May 2011 - mathfun@mailman.xmission.com

[math-fun] The Roman Pantheon = Hat Box Theorem
by Henry Baker 09 May '11

09 May '11

I just realized that the Roman Pantheon may have been designed as a illustration of Archimedes's "Hat Box Theorem". From the inside of the Pantheon, the top half is a hemisphere and the bottom half is a cylinder of the same height as the radius; i.e., the continuation of the sphere that just fit inside the dome would have just touched the floor in the middle. Hadrian is known to have been 1) fond of everything Greek; 2) an amateur architect; and 3) fond of domes. He certainly wasn't good enough to design the Pantheon, but he certainly would have appreciated the architect who proposed it to him. http://en.wikipedia.org/wiki/Pantheon,_Rome "The height to the oculus and the diameter of the interior circle are the same, 43.3 metres (142 ft), so the whole interior would fit exactly within a cube (also, the interior could house a sphere 43.3 metres (142 ft) in diameter).[33] These dimensions make more sense when expressed in ancient Roman units of measurement: The dome spans 150 Roman feet; the oculus is 30 Roman feet in diameter; the doorway is 40 Roman feet high." http://mathworld.wolfram.com/ArchimedesHat-BoxTheorem.html

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[math-fun] Chinese remainder theorem
by Bill Gosper 08 May '11

08 May '11

To save you the trouble of looking it up: Thm: The number of dim sum is always relatively prime to the number of people sharing them. --rwg If the waiter brings you a fork, ask him to show you how to use it. DM>Thanks Mike and Tom :) On 7 May 2011, at 01:06, Mike Stay wrote: > On Fri, May 6, 2011 at 3:37 PM, David Makin <makinmagic(a)tiscali.co.uk <http://gosper.org/webmail/src/compose.php?send_to=makinmagic%40tiscali.co.uk>> wrote: >> (I mean for example - what's CRT here ?) > > http://en.wikipedia.org/wiki/Chinese_Remainder_Theorem > -- > Mike Stay - metaweta(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=metaweta%40gmail.com> > http://www.cs.auckland.ac.nz/~mike > http://reperiendi.wordpress.com

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[math-fun] multidigit representations
by quad 07 May '11

07 May '11

Hey all, I'm here in Nova Scotia at a conference, and I was speaking with a professor and work she's doing. She's working on so called "multidigit representations". The basic gist is this: Let B = NxN dilation matrix (matrix with positive eigenvalues) of integers. Let D be a set of N dimensional row vectors (a particular set). Then for v1, ..., vk in D, SUM(i = 0 to k) B^i v_i is a multidigit number, which should remind you of our single dimensional system. The set D can be derived from finding the lattice points in the square {(x,y) | -1/2 < x,y <= 1/2} transformed by B. So D is the set of coset representatives Z^n/B(Z^n). An interesting choice of B is the twin dragon matrix [[1,1],[-1,1]] and the digit set {[0,0], [-1,1]}. Note that the number of digits equals the determinant of B. This base and digit set will constitute all points in Z^2. Anyone have any comments or fun things you can do with this? There are sums and matrices involved, I know one of you will get hooked. Sincerely, Robert

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[math-fun] Nearest integer function
by Adam P. Goucher 07 May '11

07 May '11

Here's a formula I created for the nearest integer function: nint(x) = x + sum[k=1..infinity] sin(2*pi*k*(x+1/2))/(pi*k) It is equivalent to the following discrete function: nint(x) = x iff 2x is an integer; nint(x) = floor(x+1/2) otherwise. http://mathworld.wolfram.com/NearestIntegerFunction.html It would be interesting to see how my formula handles complex numbers. (Unlike the MathWorld definition, my function leaves half-integers invariant.) Sincerely, Adam P. Goucher

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[math-fun] Symbolic Calculation
by Adam P. Goucher 07 May '11

07 May '11

Today I received a Casio fx-85GT calculator as a present from a couple of my friends. It gives expressions in exact forms as an alternative to the ordinary decimal output. For example, enter: sqrt(8) and it will output: 2*sqrt(2) Of course, I believed that the calculator was performing symbolic manipulation -- very impressive for a handheld scientific calculator! However, I then decided to test it: ln(640320^3+744)/sqrt(163) Instead of displaying the correct answer, 3.14159265..., a prominent image of the Greek letter pi was rendered! Hmm... I'm waiting for a version that prints "stop trying to test my symbolic calculation facilities with well-known near-miss identities!" On this topic, there is this comic: http://xkcd.com/217/ Sincerely, Adam P. Goucher

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Re: [math-fun] Symbolic Calculation
by Henry Baker 07 May '11

07 May '11

Actually, this is one book (actually _three_ books!) that you really want to buy. I marked up nearly every single page of all three volumes. I think I learned more math from Knuth than from all of the rest of my math courses combined. You can probably get the 3-volume set used if you look on EBay. Another keeper: Feynmann's Lectures on Physics. You can learn more physics from these volumes than from any six other physics or math courses. At 06:02 PM 5/6/2011, David Makin wrote: >Oh ! I guess |I'll have to read it properly at last (been meaning to for about 15 years). >Previously I've just gleaned one or two gems from it but not read the whole thing - if I remember correctly the most useful item for me from it (in the past) was the max/med/min method to get the approx, 3D magnitude., though knowing my memory nowadays I could have got that from somewhere else entirely ;) >Another visit to the library is in order. > >On 6 May 2011, at 21:03, Henry Baker wrote: > >> Knuth's Art of Computer Programming, among others. I would hope that it is accessible to UK "A" levels! >> >> At 12:37 PM 5/6/2011, David Makin wrote: >>> On 6 May 2011, at 19:53, Joerg Arndt wrote: >>> >>>> * Henry Baker <hbaker1(a)pipeline.com> [May 06. 2011 20:25]: >>>>> You know that the Casio performs an inverse lookup for sure? >>>>> >>>>> As an aside, there is an interesting trend in symbolic algebra >>>>> that's been going on for 35 years or so: doing less traditional >>>>> algebra & more numerical calculations. E.g., instead of using >>>>> bignums to compute the inverse of an integer determinant, compute >>>>> the determinant mod p for enough p's; using "black box programs" to >>>>> compute polynomials & computing the coefficients (if you really want >>>>> them) by interpolating enough point values. >>>> >>>> Note this method (compute mod enough coprime moduli, then CRT) is >>>> 1) exact >>>> 2) bloody fast >>> >>> Where can I find details of (compute mod enough coprime moduli, then CRT) ? (that someone with only UK "A" level math can understand) >>> (I mean for example - what's CRT here ?)

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[math-fun] semicircles, parabolae, catenaries
by Henry Baker 07 May '11

07 May '11

The ancient Roman's were really big into circular arches. I presume that arches were easy to draw with a string for a radius. Even the medieval churches continued to depend upon circular arcs: their windows with pointed tops used two circular arcs intersecting at the top. Somewhere along the line, parabolae became fashionable. Was it as late as the Brooklyn Bridge in the late 1800's ? Perhaps the requirement for thousands of railway bridges around the world forced the development of parabolic arches. The ancient Egyptians toyed with catenaries, but didn't really grasp the significant differences between catenaries & other kinds of arches, but they did dimly sense that they weren't the same. Catenaries seemed to finally hit the "big time" with the Sagrada FamÃlia in Barcelona in the early 20'th century, which was designed "upside down" using weights & ropes. http://en.wikipedia.org/wiki/Sagrada_Fam%C3%ADlia The ancient Romans were darn good engineers & I'm sure they didn't like to carry around more dirt & stone than was strictly necessary. Is anyone here aware of any deviation from circularity in any ancient Roman structures? Also, the Romans developed the use of concrete to a high art in the 2nd & 1st centures B.C.E. Did they ever utilize strengthening members -- e.g., "rebar"-like members?

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Re: [math-fun] Symbolic Calculation
by Bill Gosper 07 May '11

07 May '11

In the early 70s there was an effort at HP to put a subset of Macsyma on a handheld device. This was crushed by their consultant W. Kahan, then a vocal and effective opponent of computer algebra. In a talk at Symsac 72 at IBM Yorktown, he claimed the only economically legitimate use for computers was beancounting, and likened Macsyma's symbolic integrator to "The Fisher Space Pen--the pen that writes underwater!" Joel Moses, then director of Macsyma, was completely cowed. TI bought Derive from Albert Rich to try to leverage sales of their handheld calculators with a scheme where Derive on an instructor's PC would download over a cable to a student's handheld. When this flopped, they pulled the plug on Derive development, which they were never really interested in. This is a shame--Derive had fans worldwide, and was improving rapidly. You can't have too much computer algebra. --rwg apg>Yes, but MuMath *actually* performs symbolic calculation (or that's the impression I got, anyway), whereas the Casio must perform an inverse lookup to convert from a decimal approximation to a symbolic expression. Sincerely, Adam P. Goucher > Your Casio is not the first "symbolic hand calculator"; MuMath was really > the first: > > http://en.wikipedia.org/wiki/MuMATH > > At 09:46 AM 5/6/2011, Adam P. Goucher wrote: >>Today I received a Casio fx-85GT calculator as a present from a >>couple of my friends. It gives expressions in exact forms as an >>alternative to the ordinary decimal output. For example, enter: >> >>sqrt(8) >> >>and it will output: >> >>2*sqrt(2) >> >>Of course, I believed that the calculator was performing symbolic >>manipulation -- very impressive for a handheld scientific calculator! >> >>However, I then decided to test it: >> >>ln(640320^3+744)/sqrt(163) >> >>Instead of displaying the correct answer, 3.14159265..., a prominent >>image of the Greek letter pi was rendered! >> >>Hmm... >> >>I'm waiting for a version that prints "stop trying to test my symbolic >>calculation facilities with well-known near-miss identities!" >> >>On this topic, there is this comic: >> >>http://xkcd.com/217/ >> >>Sincerely, >> >> Adam P. Goucher >

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[math-fun] Convex polyominoes
by Allan Wechsler 06 May '11

06 May '11

On a previous occasion, I talked about enumerating "disklike" polyominoes, polyominoes which occurred as the intersection between a disk and a square lattice. Today I tried counting polyominoes which occurred as the intersection between any convex figure and the lattice. This implies that the convex hull of the polyomino (viewed as a set of lattice points) includes no additional lattice points. I call these polyominoes "convex", but apparently the term has already been taken by a different class, so I don't know what we should really call them. Starting from the monomino, the counts seem to be 1, 1, 2, 5, 10, 25. The sequence is not in OEIS, though I confess some uncertainty about the 25 convex hexominoes, and would like confirmation.

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Re: [math-fun] The Klein quartic
by Dan Asimov 05 May '11

05 May '11

James writes: << Also, the slides are not present in the Ferguson's chapter, so no image of the eightfold way sculpture is present in the book. Is there a version with images present on the internet? >> I think there are many images of that sculpture available on the Web. Try check Google Images for [ferguson klein quartic]. --Dan Sometimes the brain has a mind of its own.

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