December 2014 - mathfun@mailman.xmission.com

[math-fun] Re Tetris Light Configuration
by Guy Haworth 30 Dec '14

30 Dec '14

I was thinking in terms of: - using only the seven pieces from one instance of the toy, - distinguishing between pieces which are different colours (so the two Zs are different) - NOT 'identifying' particular faces of pieces (so the 'Zs' can be flipped 180 degrees) - touching faces are identically laid over one another, so there are only a finite number of solutions - NOT distinguishing between two arrangements which can be physically translated to each other The planar constraint (required to light all pieces) only applies if they are all required to light up. The question of the number of arrangements is rather prompted by the 'endless' claim on the packaging. :-) Guy

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Re: [math-fun] Just another brick in the wall
by Henry Baker 29 Dec '14

29 Dec '14

At 12:47 PM 12/26/2014, Warren D Smith wrote: >It is also possible to lay herringbone with L:1 bricks for any L>1, including >non-integer L: > http://www.mytinyworld.co.uk/images/superpic/brick-12th-0001.jpg >Apparently this bricklaying pattern was invented by Filippo Brunelleschi >(1377-1446) the architect behind the dome of the great cathedral in >Florence (and it was employed in the dome). I believe this still is >the largest masonry dome ever built. >Quite possibly it will never be exceeded because *** nobody today would be >interested in using masonry anymore.*** Actually, this statement isn't true. While rebar-reinforced masonry is stronger, it is _not_ more durable, and structures older than perhaps 100 years are a much greater risk of failure due to the inevitable corrosion and temperature-cycling effects. This is one reason why workmen are running around Manhattan *** tapping at every single brick *** to see if any are loose; this is now required after several people have been killed in the last decade or so from falling bricks. Pure Roman-style masonry is far more durable, assuming that it was properly designed with the strength limitations of masonry in mind. A proper 3-D "printed" structure in which the bricks are mechanically laid, positioned & mortared could be quite strong, inexpensive and extremely durable. The bricks could be delivered to the working surface in a rectangular tube, so long as there was some form of "lubrication", which might consist of small steel bearings which could be recovered with magnets at the point where the bricks "flow" out of this tube. Such a 3-D printer could print "voids" much more consistently than could a human bricklayer, and thereby create strong, lacey "fractal" structures which are much lighter than solid brick & mortar structures.

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[math-fun] Wechsler's Gaudi cloth/concrete/inversion technique
by Bill Gosper 28 Dec '14

28 Dec '14

WDS>I don't believe it. First of all, a cloth will always form a "developable surface" i.e. gaussian curvature=0. However, a "dome" has positive curvature, and various hyperbolicish "minimal surfaces" (minimal surface ==> rigid, which is a useful property for some constructions) have negative curvature. So the technique is mathematically wrong. It also physically sounds like bull, because I have trouble seeing how you are going to make a dome larger than a house, then "invert it." In the case of the Duomo in Florence, by the way, it weighs about 40,000 tons. Want to just "invert" that? Meanwhile, the balloon+spray on technique has been used, and continues to be used, to make domes larger than houses (some of them, in fact, are houses, storage facilities, churches, etc). And you can design the shape of the balloon to make your domes have pretty controllable and somewhat intricate shapes, not just a plain spheroid.<WDS This would make trampolines a lot less fun. --rwg

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[math-fun] Found this interesting
by David Wilson 28 Dec '14

28 Dec '14

It's arguably math and fun. http://www.cs.huji.ac.il/~yweiss/Colorization/

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Re: [math-fun] reinforced concrete
by Henry Baker 28 Dec '14

28 Dec '14

Great idea! But how accurate is this translation of the catenary idea into 2D? What you really want is to hang a parallel array of chains all parallel to the X axis, and then hang another array of chains all parallel to the Y axis, and then adjust the length of each chain until all X chains touch all Y chains at all intersection points. This way, there is no assumption about the strength of the cloth. In other words, the problem with putting a non-elastic cloth into a catenary-type shape is that the cloth doesn't have enough degrees of freedom to assume the best position. Thus, when the dried concrete shape is turned over, the stresses will not all be tangential. A rubber sheet is somewhat better, but the thickness of the rubber is inverse to what you want: the highest stress will have the thinnest rubber, but that is where you want the thickest concrete. At 02:25 PM 12/27/2014, Allan Wechsler wrote: >The technique I heard about in the 70s was to hang a cloth sheet from a >support the same shape as the intended foundation, let it settle into a >nice catenary-like shape, and gradually spray it with concrete. You have to >do it gradually, in thin, even layers, so that the heavy spots don't drag >the cloth down and distort the shape. After it dries, you invert it. >Perfect vaulted dome, ideal for bearing its own weight down to the >foundation.

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[math-fun] Wechsler's Gaudi cloth/concrete/inversion technique
by Warren D Smith 28 Dec '14

28 Dec '14

I don't believe it. First of all, a cloth will always form a "developable surface" i.e. gaussian curvature=0. However, a "dome" has positive curvature, and various hyperbolicish "minimal surfaces" (minimal surface ==> rigid, which is a useful property for some constructions) have negative curvature. So the technique is mathematically wrong. It also physically sounds like bull, because I have trouble seeing how you are going to make a dome larger than a house, then "invert it." In the case of the Duomo in Florence, by the way, it weighs about 40,000 tons. Want to just "invert" that? Meanwhile, the balloon+spray on technique has been used, and continues to be used, to make domes larger than houses (some of them, in fact, are houses, storage facilities, churches, etc). And you can design the shape of the balloon to make your domes have pretty controllable and somewhat intricate shapes, not just a plain spheroid. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] 2015 with the other six digits (in order)
by rtwainwright＠optonline.net 28 Dec '14

28 Dec '14

((3!)(.4/6)(7!))+8-9 R. Wainwright

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[math-fun] Tetris Light configurations
by Guy Haworth 27 Dec '14

27 Dec '14

So ... my wife buys Tetris Lights for some children ... http://www.firebox.com/product/5339/Tetris-Light ... made of seven tetrominoes. What is an upper bound on the number of configurations in which the components are face-to-face touching another component on at least one face so that: - all components will (A) stand stationery and (B) light up, or - one or both of conditions 'A' and 'B' are ignored. Guy

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[math-fun] reinforced concrete
by Warren D Smith 27 Dec '14

27 Dec '14

Well, one of the great fortunate numerical coincidences is, that steel and concrete have the same coefficient of thermal expansion, which is why reinforced concrete is good stuff.(Or so it claims in a book I have on this.) I suppose it must not be *exactly* the same, but pretty close. It definitely is the case that with modern bricks and mortar, the mortar tends to crack and bricks not. With ancient bricks, which may not have had as good quality control -- thermometers not invented yet, hard to control your kiln... -- that might not have been true. An interesting modern dome-making technique is, you create a balloon out of plastic sheet, inflate with fan, place rebar mesh on it held together with wires, and spray on concrete. A few days later, you have a dome, which supposedly is very strong, long lasting, and hermetically sealed. The concrete can be sprayed either on the exterior or interior of the balloon. (Foundation, doors & windows have to be created ahead of time...) -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Just another brick in the wall
by Henry Baker 27 Dec '14

27 Dec '14

In many ancient brick structures, the mortar is stronger than the brick -- at least at the time of collapse. The mortar retains its shape, and the bricks have collapsed & turned to dust. At 11:57 PM 12/26/2014, Bill Gosper wrote: >Who needs mortar? gosper.org/martinsmarbles.png --rwg > >wds> > >QUESTION: Using L:1 rectangular bricks, does there exist any pattern >such that the crack length amplification factor always is >=C, for >some C>sqrt(2), no matter what direction? What is the greatest >achievable C? > >If nonconvex tesselators are allowed, then Gosper's fractal >"snowflake" shaped "bricks" would assure crack-length amplification >factor of INFINITY... but only in the limit where the mortar thickness >tends to 0.

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