March 2017 - mathfun@mailman.xmission.com

[math-fun] Binary calculators
by James Propp 22 Mar '17

22 Mar '17

In conjunction with the Global Math Project, I'm putting together a list of videos showing interesting forms of binary calculation, using water in cups, trains on tracks, etc.: http://mathenchant.org/binary.html I'm aware that the categorization of the links is a bit inconsistent; I plan to fix that before I publicize the webpage broadly. For now I'm mostly looking for more interesting links, and I figure that some of you will know about things I'm unaware of. I also am chasing the idea (mirage?) of a robust, cheap, easy-to-build binary calculator that could be taken on as an in-class project by students around the world. (Call it the mathematical equivalent of the penny-a-shot rotavirus vaccine.) I have no idea how to build such a thing, but maybe some really clever person has already figured it out. Thanks, Jim Propp

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Re: [math-fun] Binary calculators
by Henry Baker 22 Mar '17

22 Mar '17

A modern valve trumpet/French Horn/tuba uses multiple valves to "compute" the proper length to achieve a certain set of notes. Since the small lengths of tubing "add" to one another, I claim that these are "computers". At 07:32 AM 3/21/2017, James Propp wrote: >In conjunction with the Global Math Project, I'm putting together a list of >videos showing interesting forms of binary calculation, using water in >cups, trains on tracks, etc.: > >http://mathenchant.org/binary.html > >I'm aware that the categorization of the links is a bit inconsistent; I >plan to fix that before I publicize the webpage broadly. For now I'm mostly >looking for more interesting links, and I figure that some of you will know >about things I'm unaware of. > >I also am chasing the idea (mirage?) of a robust, cheap, easy-to-build >binary calculator that could be taken on as an in-class project by students >around the world. (Call it the mathematical equivalent of the penny-a-shot >rotavirus vaccine.) I have no idea how to build such a thing, but maybe >some really clever person has already figured it out. > >Thanks, > >Jim Propp

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Re: [math-fun] Binary calculators
by Henry Baker 22 Mar '17

22 Mar '17

Ummm, ancient Greek water clocks? At 12:08 AM 3/22/2017, rcs(a)xmission.com wrote: >IIRC, someone has done a water fountain with fluid logic, >making some sort of counter -- maybe a LFSR shift register. >Has anyone seen this? > >Rich

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Re: [math-fun] Binary calculators
by Henry Baker 22 Mar '17

22 Mar '17

There is some evidence that the ancient Romans used non-trivial (at least by ancient standards) fluidic logic in some of their waterworks. If you have a stream of water coming down a hill and it meets a narrow Y, it doesn't take much effort (water and/or mechanical) to change the course from one to the other exit, so you can have *amplification*. I've always been fascinated by ancient Roman engineering, because some of these systems operated more-or-less continuously for hundreds of years. I don't know very many modern systems that can operate for more than a few years without substantial maintenance and repair. For example, I've climbed through the water channel at the top of the Pont du Gard, and the channel is about the same size as a man (ok, a short, ancient Roman-sized man), so that a man could walk through the channel even with the stone roof in place. However, this would have been impossible without first turning off the water, so there had to be a complex series of gates/diverters to enable service persons to remove debris from the channels. I noticed that every once in a while, the water channel would drop vertically and change direction; I believe this enabled any large debris in the channel to be sorted out from the water. I have also speculated that the ancient Romans utilized some of the aqueducts to transport wood from high up in the mountains in a manner similar to the way lumber was transported from the California mountains into Sacramento; the ancient Roman ovens used for baking their bread consumed prodigious amounts of wood for fuel, and post Augustus (?), delivery trucks/carts were only allowed to use the streets in the middle of the night (NYC, are you listening?). At 11:27 AM 3/22/2017, rcs(a)xmission.com wrote: >This had no moving parts, except for the water. >There must also have been a pump somewhere. > >Imagine a sloped surface, with various bumps & channels. >Water is supplied at the top, and runs down the surface, >being deflected by the topography. The logic comes from >non-linear behavior of streams & droplets. And maybe some >'sticky' effects, like a stream that sticks to the left side >of a channel for historical reasons, but can be switched to >the right side, or perhaps the center. >This would seem to lead to a fixed pattern, or a simple >repetition; to get interesting behavior, some feedback would >be necessary. I don't know the mechanism. > >Rich >-------- >Quoting Henry Baker <hbaker1(a)pipeline.com>: >>Ummm, ancient Greek water clocks? >> >>At 12:08 AM 3/22/2017, rcs(a)xmission.com wrote: >>>IIRC, someone has done a water fountain with fluid logic, >>>making some sort of counter -- maybe a LFSR shift register. >>>Has anyone seen this? >>> >>>Rich

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[math-fun] Help with some math
by David Wilson 22 Mar '17

22 Mar '17

Let f(x)^2 = f(x^2) - 1 and assume f is of the form f(x) = x + a x^-1 + b x^-3 + c x^-5 + ... What are the first few coefficients (say 20) as rational numbers? With the tools I have, it would take me a while to power through this. I assume it would be easier with a symbolic math program.

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[math-fun] circuit complexity of table lookups
by Henry Baker 21 Mar '17

21 Mar '17

OK, so you want to compute some function that is presented as a (constant) table lookup on 64-bit integers, for example. We have a simple which is 2^64 entries of 64-bits each. This can be done using a ROM, which is usually laid out as an approximate square on a chip, so we have 64 square tables each (2^32)x(2^32), or an 8x8 array of such tables, or a single table (2^35)x(2^35). Even today, this is a monster chip (113'x113' -- the footprint of a large mansion or a small hotel) at nanometer feature sizes. But if the function is quite smooth & multiply differentiable -- e.g., a typical arithmetic function, we should be able to dramatically reduce this size to something reasonable. We can use various piecewise interpolation techniques and trade off huge amounts of area for very modest increases in circuit depth. If we're quite lucky, we might even be able to use a table for a 2-argument function -- e.g., f(x,y)=x*y. The "Chebfun" system (www.chebfun.org) uses Chebyshev polynomials to dramatically reduce the effort for representing arbitrary functions with, e.g., 20-30 points instead of 1,000,000. So something like Chebfun might be used to mechanically compute a table lookup chip of reasonable size. Has anyone in academia (or anywhere else) gone into the business of accepting a large table of function values over the Internet and delivering an optimized chip for that function?

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[math-fun] Cool Heron's Formula example
by Henry Baker 21 Mar '17

21 Mar '17

>From John L. Gustafson's discussion with Kahan re floating point: Triangle with side lengths a,b,c. a=b=7/2+3*2^(-111), c=7 Units are light-years. Note that 3*2^(-111) is ~1/200 size of a proton. Area ~ 55x Earth's surface. Believe it or not, the triangle altitude is ~ 85 cm ~ 33.5" -- almost a yard. Don't try this calculation at home; it's Herondisly inaccurate.

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[math-fun] Pixel perfect Ptolemizer playpen [was: Julian has constructed a spacefill dense with sextuple points]
by Bill Gosper 21 Mar '17

21 Mar '17

On 2016-08-31 16:55, Bill Gosper wrote: Hi Vitaliy, I have a sandbox with a general Fourier animator. It takes a formula for the coefficients (amplitudes and phases as functions of frequency), speed, and coloration. It isn't very fast because it colors one pixel at a time, emulating the Symbolics mathematically correct draw-triangle ALU-add microcode necessary for smoothly drawing moving edges. Mathematically correct means that, if two triangles share an edge, there will never be missing or overwritten pixels, and if the triangle is "inside out", the "bump" will be negated. (The trick is to consistently define exactly which pixels to bump.) With this primitive, you can then animate the movement of an edge segment simply by drawing two triangles filling the quadrilateral defined by the old and new positions of its endpoints. To draw a seamless T joint, just [include] the "zero area" triangle formed by the constituent edges, which will usually add or subtract the "bump" constant to or from a few pixels. --Bill Howard Cannon has implemented a fast, full-res re-creation of the Symbolics triangle primitive, along with several lovely Ptolemaic sweeps, plus luxurious controls: http://hichacks.is-a-geek.net/triangles/artcircle E.g., for a nice picture of 2D Gibbs ringing, change the "Circle" box to Gibbs Tri, and click >. Besser illustrates frequency (over)modulation. The vector sum runs back and forth <Amplitude> radians along the circumference of the circle Howard (hic(a)iname.com) tailored the page for Chrome, and seeks your comments and suggestions. (I'll try to supply him with the coefficients for Julian's sextuple point spacefill.) --rwg

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[math-fun] Beyond Floats: Next Gen Computer Arithmetic
by Henry Baker 20 Mar '17

20 Mar '17

FYI -- https://www.youtube.com/watch?v=aP0Y1uAA-2Y Stanford Seminar: Beyond Floating Point: Next Generation Computer Arithmetic John L. Gustafson, Natl Univ of Singapore https://web.stanford.edu/class/ee380/Abstracts/170201.html A new data type called a "posit" is designed for direct drop-in replacement for IEEE Standard 754 floats. Unlike unum arithmetic, posits do not require interval-type mathematics or variable size operands, and they round if an answer is inexact, much the way floats do. However, they provide compelling advantages over floats, including simpler hardware implementation that scales from as few as two-bit operands to thousands of bits. For any bit width, they have a larger dynamic range, higher accuracy, better closure under arithmetic operations, and simpler exception-handling. For example, posits never overflow to infinity or underflow to zero, and there is no "Not-a-Number" (NaN) value. Posits should take up less space to implement in silicon than an IEEE float of the same size. With fewer gate delays per operation as well as lower silicon footprint, the posit operations per second (POPS) supported by a chip can be significantly higher than the FLOPs using similar hardware reso urces. GPU accelerators, in particular, could do more arithmetic per watt and per dollar yet deliver superior answer quality. A series of comprehensive benchmarks compares how many decimals of accuracy can be produced for a set number of bits-per-value, using various number formats. Low-precision posits provide a better solution than "approximate computing" methods that try to tolerate decreases in answer quality. High-precision posits provide better answers (more correct decimals) than floats of the same size, suggesting that in some cases, a 32-bit posit may do a better job than a 64-bit float. In other words, posits beat floats at their own game. http://ee380.stanford.edu/Abstracts/170201-slides.pdf

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[math-fun] Insolation problem
by Keith F. Lynch 20 Mar '17

20 Mar '17

Inspired by the equinox that's coming in about seven hours: Higher latitudes get less insolation (watts of sunlight per square meter of level surface area averaged over a year) than lower latitudes. How high would the Earth's obliquity (axial tilt) have to be for the poles to get more insolation than the equator? Or could this never happen? Assuming Earth's orbit was perfectly circular and that the insolation is exactly 1 kilowatt per square meter when the sun is directly overhead, is there a simple formula for insolation as a function of latitude and obliquity?

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