September 2012 - mathfun@mailman.xmission.com

[math-fun] Tour the Solar System from home
by Henry Baker 12 Sep '12

12 Sep '12

FYI -- NASA Java (not Javascript) browser app that allows you to tour the solar system, including the location of all the NASA missions. It takes a while to download the info to display, but it's pretty cool. http://solarsystem.nasa.gov/

1 0

[math-fun] Strogatz on the topology of fingerprints
by Henry Baker 11 Sep '12

11 Sep '12

FYI -- Very nice article: http://opinionator.blogs.nytimes.com/2012/09/10/singular-sensations/?hp

2 1

Re: [math-fun] Symbolic Functions
by Henry Baker 11 Sep '12

11 Sep '12

You might find the recent exchange about differentiation on the Maxima mailing list interesting. I believe that this definition extension made it through all of the Maxima test cases and has been incorporated into the most recent version of Maxima. At 11:12 AM 2/16/2012, Stavros Macrakis wrote: >Actually, a little reflection makes it clear that diff(f(x),g(x)) should not be an error or a noun form, because in fact it is perfectly well-defined and meaningful: > > diff(A,B) == ev(diff(A)/diff(B),del) > >For example, > > diff(sin(x),x) == cos(x)*del(x)/del(x) == cos(x) > << diff(...,V) reduces to normal differentiation > diff(sin(x),cos(x)) == cos(x)*del(x)/(-sin(x)*del(x)) == -cos(x)/sin(x) > << in diff(f(x),g(x)), del(x)/del(x) => 1 > diff(x*y,x^2+y^2) == (x*del(y)+y*del(x)) / (2*y*del(y) + 2*x*del(x)) > << consistent handling of multiple-variable case > >How useful is this? I'll leave that to others.... > >Right now, Maxima doesn't do much of interest with del(...) -- not even giving an error for del(1) -- but it does do ev(del(x^2),del) == diff(x^2) == 2*x*del(x). You can even try ratsubst('diff(x,y),del(x)/del(y),...) on del expressions and get things that look reasonable (and useful?).... > >One problem: diff(del(x)) => 'diff(del(x),x)*del(x), which is I suppose correct, but ugly and hard to work with. Can be deferred for now. > >Any objections to this definition of diff with a second argument which is an expression? > > -s

1 0

[math-fun] Symbolic Functions
by quad 11 Sep '12

11 Sep '12

Hey all: I thought I might share a little thing I've been hacking on called "symbolic functions", since Henry Baker recently told math-fun about chebfun (IIRC). First, define a pythag/distance function because we'll need it later: > (defun pythag (x y) (sqrt (+ (* x x) (* y y)))) PYTHAG Now we define a symbolic function UNIT-SINE: > (define-symbolic-function unit-sine (x) (sin (* (* 2 pi) x))) UNIT-SINE Of course this is thew sine function with a period of 1. We can evaluate it as normal: > (unit-sine 1.0) -2.4492935982947064d-16 ;; approximately 0.0 It can be used in every way a traditional Lisp function can be used. Mapped, reduced, composed, disposed, etc. But it has a special property. We can do cool operations with it. For example, we can differentiate it: > (pythag (unit-sine 0.75) (funcall (differentiate #'unit-sine) 0.75)) 1.0d0 which is to be expected from sin^2 + cos^2 = 1. But of course, maybe I'm just using the difference quotient. But: > (loop :for i :from 0.0 :to 1.0 :by 0.1 :collect (- (* 2 pi (cos (* 2 pi i))) (funcall (differentiate #'unit-sine) i))) (0.0d0 0.0d0 0.0d0 0.0d0 0.0d0 0.0d0 0.0d0 0.0d0 0.0d0 0.0d0) Surely with something like the difference quotient, we'd get some numerical error. Of course, there's a trick, and it's what I call "symbolic functions". They're likely not a new idea at all, but I've never seen them like this. A symbolic function is a regular ol' executable function equipped with its symbolic form under the hood. It has the appearance of a regular function and can be used in every way a regular function can be used, so it's a transparent thing. But evaluating one by itself exposes the information: > #'unit-sine #<X -> (SIN (* (* 2 PI) X))> Like I said, this is a regular, *COMPILED* function. When you provide a value for X, it is not re-interpreted to get a numerical value. It's calling the compiled form of that. However, numerical evaluation isn't the interesting thing. We can also produce new kinds of function using SLAMBDA (Symbolic LAMBDA): > (slambda (x) (* x x)) #<X -> (* X X)> which work exactly like LAMBDA functions. What happens if we differentiate? > (differentiate (slambda (x) (* x x))) #<X -> (+ (* 1 X) (* X 1))> We get a brand new SLAMBDA, recompiled on-the-fly (so a poor man's JIT!). DIFFERENTIATE is really just a classical (and very naive) symbolic differentiator: https://bitbucket.org/tarballs_are_good/symbolic-function/raw/fa394a19d24e/… Also notice that DIFFERENTIATE never required any "variable" parameter. Since differentiation occurs on *functions* and not *expressions*, we know what we are differentiating with respect to. (This is in my opinion the right way to differentiate.) What does differentiating UNIT-SINE produce? > (differentiate #'unit-sine) #<X -> (* (+ (* (+ (* 0 PI) (* 2 0)) X) (* (* 2 PI) 1)) (COS (* (* 2 PI) X)))> Of course, that's very ugly, but when compiled, that ugliness gets washed away through constant folding and friends. However (!), it would not be difficult to write a simple symbolic simplifier for symbolic functions either. Anyway, the code is here: Main Meat: https://bitbucket.org/tarballs_are_good/symbolic-function/raw/fa394a19d24e/… Repository: https://bitbucket.org/tarballs_are_good/symbolic-function There are some outstanding "problems" really. For example, it's not immediately clear to me how to symbolically manipulate constructs like LET and friends in a mathematical fashion, or deal with symbolic functions that stray away from "symbolicness" (like functions which use mutation, etc.) But I think the biggest advantage is that you get truly fast numerical code which is transparently done with symbolic operations, sort of like chebfun (except chebfun is more numeric than symbolic: it just has function approximations). If you're interested in running the code, you need a dependency (the metaobject protocol), and you need to know how to use ASDF (or Quicklisp). (I'd be more than happy to describe those steps.) Any comments, history, suggestions, fan fiction, love letters, death threats? Cheers, Robert Smith

3 3

[math-fun] Magic: The Gathering is Turing Complete
by Paul Reiners 11 Sep '12

11 Sep '12

http://www.toothycat.net/~hologram/Turing/

1 0

[math-fun] Squared rectangles from Platonic and Archimedean graphs.
by Stuart Anderson 11 Sep '12

11 Sep '12

In his book 'More Mathematical Puzzles and Diversions' Martin Gardner posed the following puzzle "The electrical puzzle involves the network depicted in Figure 4. If each edge of the cube has a resistance of one ohm, what is the resistance of the entire structure when current flows from A to B? " (a copy of the relevant chapter 'The Five Platonic Solids' is available here; http://assets.cambridge.org/97805217/56105/excerpt/9780521756105_excerpt.pdf) In the same book, in a different chapter titled 'Squaring the Square', written by W.T. Tutte is an explanation of how the square was squared using electrical network theory. Tutte mentions "it was a pleasing recreation to work out perfect rectangles corresponding to networks with a high degree of symmetry", this activity and the ensuing analysis of network symmetry led ultimately to the solution of the problem of squaring the square. However we never heard anything further about the squared rectangles of networks with a high degree of symmetry, except for some tantalizing hints from time to time , eg http://www.math.indiana.edu/gallery/color.phtml So I decided to create the squared rectangles of some symmetrical networks. One way of creating a squared rectangle is to select a polyhedral graph, with unit resistances in each edge, choose one of the edges of the graph, replace that edge with an electromotive force and calculate the node voltages and edge currents to give the positions and sizes of squares. The obvious place to start is with regular polyhedra such as the Platonic Solids, as they are highly symmetrical, and there is really only one kind of edge under symmetry, so they (and their duals) will produce only one squared rectangle each, being the same squared rectangle for graph and dual. Following wikipedia; "In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged. Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive) and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral; their duals are isohedral and isotoxal, but not isogonal. Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. For instance, the truncated icosahedron (the familiar soccerball) has two types of edges: hexagon-hexagon and hexagon-pentagon, and it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge." Perhaps we can call regular polyhedra which are not isotoxal, 2-isotoxal or 3-isotoxal depending on the number of edge transitive classes that the edges belong to. I have been able to work out the squared rectangles belonging to the Platonic solids, for these, there are only 3 rectangles in total for all 5 polyhedra. They are not perfect (meaning the squares are not all different sizes). In the terminology of squared rectangles, they are often imperfect and compound. Due to symmetry some of the square sizes come out as zero, reducing the number of squares in the rectangles. One edge is the battery edge and does not appear in any tiling, unless one adds it back in to create a trivial squared cylinder. Next I tried the Archimedean solids. I took the graphs from here ; http://mathworld.wolfram.com/ArchimedeanGraph.html The squared rectangles I created are here; http://www.squaring.net/sq/sr/platonic--archimedean-sq-rects.pdf The only one I havnt worked out yet is the Great Rhombicosidodecahedron, my C++ software is failing due to numerical precision errors. I have a Mathematica program which should calculate the square sizes correctly, this version calculates Duijvestijn's squared square, I will need to insert the graph of the Great Rhombicosidodecahedron in place of Duijvestijn's graph; Text[" Incidence matrix Aa[i,j] -> graph G(i,j); i nodes, j edges Aa[i,j]={ 1, if edge j is directed towards node i, Aa[i,j]={-1, if edge j is directed from node i, Aa[i,j]={ 0, if edge j is not incident to node i. "] a = { {-1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1}, { 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, { 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, { 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, 0, 1, 1, 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, -1, 0, 0, 0, 0, -1, 0}, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, -1, 0, -1, 0, 0}, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0}, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1}}; A = Drop[Aa, {1, 1}] ; MatrixForm[A] Text["Multiply A by its transpose, Transpose[A] to get the Laplacian \ (or Kirchhoff) matrix K"] K = A.Transpose[A]; MatrixForm[K] Text["Perform LU decomposition on K, to factor it into an upper (u) \ and a lower (l) triangular matrix and diagonal matrix"] {lu, p, c} = LUDecomposition[K]; det = Tr[lu, Times] Text["The V matrix is equal to determinant* Inverse matrix of K and \ gives the (full)voltages for nodes satisfying Kirchhoff's 1st law"] V = det*Inverse[K]; MatrixForm[V] Text["Dividing rows or columns by GCD , we get reduced voltages - \ this step is not necessary"] W = Apply[GCD, V, {1}] Y = MatrixForm[V/W] Text["The triple matrix product A.V.A gives the edge full current \ solutions matrix F"] F = Transpose[A].V.A; MatrixForm[F] Text["Dividing rows or columns of F by the GCD of each row we get a \ reduction vector, needed to reduce the full currents so we get \ tilings in smallest solutions"] R = Apply[GCD, F, {1}] Text["Dividing the full currents matrix F by reduction vector R gives \ the reduced currents matrix B, which provides all the solutions in \ terms of the sizes of the edge currents. Edge currents become square \ sizes in the squared square. The last row of the B matrix is \ Duijvestijn's 112 squared square, with all the square sizes in \ reduced (GCD=1) size. The other non-diagonal rows are squared \ rectangle element sizes, and the width of the rectangle is the \ diagonal entry in each row, once we know the width we can work out \ the height from the determinant. In the case of a squared square w = \ h = diagonal entry."] Text["Now we substitute the values of the edge currents from matrix \ B, one row at a time, into the edges of the graph we started with, \ this gives us the 'Smith Diagram' of the squared square, and we can \ now easily draw or construct the squared square."] B = MatrixForm[F/R]

1 0

[math-fun] Possible Proof of ABC Conjecture
by Ray Tayek 11 Sep '12

11 Sep '12

http://science.slashdot.org/story/12/09/10/226217/possible-proof-of-abc-con… Posted by <http://unknownlamer.org/>Unknown Lamer on Monday September 10, @06:54PM from the lord-of-the-proof dept. <http://blog.submeta.org/>submeta writes "Shinichi Mochizuki of Kyoto University has released a <http://www.kurims.kyoto-u.ac.jp/%7Emotizuki/Inter-universal%20Teichmuller%2…>paper which claims to prove the decades-old <http://en.wikipedia.org/wiki/Abc_conjecture>ABC conjecture, which involves the <http://www.nature.com/news/proof-claimed-for-deep-connection-between-primes…>relationship between prime numbers, addition, and multiplication. His solution involves thinking of numbers not as members of sets (the standard interpretation), but instead as objects which exist in '<http://www.newscientist.com/article/dn22256-fiendish-abc-proof-heralds-new-…>new, conceptual universes.' As one would expect, the proof is extremely dense and difficult to understand, even for experts in the field, so it may take a while to verify. However, Mochizuki has a strong reputation, so this is likely to get attention. Proof of the conjecture could potentially lead to a revolution in number theory, including a greatly simplified proof of <http://en.wikipedia.org/wiki/Fermat%27s_last_theorem>Fermat's Last Theorem." --- co-chair http://ocjug.org/

1 0

[math-fun] Khan Academy fun
by Mike Stay 10 Sep '12

10 Sep '12

Khan Academy recently released a JavaScript + Processing.js integrated development environment for teaching kids to program. It's a lot of fun. Here are a bunch of very simple demos I've written in the past week or so. Explore the limit sets of the Mandelbrot iteration http://www.khanacademy.org/cs/mandelbrot-spirals/1026049545 (without the map) http://www.khanacademy.org/cs/mandelbrot-spirals-2/1030775610 (with the map) A fractal tree that grows and changes shape http://www.khanacademy.org/cs/tree/1029209629 Playing with color and shape http://www.khanacademy.org/cs/traces/1029027232 http://www.khanacademy.org/cs/traces-2/1029086864 Doubling-angle spirals with turtle graphics http://www.khanacademy.org/cs/spirals/1023512142 -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

2 2

[math-fun] Circle question
by David Wilson 10 Sep '12

10 Sep '12

Consider a circle whose diameter lies on a fixed axis line passing obliquely through a plane (not perpendicular to the plane) Now rotate the circle about the axis line subject to the following conditions: - The plane position is fixed. - The circle radius is fixed. - The axis is parallel to its initial position. - The circle is tangent to the plane, on the same side of the plane as its initial position. Can the circle be made to move without slippage along the plane? What is the locus of the point of circle-plane contact?

4 7

[math-fun] Counting English Number Expression Lengths
by Hans Havermann 10 Sep '12

10 Sep '12

A little combinatorics exercise for those of you who still like to count: The 21 3k-powers-of-ten words (thousand, million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, undecillion, duodecillion, tredecillion, quattuordecillion, quindecillion, sexdecillion, septendecillion, octodecillion, novemdecillion, vigintillion) may be combined with the 999 positive-integers-less-than-1000 words (let's call them n-words) to express all numbers < 10^66 (U.S. notation). Any given power word may be used zero or one time, but every power word used once must be accompanied by a single n-word. In addition, we have the option of adding one extra single n-word. Finally, if no power words at all are used, then we must choose a single n-word. Let the length of an English number expression be the total number of letters in its written form; i.e., separators (spaces, commas) and conjunctives (hyphens, the word 'and') are ignored. The lengths of the above power words are: 8, 7, 7, 8, 11, 11, 10, 10, 9, 9, 9, 11, 12, 12, 17, 13, 12, 15, 13, 14, 12. Tallying these, we get lengths of 7-17 occurring with a frequency (respectively) of: 2, 2, 3, 2, 3, 4, 2, 1, 1, 0, 1. Tallying the n-words, we get lengths of 3-24 occurring with a frequency (respectively) of: 4, 3, 6, 6, 3, 13, 22, 27, 18, 6, 12, 21, 39, 45, 45, 66, 114, 177, 183, 126, 54, 9. So, for example, the longest number expression < 10^66 should include every power word (with a total of 230 letters) plus 22 instances of a 24-letter n-word: 22*24 + 230 = 758. How many 758-letter number expressions < 10^66 are there? 9^22. Correct? How many 42-letter number expressions < 10^66 are there?

1 0