May 2019 - mathfun@mailman.xmission.com

[math-fun] Impossible triangle raytracer
by Neil Bickford 11 May '19

11 May '19

Here's a neat visual trick: This shader program (by yx) renders a fully illuminated Reutersvard*/Penrose impossible triangle: https://www.shadertoy.com/view/ttXGWr All of the GLSL code used to generate the image on the left (minus Shadertoy's framework) is listed and editable on the right, and you can edit the code on the right to see what it does. Note how the geometry is locally consistent, and how the shadows between cubes, other cubes, and the ground all match up without any hard seams. Here's some context behind what makes this especially interesting. This program simulates global illumination using Monte Carlo path tracing: For each on- screen pixel, we trace a ray from a virtual camera into the scene, find the first place where the ray intersects the scene geometry, choose a new direction for the ray, trace the ray until it intersects the scene geometry again, and repeat until the ray reaches a light source (in this case, the sky) or reaches some maximum number of bounces (in this case, 4). (There's a lot I'm leaving out here, but this is sort of the basis behind it.) For geometry in R^3, this makes sense; finding ray-object intersections can sometimes be difficult, but are usually feasible. But here's the impossibility: how can you find the intersection of a ray and an object that can't exist? Left-click and drag over the image on the left (ideally while the shader is playing) to see a hint of how the shader works. *With a diaresis over the a: Reutersvärd --Neil Bickford (encoded partial solution (swaps a<->z): FromCharacterCode[219 - ToCharacterCode["zeviztv{tvlnvgib{levi{grnv"]].)

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[math-fun] Lots of Substitution Tilings
by Ed Pegg Jr 10 May '19

10 May '19

I've been parameterizing substitution tilings by their algebraic spaces in my program at http://demonstrations.wolfram.com/SubstitutionTilings/ I've put a bunch of pictures at https://community.wolfram.com/groups/-/m/t/1679969 Additions, Comments, Corrections are all welcome. Ed Pegg Jr

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Re: [math-fun] 170
by Dan Asimov 10 May '19

10 May '19

There are also a few fun facts here: https://en.wikipedia.org/wiki/170_(number)#In_mathematics , like that 170 is the smallest number whose phi(170) (= 64) and whose sum of divisors, sigma(170) (= 324), are perfect squares. —Dan

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[math-fun] The minimal polynomial for e^(i π/170) has a "balanced ternary" coefficient list.
by Bill Gosper 10 May '19

10 May '19

In[87]:= CoefficientList[MinimalPolynomial[E^(I π/170), x], x] Out[87]= {1, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1} (Presumably unexceptional.) In[88]:= FromDigits[%, x] (1 - x^2 + x^4 - x^6 + x^8 + x^34 - x^36 + x^38 - x^40 + x^42) // Expand Out[88]= 1 + x^170 —rwg

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[math-fun] Fwd: Re: 170
by Bill Gosper 10 May '19

10 May '19

Interestingly, In[78]:= Arg[(((Sqrt[5] + 1)*(-Sqrt[Sqrt[17] - 3]* Sqrt[4*Sqrt[17] - 2*Sqrt[2*Sqrt[17] + 34]] + Sqrt[2*Sqrt[17] + 34] + Sqrt[17] + 1)/ 64) - ((Sqrt[5 - Sqrt[5]]* Sqrt[Sqrt[Sqrt[17] - 3]* Sqrt[2*Sqrt[2*Sqrt[17] + 34] + 4*Sqrt[17]] - Sqrt[2*Sqrt[17] + 34] + Sqrt[17] + 17])/16))* I + (((Sqrt[5] + 1)* Sqrt[Sqrt[Sqrt[17] - 3]* Sqrt[2*Sqrt[2*Sqrt[17] + 34] + 4*Sqrt[17]] - Sqrt[2*Sqrt[17] + 34] + Sqrt[17] + 17])/(16* Sqrt[2])) + ((Sqrt[ 5 - Sqrt[5]]*(-Sqrt[Sqrt[17] - 3]* Sqrt[4*Sqrt[17] - 2*Sqrt[2*Sqrt[17] + 34]] + Sqrt[2*Sqrt[17] + 34] + Sqrt[17] + 1))/(32*Sqrt[2]))] // FullSimplify // tim During evaluation of In[78]:= 9.430501 [seconds], 2 [multiplicands] Out[78]=π/170 is a constructible angle, whereas In[79]:= UnitConvert[Quantity[π/180, "Radians"], "Degrees"] /.n_Integer :> IntegerName@n Out[79]= Quantity["one", º] is not. —rwg -------- Original Message -------- Subject: Re: [math-fun] 170 Date: 2019-05-08 08:50 From: James Propp <jamespropp(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> That's perfect! I see that in particular the Gray-code-based puzzle "The Brain Puzzler" takes 170 moves to solve: https://everything2.com/title/The+Brain+Puzzler Thanks, Michael! Jim On Wed, May 8, 2019 at 11:06 AM Michael Kleber <michael.kleber(a)gmail.com> wrote: > Membership in http://oeis.org/A000975 is definitely the way to go. The > description of the sequence makes it clear that it's just saying 170 > is 10101010 in binary, but the comments section offers plenty of > examples > of cool things about those alternating-0s-and-1s numbers. > > --Michael > > > On Wed, May 8, 2019 at 10:41 AM Mike Stay <metaweta(a)gmail.com> wrote: > > > http://oeis.org/search?q=170 > > > > On Wed, May 8, 2019 at 8:29 AM James Propp <jamespropp(a)gmail.com> wrote: > > > > > > I’ve agreed to host Carnival of Mathematics #170 next month, and I’m > > > wondering whether the number 170 has any especially nice properties > that > > > might deserve mention. > > > > > > The number 170 has its own Wikipedia page, but none of the mathematical > > > properties listed there appealed to me. > > > > > > I might say a little about writing numbers as sums of two squares, but > > I’d > > > prefer something a little more exciting. > > > > > > Jim Propp > > > ________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > -- > Forewarned is worth an octopus in the bush.

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[math-fun] Egan's _Dichronauts_
by Keith F. Lynch 10 May '19

10 May '19

I've mentioned Greg Egan's "Orthogonal" trilogy (_The Clockwork Rocket_, _The Eternal Flame_, and _The Arrows of Time_) in which the interval between two events isn't x^2 + y^2 + z^2 - t^2 as in our universe, but x^2 + y^2 + z^2 + t^2. In other words, there is no distinction between the time and space dimensions. The perceived time dimension is simply the direction of your worldline, which can point in literally any direction. As in our universe, people traveling at close to the same speed will almost perfectly agree on the direction of the time axis. He has since written _Dichronauts_, in which the interval between two events is x^2 + y^2 - z^2 - t^2. Roughly speaking, there are two space axes and two time axes. But people experience linear time, just as in our universe and in the "Orthogonal" universe. The main difference from our universe is that the three axes that aren't your personal time axis don't all act alike. For instance objects can't rotate around a vertical axis by 45 degrees or more. Smaller rotations result in the object elongating. Some people are born facing east, others facing west. To look in the opposite direction you need to bend way forwards or way backwards. There would be no point in facing north or south even if you could, since no light travels in those directions. Also, energy works rather differently than in our universe. See http://www.gregegan.net/DICHRONAUTS/00/DPDM.html for more information on the physics. There's a similar page for the physics of the Orthogonal trilogy, http://www.gregegan.net/ORTHOGONAL/00/PM.html I could be wrong, but I think that swapping all the signs makes no difference, so there are only the three possibilities, all the same signs (Orthogonal), three and one (our universe), and two and two (Dichronauts). Those novels are perfect for people who enjoyed learning physics so much that they wish they could do it all again.

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Re: [math-fun] Impossible triangle raytracer
by Dan Asimov 09 May '19

09 May '19

The impossible triangle most definitely exists, in that there is a solid object (many!) in space that looks exactly like the drawings of it (in one view). —Dan ----- how can you find the intersection of a ray and an object that can't exist? -----

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[math-fun] ROITMAN: Zen math will never add up.
by bradklee＠gmail.com 09 May '19

09 May '19

https://www.lionsroar.com/it-will-never-add-up/ Re: Kong-an about dog—does Buddha nature have a dog? And if yes, does Buddha nature have a Viola Cryana? I am worried, very worried, terribly worried, about what the latest ipbes Report on biodiversity implies to our understanding of this “Shunyata”. d/dx F(x)=-k*F(x) — BrAd.XR

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[math-fun] DGF --> OGF?
by Mike Stay 09 May '19

09 May '19

Can every suitably nice function A(s) be "inverted" to give an arithmetic function a(n) such that A(s) = sum_{n >= 1} a(n)/n^s? (E.g. by Perron's formula?) In other words, given ordinary generating functions A_i(x) = sum_{n >= 0} a^i_n x^n, I can combine them into a Dirichlet generating function A(s) = prod_{i >= 1} A_i(p_i^{-s}). My question is whether given a suitably nice but otherwise arbitrary function of s and think of it as a Dirichlet generating function, can I decompose it into a collection of ordinary generating functions? If so, what constitutes "suitably nice"? Merely convergence for Re(s) > sigma? -- Mike Stay - metaweta(a)gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com

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Re: [math-fun] Integers of the form K*K +K*L + L*L
by Keith F. Lynch 09 May '19

09 May '19

Dan Asimov <dasimov(a)earthlink.net> wrote: > I made a table of the Loeschian numbers Lo(K,L) for K >= L. > and then tried to *connect each one with the next largest one* > with a line segment on paper. > But it was hard to see any pattern in how they connect (below). > So the question boils down to: > Is there a concise formula for the "next largest number" in a set > like Lo = {Lo(K,L) | K, L in Z} ??? I.e., f(0) = 1, f(1) = 3, f(3) > = 4, .... I wouldn't think so. There are lots of examples of two-dimensional integer sequences, aka tables, where there's a simple pattern to every row and to every column, but no pattern to all the numbers sorted in order. Examples include: * The multiplication table (excluding the 1s row and column). The entries are the composite numbers (A002808), which are of course exactly as patternless as the primes. * Power numbers (A001597), i.e. the union of the squares, cubes, etc. Only in 2002 was it even proven that 8 and 9 are the only adjacent members of the set. * Numbers in Pascal's triangle (other than the first two rows and columns), aka the binomial coefficients (A006987). * Nontrivial polygonal or figurate numbers, i.e. the union of the triangular numbers (other than 1 and 3), the squares (other than 1 and 4), the pentagonal numbers (other than 1 and 5), etc. (A090466). * Numbers that are a power of 2 plus a power of 3 (A004050). Similarly with any other two primes. * Numbers that are a power of 2 times a power of 3 (A003586). Similarly with any other two primes. Can anyone think of a nontrivial example of such a table sequence that *does* have a simple formula for finding the next one? I can't. (By "trivial," I think I just mean tables that contain all numbers or that contain only a small number of distinct numbers.)

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