map was suggested ~30 years ago as a way to focus attention/computational cycles on areas of greatest interest. For example, the fovea of the human eye has the highest density of pixels (as well as all of the color pixels), and the density of pixels falls off pretty quickly the further from the fovea that one goes. By utilizing a logarithmic map, one can theoretically achieve any desired resolution, at the cost of moving the center point -- either mechanically or digitally. I believe that a number of common geometric objects (circles, lines, etc.) were analyzed by these researchers to understand how they mapped under this logarithmic transformation. At 02:24 PM 12/5/2010, Robert Munafo wrote: >I was working on a page about the Mandelbrot set as seen through an >exponential (or logarithmic) coordinate transformation: > > http://mrob.com/pub/muency/exponentialmap.html > >and I ran across the need to describe the shape of an offset circle after >its logarithm is taken. To be more precise: > >If A is a circle (viewed as a set of points on the complex plane) whose >distance from the origin is greater than its radius (i.e. the origin is >outside the circle), and if B is the set of points you get by taking the >(complex-valued) natural logarithm of each point in A, then what type of >shape is B? > >Is B an ellipse, some kind of superellipse (using a transcendental function >perhaps)? If I need to give it a name, is there any name (like >"quasi-ellipse") that doesn't already have some other meaning that would >confuse my readers? > >-- > Robert Munafo -- mrob.com > Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - >youtube.com/user/mrob143 - rilybot.blogspot.com

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[math-fun] Exponential map of the Mandelbrot set, and the "logarithm of a circle"
by Robert Munafo 06 Dec '10

06 Dec '10

I was working on a page about the Mandelbrot set as seen through an exponential (or logarithmic) coordinate transformation: http://mrob.com/pub/muency/exponentialmap.html and I ran across the need to describe the shape of an offset circle after its logarithm is taken. To be more precise: If A is a circle (viewed as a set of points on the complex plane) whose distance from the origin is greater than its radius (i.e. the origin is outside the circle), and if B is the set of points you get by taking the (complex-valued) natural logarithm of each point in A, then what type of shape is B? Is B an ellipse, some kind of superellipse (using a transcendental function perhaps)? If I need to give it a name, is there any name (like "quasi-ellipse") that doesn't already have some other meaning that would confuse my readers? -- Robert Munafo -- mrob.com Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com

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[math-fun] pendulum harmonics
by Bill Gosper 05 Dec '10

05 Dec '10

Better tested and less fishily written, they still look like closed forms for the Fourier coefficients of am, dn, and sc: http://gosper.org/fourndulum.pdf Familiar? --rwg

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[math-fun] Masters of Math, From Old Babylon
by Henry Baker 05 Dec '10

05 Dec '10

FYI -- November 26, 2010 Masters of Math, From Old Babylon By EDWARD ROTHSTEIN If the cost of digging a trench is 9 gin, and the trench has a length of 5 ninda and is one-half ninda deep, and if a workerÂs daily load of earth costs 10 gin to move, and his daily wages are 6 se of silver, then how wide is the canal? Or, a better question: if you were a tutor of Babylonian scribes some 4,000 years ago, holding a clay tablet on which this problem was incised with cuneiform indentations Â the very tablet that can now be seen with 12 others from that Middle Eastern civilization at the Institute for the Study of the Ancient World Â what could you take for granted, and what would you need to explain to your students? In what way did you think about measures of time and space? How did you calculate? Did you believe numbers had an abstract existence, each with its own properties? And how would you have figured out the width of that canal (which, the tablet tells us, is one-and-a-half ninda)? Spend some time at this modest yet thoroughly intriguing exhibition, ÂBefore Pythagoras: The Culture of Old Babylonian Mathematics,Â and you begin to realize that the answers can be far more cryptic than these tablets were before great scholars like Otto E. Neugebauer began to decipher them during the first half of the 20th century. The institute, part of New York University, has gathered together a remarkable selection of Old Babylonian tablets from the collections of three universities Â Columbia, Yale and the University of Pennsylvania Â that cover a wide mathematical range. Made between 1900 and 1700 B.C., they include student exercises, word problems and calculation tables, as well as more abstract demonstrations. Under the curatorship of Alexander Jones, a professor at N.Y.U., and Christine Proust, a historian of mathematics, the tablets are used to give a quick survey of Babylonian mathematical enterprise, while also paying tribute to Neugebauer, the Austrian-born scholar who spent the last half of his career teaching at Brown University and almost single-handedly created a new discipline of study through his analysis of these neglected sources. Only about 950 mathematically oriented tablets survived two millenniums of Babylonian history, and since their discovery, debate has raged over what they show us about that lost world. Every major history of Western mathematics written during the last 70 years has at least started to take Babylonians into account. Generally, their systems have been seen as precursors to the theoretical flowering of Greek mathematics, out of which our own mathematical approaches have grown. But Neugebauer, and then his many students and rivals, also showed how sophisticated Babylonian mathematics was and how many similarities existed to later Western systems Â if, that is, you counted using 60 fingers (as we often seem to, thanks to the Babylonians, when dealing with seconds and minutes and, in part, even when measuring angles). Examining the surviving tablets, including one multiplication table on display here, scholars decoded the birdÂs feet of Babylonian numerals, showing that the Babylonians, like us, used the same symbol to represent different numerical values. (The same digit for us has a different value if it is in the 1Âs column, the 10Âs column, or the 100Âs column; the Babylonians could use the same sign, depending on context, to represent a 1 or a 60 or a 3,600.) The most famous tablets here Â one showing a square with two diagonals, and another, known as Plimpton 322, containing a table of numerical symbols Â suggest that the Babylonians knew at least some of the consequences of the theorem that now bears the name of the Greek mathematician Pythagoras, who lived some 1,500 years after these works were chiseled. But did the Babylonians conceive of it as a ÂtheoremÂ Â a timeless truth subject to proof based on accepted principles? Or was it thought of as a property of areas of land that were mapped out by surveyors? Or, as one scholar recently wrote, was Plimpton 322 Âa teacherÂs catalog of parametersÂ for calculation? Or something else completely? Aside from the fact that the analysis of these tablets is relatively recent, one of the problems is that much of their context is hypothetical, because of the almost haphazard way in which early modern explorers pulled these artifacts out of the layered rubble of ancient mounds of detritus. In a fascinating 2008 book, ÂMathematics in Ancient Iraq: A Social HistoryÂ (Princeton), Eleanor Robson even suggests that many tablets like these of the second millennium B.C. Âwere essentially ephemera, created to aid and demonstrate recall, destined almost immediately for the recycling bin.Â But as Ms. Robson also points out, these tabletsÂ word problems about digging and construction, their use in teaching record keeping and calculation, and their implicit affirmation of the importance of scribes and teachers, also reveal a highly organized, bureaucratic society, an Âordered urban state, with god, king and scribe at its center.Â At the exhibition itself, it would have helped if some of the translations and interpretations of the tablets provided in an accompanying brochure had been made available directly on the labels themselves, so you could both look at the artifact and see how to interpret it. It might have helped, too, if the show suggested the kinds of debates that have arisen about the status of Babylonian mathematics, which, in many ways, parallel larger debates about Western and non-Western cultures. In this case those debates become all the more charged because of the geographical region involved. While once, for example, Babylonian mathematics was clearly seen as a precursor to the Greeks and to Western systems, there is an impulse now to see it as a historical victim of Western cultural perspectives. That is partly how Ms. Robson portrays it, even (unconvincingly) evoking Edward SaidÂs book ÂOrientalismÂ; a hint of that sentiment may even be one of the subtle draws of this exhibition. But for an outsider surveying these objects and the history, the response is at once awe at the ancient minds that created such powerful systems for understanding and ordering experience, and still more amazement at the mathematical world that later developed and went so much further: explicitly turning individual examples into theorems, earthly practices into abstract principles and outlining new methods for understanding that are still being applied. This is the trench solution as it appears on the Babylonian tablet. The translation, by Alexander Jones, a curator of the ÂBefore PythagorasÂ exhibition, is in the showÂs brochure: ÂSolution: Multiply the length and the depth, and you get 30. Take the reciprocal of the workload, multiply by 30 and you will get 3. Multiply the wages by 3, and you will get 6. Take the reciprocal of 6, and multiply it by 9, the total cost in silver, and you will get its width. One and a half ninda is the width. Such is the procedure.Â (A reciprocal in Babylonian arithmetic is in relation to 60, so the reciprocal of 10 is 6.) ÂBefore Pythagoras: The Culture of Old Babylonian MathematicsÂ is on view through Dec. 17 at the Institute for the Study of the Ancient World, New York University, 15 East 84th Street, Manhattan; nyu.edu/isaw.

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[math-fun] The Fibonomial Triangle
by Richard Guy 05 Dec '10

05 Dec '10

Dear all, How much of the following is known to those who well know it? I haven't yet been able to consult Knuth, vol.1, p.85, so it may be there. Or in Duke Math J 29(1962) page numbers need correcting in some? of [A000012, A000045, A007598], A056570--4, A056585--7. The characteristic polynomials for these sequences are x - 1 x^2 - x - 1 x^3 - 2x^2 - 2x + 1 x^4 - 3x^3 - 6x^2 + 3x + 1 x^5 - 5x^4 - 15x^3 + 15x^2 + 5x - 1 x^6 - 8x^5 - 40x^4 + 60x^3 + 40x^2 - 8x - 1 x^7 - 13x^6 - 104x^5 + 260x^4 + 260x^3 - 104x^2 - 13x + 1 x^8 -21 -273 +1092 +1820 -1092 -273 +21 +1 x^9 -55 -1870 +19635 +85085 -136136 -85086 +19635 +1870 -55 -1 x^10 -89 -4895 +83215 +582505 -1514513 -1514513 +582505 + - - + which (it is known by some) factor into x-1 x^2-x-1 (x+1)(x^2-3x+1) (x^2+x-1)(x^2-4x-1) (x-1)(x^2+3x+1)(x^2-7x+1) (x^2-x-1)(x^2+4x-1)(x^2-11x-1) (x+1)(x^2-3x+1)(x^2+7x+1)(x^2-18x+1) (x^2+x-1)(x^2-4x-1)(x^2+11x-1)(x^2-29x-1) (x-1)(x^2+3x+1)(x^2-7x+1)(x^2+18x+1)(x^2-47x+1) (x^2-x-1)(x^2+4x-1)(x^2-11x-1)(x^2+29x-1)(x^2-76x-1) (x+1)(x^2-3x+1)(x^2+7x+1)(x^2-18x+1)(x^2+47x+1)(x^2-123x+1) where the middle coefficients are, of course, Lucas numbers. If we make a triangle of the coeffs of the unfactored polynomials, we find that, apart from signs, they are e (F0/F1) e F1/F1 e F2/F1 F2F1/F1F2 e F3/F1 F3F2/F1F2 F3F2F1/F1F2F3 e F4/F1 F4F3/F1F2 F4F3F2/F1F2F3 F4F3F2F1/F1F2F3F4 e F5/F1 F5F4/F1F2 F5F4F3/F1F2F3 .... e F6/F1 F6F5/F1F2 F6F5F4/F1F2F3 F6F5F4F3/F1F2F3F4 '''' e F7/F1 F7F6/F1F2 F7F6F5/F1F2F3 F7F6F5F4/F1F2F3F4 .... where e = 1 is the empty product, and F7F6F5/F1F2F3, for example, is 13*8*5/1*1*2 = 260, a sort of `binomial coefficient' of Fibonacci numbers F1=F2=1, F3=2, F4=3, ... Of course, these sequences are all divisibility sequences. R.

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