September 2013 - mathfun@mailman.xmission.com

[math-fun] near regular dodecahedron with integer vertices
by James Buddenhagen 09 Sep '13

09 Sep '13

A while back the fact that no regular dodecahedron can be placed in R^3 with all vertices having integer coordinates came up here. Here, just for fun, is a near miss: The faces are congruent planar non-regular pentagons each with face angles: [108.0000197, 107.9999984, 107.9999918, 107.9999918, 107.9999984], as compared to a regular pentagon with all face angles 108 degrees. This irregular dodecahedron has vertices with integer x,y,z coordinates: 2550409 2550409 2550409 4126648 0 1576240 1576240 4126648 0 2550409 2550409 -2550409 4126648 0 -1576240 0 1576240 4126648 0 -1576240 4126648 2550409 -2550409 2550409 -1576240 4126648 0 -2550409 2550409 2550409 0 1576240 -4126648 0 -1576240 -4126648 2550409 -2550409 -2550409 -2550409 2550409 -2550409 -2550409 -2550409 2550409 -4126648 0 1576240 -2550409 -2550409 -2550409 -1576240 -4126648 0 1576240 -4126648 0 -4126648 0 -1576240 and is the convex hull of those points.

2 2

[math-fun] Incomparable integers
by James Propp 04 Sep '13

04 Sep '13

Are there (positive) integers m,n definable in PA whose existence is provable in PA such that m \geq n and n \geq m are undecidable in PA? (I think that's the right way to ask the question I have in mind, but if my wording evinces misunderstanding of foundational issues, please enlighten me!) What if we replace PA by a stronger theory? The underlying intuition is that if there are incomprehensibly big numbers, at some point even comparative notions of bigness should start to fail us, so that, in a certain sense, the well-ordering of the natural numbers should become problematical. But my intuitions may be totally off-base... Jim Propp

6 11

[math-fun] Correction Re: exp(Pi*sqrt(d)) [was one-ninthish]
by elkies＠math.harvard.edu 04 Sep '13

04 Sep '13

Correction: I wrote << you can also use 163/9, 67/3, and (barely) 43/3. That is, 163/9, 67/9, and (barely) 43/9, i.e. the cube roots of exp(Pi*sqrt(d)) for d=163, 67, and 43 -- as you might have guessed from << [I've suggested that a CM mile should be exactly exp(Pi*sqrt(67)/3) feet long. :-)] NDE

1 0

Re: [math-fun] another slight design flaw... skyscraper cooks cars
by Bill Gosper 03 Sep '13

03 Sep '13

Google Las Vegas death ray . Ah, Plywood Ranch. I can tell you their phone number in the late 60s: 617 484 3232, kindly reassigned by New England Failtone to Jack Holloway, Stew Nelson, and me when we lived in Belmont. *Most* of our calls were for Plywood Ranch. "Plywood Raunch." "Yes, we make naughty cutouts." "Where is your brothel located?" --rwg VMiller>This makes me think of the notorious Hancock Tower in Boston. It was being built in the early 1970s, and had many problem (see below for an excerpt from Wikipedia). I was teaching one of my classes in a building only a block from it when it was a 70 story building clad in black painted plywood. Boy was it ugly. What was even worse, and what Wikipedia doesn't mention, is that when the new glass was installed, it was much more reflective than the old glass, and for a few months of the year it would blind all of the morning commuters driving in on Storrow Drive. Victor Inventing a way to use the blue reflective glass< http://en.wikipedia.org/wiki/Reflective_glass>in a steel tower came at a high price. Entire 4' × 11', 500-lb (1.2 × 3.4 m, 227 kg) windowpanes detached from the building and crashed to the sidewalk hundreds of feet below. Police had to close off surrounding streets whenever winds reached 45 mph (72 km/h). According to the *The Boston Globe < http://en.wikipedia.org/wiki/The_Boston_Globe>*, a scale model of the entire Back Bay was built in MIT<http://en.wikipedia.org/wiki/MIT>'s Wright Brothers wind tunnel <http://en.wikipedia.org/wiki/Wind_tunnel> to identify the problem. The research raised questions about the structural integrity of the entire building (due to unanticipated twisting of the structure), but it did not account for the loss of the glass panels. An independent laboratory eventually confirmed that the failure of the glass was due to oscillations and repeated thermal stresses< http://en.wikipedia.org/wiki/Thermal_stress>caused by the expansion and contraction of the air between the inner and outer glass panels which formed each window; the resilient bonding between the inner glass, reflective material, and outer glass was so stiff that it was transmitting the force to the outer glass (instead of absorbing it), thus causing the glass to fail.[5]< http://en.wikipedia.org/wiki/John_Hancock_Tower#cite_note-levy-5> In October 1973, I.M. Pei & Partners announced that all 10,344 window panes would each be replaced by a single pane, heat-treated panel at a total cost between $5 million and $7 million.[5]< http://en.wikipedia.org/wiki/John_Hancock_Tower#cite_note-levy-5>Approximately 5,000 of the original glass panes were removed intact, and were later reused by artists <http://en.wikipedia.org/wiki/Glass_art>.[6]< http://en.wikipedia.org/wiki/John_Hancock_Tower#cite_note-NYT1988-6> During the many months it took to diagnose and repair the building, sheets of plywood <http://en.wikipedia.org/wiki/Plywood> replaced many of the missing glass windows of the building, earning it the nicknames *Plywood Palace* and *Plywood Ranch* (the same name as a local lumber yard< http://en.wikipedia.org/wiki/Lumber_yard>chain at the time). The joke was that the Hancock Tower was *the world's tallest plywood building*. The building's upper-floor occupants suffered from motion sickness< http://en.wikipedia.org/wiki/Motion_sickness>when the building swayed in the wind. To stabilize the movement, contractors installed a tuned mass damper<http://en.wikipedia.org/wiki/Tuned_mass_damper>on the 58th floor. [7] <http://en.wikipedia.org/wiki/John_Hancock_Tower#cite_note-campbell-7>As described by Robert Campbell, architecture critic for *The Boston Globe*: Two 300-ton weights sit at opposite ends of the 58th floor of the Hancock. Each weight is a box of steel, filled with lead, 17 ft (5.2 m) square by 3 ft (0.91 m) high. Each weight rests on a steel plate. The plate is covered with lubricant so the weight is free to slide. But the weight is attached to the steel frame of the building by means of springs and shock absorbers. When the Hancock sways, the weight tends to remain still, allowing the floor to slide underneath it. Then, as the springs and shocks take hold, they begin to tug the building back. The effect is like that of a gyroscope, stabilizing the tower. The reason there are two weights, instead of one, is so they can tug in opposite directions when the building twists. The cost of the damper was $3 million. The dampers are free to move a few feet relative to the floor. According to Campbell, engineers discovered that—despite the mass damper—the building could have fallen over under a certain kind of wind loading. The structure was assessed as more unstable on its narrow sides than on the big flat sides. Some 1,500 tons of diagonal steel bracing, costing $5 million, were added to prevent such an event.[7]< http://en.wikipedia.org/wiki/John_Hancock_Tower#cite_note-campbell-7> On Tue, Sep 3, 2013 at 1:31 PM, Warren D Smith <warren.wds(a)gmail.com> wrote: http://www.bbc.co.uk/news/magazine-23944679

2 1

Re: [math-fun] Incomparable integers
by Henry Baker 03 Sep '13

03 Sep '13

You might contact Jeanne Ferrante, who wrote the following with Charles Rackoff: The Computational Complexity of Logical Theories. Lecture Notes in Mathematics 718 Pressburger Arithmetic is pretty well behaved, and some fragments are now used for proving theorems about index arithmetic in 'for' loops in compilers. http://en.wikipedia.org/wiki/Presburger_arithmetic You might also look at "S1S" and "S2S" as possible inspirations: From: http://www.cs.tau.ac.il/~rabinoa/seminar11/s2s.pdf Size: 1.2 MB (1,239,226 bytes) At 11:05 AM 9/3/2013, James Propp wrote: >Are there (positive) integers m,n definable in PA whose existence is >provable in PA such that m \geq n and n \geq m are undecidable in PA? (I >think that's the right way to ask the question I have in mind, but if my >wording evinces misunderstanding of foundational issues, please enlighten >me!) > >What if we replace PA by a stronger theory? > >The underlying intuition is that if there are incomprehensibly big numbers, >at some point even comparative notions of bigness should start to fail us, >so that, in a certain sense, the well-ordering of the natural numbers >should become problematical. > >But my intuitions may be totally off-base... > >Jim Propp

1 0

Re: [math-fun] another slight design flaw... skyscraper cooks cars
by Henry Baker 03 Sep '13

03 Sep '13

'In 2003, the opening of the Walt Disney Concert Hall in Los Angeles, designed by architect Frank Gehry, hit a similar stumbling block. '"The building was clad from head to toe, right down to the pavement, in stainless steel panels, and they would send the sun dazzling across the sidewalks to hotspots where people were. It was measured up to 60C (140F). '"Local people living there complained they were having to crank their air conditioning up to maximum to cool things down," he says. 'Blinding glare also affected drivers passing the building. 'After computer models and sensor equipment identified the panels causing the problem, *** they were sanded down *** to break up the sun's rays.' *** Good luck with that 'sanding down' effort. We're talking about 10 _octaves_ of infrared light, so the imperfections caused by the sanding have to be on the order of 1 mm to be effective. *** I think what Disney Hall really did was to eliminate the _visible_ evidence of the hotspot, but the heat is mostly still there. "Infrared (IR) light is electromagnetic radiation with longer wavelengths than those of visible light, extending from the nominal red edge of the visible spectrum at 700 nanometres (nm) to 1 mm." "Slightly more than half of the energy from the Sun arrives on Earth in the form of infrared radiation." http://en.wikipedia.org/wiki/Infrared At 10:31 AM 9/3/2013, Warren D Smith wrote: >http://www.bbc.co.uk/news/magazine-23944679

1 0

[math-fun] another slight design flaw... skyscraper cooks cars
by Warren D Smith 03 Sep '13

03 Sep '13

http://www.bbc.co.uk/news/magazine-23944679

2 1

[math-fun] That one-ninthish sextic
by Bill Gosper 03 Sep '13

03 Sep '13

Our sextic solver fervently assumes (I think with Piezas's blessing) that the solvable ones all factor into either cubics with quadratic surd coefficients, or vice versa. In[343]:= Factor[27*x^6 + 3087*x - 343, Extension -> Sqrt[21]] Out[343]= -(1/12) (-147 - 35 Sqrt[21] + (-21 + 21 Sqrt[21]) x + 6 Sqrt[21] x^2 - 18 x^3) * (147 - 35 Sqrt[21] + (21 + 21 Sqrt[21]) x + 6 Sqrt[21] x^2 + 18 x^3) is the former. The interesting real root is Out[218]= -Sqrt[(7/3)]/3 - 1/3 Sqrt[7/3] (1/2 (-70 + 9 Sqrt[21] + 9 Sqrt[85 - 12 Sqrt[21]]))^(1/3) + (Sqrt[7] (1 + 3 Sqrt[21]))/(3 2^(2/3) Sqrt[3] (-70 + 9 Sqrt[21] + 9 Sqrt[85 - 12 Sqrt[21]])^(1/3)) In[219]:= Factor[27*x^6 + 3087*x - 343 - 3^-9] Out[219]= ((-1 + 9 x) (6751270 + 9 x + 81 x^2 + 729 x^3 + 6561 x^4 + 59049 x^5))/19683 In[220]:= ContinuedFraction[%%, 33] Out[220]= {0, 9, 750141, 1, 1, 4, 50009, 4, 3, 4, 6, 1, 389, 1, 5, 7, 7, 2, 5, 1, 1, 1, 4, 64, 2, 1, 1, 1, 1, 1, 1, 2, 7} NDE>Are you suggesting there's some significance to the large convergents? Well, it can't hold a candle to Brillhart's In[228]:= ContinuedFraction[(15 Sqrt[3]-Sqrt[163])^(1/3)/Sqrt[3]+(15 Sqrt[3]+Sqrt[163])^(1/3)/Sqrt[3],239] Out[228]= {3,3,7,4,2,30,1,8,3,1,1,1,9,2,2,1,3,22986,2,1,32,8,2,1,8,55,1,5,2,28,1,5,1,1501790,1,2,1,7,6,1,1,5,2,1,6,2,2,1,2,1,1,3,1,3,1,2,4,3,1,35657,1,17,2,15,1,1,2,1,1,5,3,2,1,1,7,2,1,7,1,3,25,49405,1,1,3,1,1,4,1,2,15,1,2,83,1,162,2,1,1,1,2,2,1,53460,1,6,4,3,4,13,5,15,6,1,4,1,4,1,1,2,1,16467250,1,3,1,7,2,6,1,95,20,1,2,1,6,1,1,8,1,48120,1,2,17,2,1,2,1,4,2,3,1,2,23,3,2,1,1,1,2,1,27,325927,1,60,1,87,1,2,1,5,1,1,1,2,2,2,2,2,17,4,9,9,1,7,11,1,2,9,1,14,4,6,1,22,11,1,1,1,1,4,1,3,2,1,2,1,1,2,4,2,1,5,1,8,2,2,5,1,2,1,1,1,1,1,3,1,2,6,10,1,3,1,3,2,2,1,1,2,1} (A002937) But I don't recall the phenomenon in sextics. (Is H. M. Stark, ‘An explanation of some exotic continued fractions found by Brillhart, in A. O. L. Atkin and B. J. Birch eds., Computers in Number Theory, Academic Press, London and New York, 1971 available online? Hmm, √163.) NDE>No septics here. This family of octics factors over quadratic extensions, so it's "just" solving a quartic over Q(sqrt(d)). Yow, it Gaussian-factors! In[338]:= Factor[4 x^8 + 8 x + 13, Extension -> I] Out[338]= ((3 - 2 I) + (4 + 4 I) x - (4 - 2 I) x^2 - 4 I x^3 + 2 x^4) * ((3 + 2 I) + (4 - 4 I) x - (4 + 2 I) x^2 + 4 I x^3 + 2 x^4) --rwg

3 4

[math-fun] 18-month-old news re encryption
by Henry Baker 03 Sep '13

03 Sep '13

Given more recent news, perhaps it's time to revisit assumptions. http://www.wired.com/threatlevel/2012/03/ff_nsadatacenter/ James Bamford 03.15.12 "According to another top official also involved with the program, the NSA made an enormous breakthrough several years ago in its ability to cryptanalyze, or break, unfathomably complex encryption systems employed by not only governments around the world but also many average computer users in the US."

1 0

Re: [math-fun] Julian's dodecahedron
by Adam P. Goucher 02 Sep '13

02 Sep '13

The legendary Bill Gosper wrote: > is made by replacing ten of the regular pentagons of a dodecahedron with > concave equilateral pentagons, preserving the topology: > http://gosper.org/dodohedron.png > ( !#%&*@! was deleted again. Apologies to any of you who looked for it.) > > http://en.wikipedia.org/wiki/Dodecahedron glaringly omits it: Even worse, the Wikipedia article omits the much more well-known _endo-dodecahedron_ (the Symmetries of Things, page 328), which has concave regular pentagonal faces and is *face-transitive*. Its symmetry group is pyritohedral (3*2), and it is defined as: "The spaces left over by positioning dodecahedra of maximum size in the natural orientation, centred on the points of the FCC lattice" The faces are different from the concave pentagons of the tympanohedron, having internal angles of {7pi/5, pi/5, 3pi/5, 3pi/5, pi/5} in cyclic order. Sincerely, Adam P. Goucher http://cp4space.wordpress.com

1 0