May 2019 - mathfun@mailman.xmission.com

[math-fun] Find a polyhedron with all vertices on its circumsphere, but without all faces touching its insphere
by Dan Asimov 27 May '19

27 May '19

There is one with 6 faces, anyhow. (Far below if you want to try to find one without spoilage.) —Dan Let T be a skinny isosceles triangle with apex A. Let the polyhedron P be the result of truncating [the product T x [0, 1] of T with the unit interval] by cutting off a neighborhood N of {A} x [0, 1], thereby creating a new face — a skinny rectangle R. Then P has all 8 of its vertices on its circumsphere C. But clearly C would not touch the rectangle R if N is small enough. —Dan

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[math-fun] Loose dissection with pentominos
by Scott Kim 27 May '19

27 May '19

Here's a new type of polyomino problem that came up as a practical problem. Fit the twelve pentominoes, each made of 1" squares, on a standard 8.5" by 11" sheet, so there is at least a 1/4" border between every pentomino and its neighbors, and the edge of the paper. In other words, pentominoes never touch each other, or the border of the paper. Note that given the spacing constraint, no pentomino is allowed to be inside the concave space of the U pentomino. It's easy to fit 11 pieces, but 12 is challenging. It should be fairly straightforward to write a solver, but the loose fitting constraint does make things trickier. I did a find a solution by hand...which cannot be compressed to fit on a smaller sheet. The next question is what is the smallest area that can fit all 12 pentominoes with quarter inch spacing?

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[math-fun] about bugs in mathematica and pari-gp
by Simon Plouffe 27 May '19

27 May '19

Hello, here is a strange bug, type Prime[8200000000] or Prime[93000000000] in mathematica and your machine will hang forever.... and by a bizarre coincidence, the same thing happens with pari-gp with the value prime(8200000000), I could not test 93 000 000 000 because it is out of reach for pari-gp (I think). Does anybody has any idea why the programs are bugging on these particular values ? Best regards and have a nice day. Simon Plouffe

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[math-fun] Fwd: Re: Fwd: about bugs in mathematica and pari-gp
by Bill Gosper 27 May '19

27 May '19

Subject: Re: [math-fun] Fwd: about bugs in mathematica and pari-gp Date: 2019-05-26 07:21 From: Victor Miller <victorsmiller(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> I think that macsyma is wrong about 93 * 10^9: time ./primecount -n 93000000000 2560462462787 real 0m0.143s user 0m0.106s sys 0m0.004s time ./primecount 2560462462787 93000000000 real 0m0.106s user 0m0.101s sys 0m0.004s time ./primecount 2560462462786 92999999999 real 0m0.106s user 0m0.101s sys 0m0.004s rwg> ? It agrees with (somewhat cajoled) Mathematica: In[159]:= NextPrime@Prime[9299999999] // tim During evaluation of In[159]:= 0.56919,0 Out[159]= 233746743437 The system you are quoting invites confusion of a function with its inverse. I now recall (in the mid 80s) Macsyma computing prime(10^13). Took most of a weekend. I gave the printout (including a garbage collection message) to a class clown type and he surprised me by taking it to school and getting extra credit. —rwg On Sun, May 26, 2019 at 3:54 AM Bill Gosper <billgosper(a)gmail.com> wrote: > My old Macsyma gets > > (c71) PRIME(8200000000); > > (d71) 205021987301 > > (c72) PRIME(9300000000); > > (d72) 233746743437 > > SPlouffe>Hello, > > here is a strange bug, > > type Prime[8200000000] or Prime[93000000000] > in mathematica and your machine will hang forever.... > > rwg> No, mine is powered by PG&E. > > and by a bizarre coincidence, the same thing happens > with pari-gp with the value prime(8200000000), > I could not test 93 000 000 000 because it is out of reach > for pari-gp (I think). > > Does anybody has any idea why the programs are bugging on > these particular values ? > > rwg>NSA back door. > > >In[156]:= Prime[8200000001] // tim > > >During evaluation of In[156]:= 0.539572 (secs) > > >Out[156]= 205021987381 > >Macsyma took minutes. > > Best regards and have a nice day. > Simon Plouffe

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Re: [math-fun] 2300 yr old news
by Dan Asimov 27 May '19

27 May '19

I'm not sure how polar reciprocation determines the exact size of a dual polyhedron. —Dan Mike Stay wrote: ----- On Sun, May 26, 2019 at 6:10 PM Dan Asimov <dasimov(a)earthlink.net> wrote: > > I've long been confused about the exact metric definition of > "dual polyhedron". Sure, define one "new vertex" for each old > face, and one "new edge" for each pair of old faces that share > an edge — that gives the dual combinatorially. But as for the > exact shape & size, I'm deeply confused. (We're in R^3.) Polar reciprocation: https://math.wikia.org/wiki/Dual_polyhedron -----

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Re: [math-fun] 2300 yr old news
by Dan Asimov 27 May '19

27 May '19

I've long been confused about the exact metric definition of "dual polyhedron". Sure, define one "new vertex" for each old face, and one "new edge" for each pair of old faces that share an edge — that gives the dual combinatorially. But as for the exact shape & size, I'm deeply confused. (We're in R^3.) But nevertheless, Definition of "round polyhedron": A convex polyhedron P in R^3 is "round" if ----- a) all vertices of P lie on its circumsphere, and b) all faces of P touch its insphere. ----- Which convex polyhedra are round? (E.g., Platonic solids are round.) Vert(P) determines P uniquely, and Vert(P) is a finite subset Vert(P) = {z_j in C u {oo} | 1 <= j <= n} of the Riemann sphere S^2 C u {oo} on which much complex analysis takes place. Question: Suppose Vert(P) is a subset of the unit sphere S^2. ----- Does it follow that the insphere touches all faces? Or equivalently, Does it follow that P is round? Or vice versa. —Dan Let R = inradius, S = circumradius, so R < S as of course it should be (since alphanumeric inequality inconsistencies are rarely mnemonic). Now suppose again we sandwich a polyhedron P between its insphere and its circumsphere. Bill Gosper wrote: ----- ... If a Platonic solid is scaled to have the same inradius as its dual, then they will also have the same circumradius. ... -----

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[math-fun] 2300 yr old news
by Bill Gosper 26 May '19

26 May '19

This just in: If a regular dodecahedron and icosahedron are inscribed in the same sphere, the ratios of their areas and volumes are both √(√5 GoldenRatio/3). (If true, does anyone have a moldy-oldie reference?) —rwg

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[math-fun] Poncelet's porism overpush?
by Bill Gosper 26 May '19

26 May '19

https://cp4space.wordpress.com/2014/04/19/poncelets-porism-the-socratic-dia… APG: By the way, the result can be generalised to ellipses instead of circles by applying a projective transformation, although of course this would be beyond Ancient Greek mathematics so I decided against including it in the Socratic dialogue. *Arbitrary* (cleanly nested) ellipses? Don't they both need to be circles under the same projection? Not according to https://www.youtube.com/watch?v=Mf46H0kwXkU . (I'd be more convinced by higher eccentricity illustrations.) Also, according to http://quadrivium.info/MathInt/Notes/Apollonius.pdf , (tnx Brad Klee), Apollonius was closer to projective satori than I would have guessed. —rwg

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[math-fun] Fwd: about bugs in mathematica and pari-gp
by Bill Gosper 26 May '19

26 May '19

My old Macsyma gets (c71) PRIME(8200000000); (d71) 205021987301 (c72) PRIME(9300000000); (d72) 233746743437 SPlouffe>Hello, here is a strange bug, type Prime[8200000000] or Prime[93000000000] in mathematica and your machine will hang forever.... rwg> No, mine is powered by PG&E. and by a bizarre coincidence, the same thing happens with pari-gp with the value prime(8200000000), I could not test 93 000 000 000 because it is out of reach for pari-gp (I think). Does anybody has any idea why the programs are bugging on these particular values ? rwg>NSA back door. >In[156]:= Prime[8200000001] // tim >During evaluation of In[156]:= 0.539572 (secs) >Out[156]= 205021987381 >Macsyma took minutes. Best regards and have a nice day. Simon Plouffe ___________________________

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[math-fun] Fwd: Re: "Solid" geometry
by Bill Gosper 21 May '19

21 May '19

Thanks wfl, brad klee, etc for all the (resoundingly affirmative) responses. When you all work on such a pure math-y question, do you feel a twinge of irrelevance? I mean, who sections elliptical cones? Dishware from Dong Lai Shun Restaurant in Mountain View: https://s3-media4.fl.yelpcdn.com/bphoto/We4TAw4FEsYATsaBogX9Xw/o.jpg The symmetry of the elongated ellipse from slicing an already elliptical cone is perhaps even more surprising than in the circular conic construction, with no Dandelin spheres to save you. —rwg -------- Original Message -------- Subject: Re: [math-fun] "Solid" geometry Date: 2019-05-20 17:54 From: Dan Asimov <dasimov(a)earthlink.net> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: Dan Asimov <dasimov(a)earthlink.net>, math-fun <math-fun(a)mailman.xmission.com> If we apply a linear transformation to the elliptical cone to get a round one, the sectioning plane goes to another plane cutting a conic section, so the inverse transformation shows the answer is Yes, since linear images of conic sections are still conic. —Dan ----- Is a section of an elliptical cone a Conic Section? ----- _______________________________________________

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