December 2014 - mathfun@mailman.xmission.com

Re: [math-fun] masonry domes
by Henry Baker 27 Dec '14

27 Dec '14

St. Paul's dome in London definitely has a chain around its bottom circumference. The Pantheon dome in Rome definitely does not have a chain, but sits on an absolutely massive wall structure to keep it from spreading outwards. Hagia Sophia's dome in Istanbul definitely did not have a chain when it was built, but has numerous metal reinforcements today. The original Hagia Sophia's dome fell down about 21 years after it was built due to damage from an earthquake, but quickly rebuilt. https://en.wikipedia.org/wiki/Hagia_Sophia Brunelleschi's dome may have some sort of a "chain", but there has been intense controversy about this ever since it was built. https://en.wikipedia.org/wiki/Florence_Cathedral Note that a simple chain which encircles a structure, but isn't embedded into the mortar won't have some of the same problems as a reinforcing bar. Nevertheless, such a chain will still undergo temperature cycles which will cause expansion/contraction of the chain relative to the bricks & mortar, and thus affect the lifetime of the structure. I don't know if a dome-encircling chain would require having its tension adjusted over the years. This would be an interesting question for the people in charge of St. Paul's. --- There is an interesting ancient feature found in some of the domed Islamic mosques: a number of "pillars" which can be easily rotated by hand. These pillars are several feet long, and _measure_, in effect, the distance between two attachment points. If these pillars cannot be easily rotated by hand, then it is clear that the distance between the two attachment points has decreased, and this indicates that the surrounding structure has been compromised and is starting to be crushed. This alerts the maintenance engineers to look for cracks & begin repairs. These rotating pillars seem to be unique to the Arabic domes, and are apparently not a feature of the older Roman domes. At 08:04 PM 12/26/2014, Warren D Smith wrote: >Incidentally it is not known whether Brunelleschi employed metal reinforcements. >Apparently at least some plans said they were to be used but he may have decided >during construction they were not needed. If he did use them, they >are deeply embedded inside his masonry where nobody can detect them. >The techniques B used in building >the Duomo apparently are disputed and there is no consensus how he did it.

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[math-fun] Just another brick in the wall
by Bill Gosper 27 Dec '14

27 Dec '14

Who needs mortar? gosper.org/martinsmarbles.png --rwg wds> The usual "running bond" brick pattern (if 2:1 rectangular 2D bricks) forces any vertical crack in the wall to (if it only cracks mortar, not bricks) have length at least 3 times the straight line length. However, a horizontal crack can go straight, and a diagonal staircase crack will have length magnification factor sqrt(2). Here the factor 3 seems max possible and is a sense in which this brick pattern provides optimum resistance to vertical cracking. Not only that, the running bond pattern with L:1 bricks seems optimal for any L>1 in this sense (distance amplification factor=L/2+1), and also not only over rectangular bricks, but indeed I suspect over ALL convex tesselator shapes with unit area and given perimeter, the optimum brick shape is a rectangle and the optimal brick laying pattern is running bond, for the purpose of resisting vertical cracking. (Can you prove or disprove?) It is pointless to make L too large, though, since beyond some threshold it becomes energetically easier to crack the bricks, not the mortar. But running bond sucks as far as its crack resistance in certain other (non-vertical) directions is concerned. The "herringbone" bricklaying pattern with 2:1 bricks (pictures:http://oregonbrick.com/brick%20patterns.html) forces crack length amplification factor>=1.5 for both vertical and horizontal crack directions, albeit again sqrt(2) for 45-degree diagonal cracks. 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. The crack-length amplification factor for this pattern for integer L and either horizontal or vertical direction of cracking, is (2L-1)/L, but it seems to me this can be increased a bit if we use non-integer L. Still, though, for 45-degree diagonal direction of cracking, the amplification factor remains only sqrt(2). 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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[math-fun] masonry domes
by Warren D Smith 27 Dec '14

27 Dec '14

Re HB claiming pure masonry is better than rebar-reinforced concrete... at least if lifespans>100 years are considered... well maybe, but big domes nowadays are made neither of masonry nor reinforced concrete. The largest reinforced concrete dome was the Seattle "king dome" which had severe problems and was demolished in 2000 after only 24 years. They are still paying off the debt for it, it was never able to pay for itself. The 2nd largest such dome (now the largest after king dome demolished) still apparently in fine shape in Norfolk VA. It's an interesting claim that pure masonry is the most long lived construction method. Incidentally it is not known whether Brunelleschi employed metal reinforcements. Apparently at least some plans said they were to be used but he may have decided during construction they were not needed. If he did use them, they are deeply embedded inside his masonry where nobody can detect them. The techniques B used in building the Duomo apparently are disputed and there is no consensus how he did it.

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Re: [math-fun] Dimension where unit diameter ball has maximum content
by Bill Gosper 26 Dec '14

26 Dec '14

Mma solution: In[576]:= D[Log[\[Pi]^(d/2)/2^d/(d/2)!], d] // FullSimplify Out[576]= 1/2 (EulerGamma - HarmonicNumber[d/2] + Log[\[Pi]/4]) In[577]:= Solve[% == 0] During evaluation of In[577]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >> Out[577]= {{d -> 2 Root[{-EulerGamma + HarmonicNumber[#1] - Log[\[Pi]/4] &, 0.238291291153042647603807884429}]}} Contrary to appearances, this is an exact, manipulable quantity: In[578]:= N[%[[1]], 69] Out[578]= {d -> 0.476582582306085295207615768858823240301645515180497569319595172372413} Solving manually instead: In[579]:= 2*InverseFunction[HarmonicNumber][EulerGamma + Log[\[Pi]/4]] Out[579]= 2 Root[{-EulerGamma + HarmonicNumber[#1] - Log[\[Pi]/4] &, 0.238291291153042647603807884429}] --rwg (Let's see how many days this delivery takes.) rwg>This exposes a bug in the terminology "unit ball", which really ought to mean "unit diameter ball". There is no local content maximum when the ball is inscribed in a unit cube.<rwg Is that right? Using diameter = 1 instead of radius = 1, I'm getting for A(d) / 2^(d-1) a maximum at d = 2.4765825823060852952076157688588232403016455151805+, and for V(d) / 2^d a maximum at d = 0.4765825823060852952076157688588232403016455151805+ . --Dan Yikes, I never considered 0<d<1, whereat the "sphere" exceeds the "bounding cube"! --rwg Question: Is that Amax-Vmax = 2 (we couldn't even write this if dimension weren't dimensionless!) observation original? Did you use Area = d Volume/dr? --rwg DanA> Pretending that spheres, balls, and Euclidean spaces can have real dimensions: * let d_Amax := the real dimension d where the formula A(d) = 2 pi^(d/2) / Gamma(d/2) for the (d-1)-dimensional content of the unit (d-1)-sphere in R^d takes its maximum. -and- * let d_Vmax := the real dimension d where the formula V(d) = pi^(d/2) / Gamma(d/2 + 1) for the d-dimensional content of the unit d-ball in R^d takes its maximum Then d_Amax = 7.256946404860576780132838388690769236619+ and d_Vmax = 5.256946404860576780132838388690769236619+.

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[math-fun] Just another brick in the wall
by Warren D Smith 26 Dec '14

26 Dec '14

The usual "running bond" brick pattern (if 2:1 rectangular 2D bricks) forces any vertical crack in the wall to (if it only cracks mortar, not bricks) have length at least 3 times the straight line length. However, a horizontal crack can go straight, and a diagonal staircase crack will have length magnification factor sqrt(2). Here the factor 3 seems max possible and is a sense in which this brick pattern provides optimum resistance to vertical cracking. Not only that, the running bond pattern with L:1 bricks seems optimal for any L>1 in this sense (distance amplification factor=L/2+1), and also not only over rectangular bricks, but indeed I suspect over ALL convex tesselator shapes with unit area and given perimeter, the optimum brick shape is a rectangle and the optimal brick laying pattern is running bond, for the purpose of resisting vertical cracking. (Can you prove or disprove?) It is pointless to make L too large, though, since beyond some threshold it becomes energetically easier to crack the bricks, not the mortar. But running bond sucks as far as its crack resistance in certain other (non-vertical) directions is concerned. The "herringbone" bricklaying pattern with 2:1 bricks (pictures: http://oregonbrick.com/brick%20patterns.html) forces crack length amplification factor>=1.5 for both vertical and horizontal crack directions, albeit again sqrt(2) for 45-degree diagonal cracks. 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. The crack-length amplification factor for this pattern for integer L and either horizontal or vertical direction of cracking, is (2L-1)/L, but it seems to me this can be increased a bit if we use non-integer L. Still, though, for 45-degree diagonal direction of cracking, the amplification factor remains only sqrt(2). 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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Re: [math-fun] Dimension where unit ball has maximum content
by Bill Gosper 25 Dec '14

25 Dec '14

rwg>This exposes a bug in the terminology "unit ball", which really ought to mean "unit diameter ball". There is no local content maximum when the ball is inscribed in a unit cube.<rwg Is that right? Using diameter = 1 instead of radius = 1, I'm getting for A(d) / 2^(d-1) a maximum at d = 2.4765825823060852952076157688588232403016455151805+, and for V(d) / 2^d a maximum at d = 0.4765825823060852952076157688588232403016455151805+ . --Dan Yikes, I never considered 0<d<1, whereat the "sphere" exceeds the "bounding cube"! --rwg Question: Is that Amax-Vmax = 2 observation original? Did you use Area = d Volume/dr? --rwg DanA> Pretending that spheres, balls, and Euclidean spaces can have real dimensions: * let d_Amax := the real dimension d where the formula A(d) = 2 pi^(d/2) / Gamma(d/2) for the (d-1)-dimensional content of the unit (d-1)-sphere in R^d takes its maximum. -and- * let d_Vmax := the real dimension d where the formula V(d) = pi^(d/2) / Gamma(d/2 + 1) for the d-dimensional content of the unit d-ball in R^d takes its maximum Then d_Amax = 7.256946404860576780132838388690769236619+ and d_Vmax = 5.256946404860576780132838388690769236619+.

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[math-fun] Dimension where unit ball has maximum content
by Bill Gosper 25 Dec '14

25 Dec '14

This exposes a bug in the terminology "unit ball", which really ought to mean "unit diameter ball". There is no local content maximum when the ball is inscribed in a unit cube. Question: Is that Amax-Vmax = 2 observation original? Did you use Area = d Volume/dr? --rwg DanA> Pretending that spheres, balls, and Euclidean spaces can have real dimensions: * let d_Amax := the real dimension d where the formula A(d) = 2 pi^(d/2) / Gamma(d/2) for the (d-1)-dimensional content of the unit (d-1)-sphere in R^d takes its maximum. -and- * let d_Vmax := the real dimension d where the formula V(d) = pi^(d/2) / Gamma(d/2 + 1) for the d-dimensional content of the unit d-ball in R^d takes its maximum Then d_Amax = 7.256946404860576780132838388690769236619+ and d_Vmax = 5.256946404860576780132838388690769236619+. In particular they have the same fractional part: upsilon := 0.256946404860576780132838388690769236619+ QUESTION: Is anything known about the number theoretic properties of upsilon? Is it known to be irrational or transcendental? Or related to other numbers, like Euler gamma, whose number-theoretic properties are unknown? --Dan

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Re: [math-fun] Dimension where unit ball has maximum content
by Bill Gosper 24 Dec '14

24 Dec '14

I neglected to copy the Subject line, which seems to have discouraged math-fun delivery. Presumably, the e[a]vesdroppers got it. On Wed, Dec 24, 2014 at 1:06 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > = Daniel Asimov <asimov(a)msri.org> wrote: > > Pretending that spheres, balls, and Euclidean spaces can have real > dimensions: > > > ="Olivier GERARD" <olivier.gerard(a)gmail.com> > Very delicate hypothesis. You would have to prove that > there is a continuous > > family of geometrical objects, to whom volume and > > surface make sense, etc. > > and according precisely to these formulae. > > > > > ="Bill Gosper" <billgosper(a)gmail.com> > This exposes a bug in the terminology "unit ball", which really ought > to mean "unit diameter ball". > > MLeBrun> > I don't know balls, but, prompted by this mysterious maximum, a number of > years ago I asked this list if it was always possible to construct a fractal > of any given real dimension (ie we're not just restricted to values of the > "classical" type of log-mumble expressions), and y'all said "yes"--and I > think moreover even gave an explicit method. > > Starting with such a fractal as the "space", can you then hand-wave up a > suitably-constructed "unit subset" that degenerates to the Euclidean ball at > integer dimensions? > > The content formulae over this continua of constructions will of course > likewise degenerate to the expression under discussion here at integer > dimensions, but it might have additional twiddle factors that vanish at > those values--hence different maxima as it interpolates in-between. > > So another tack, symbolic instead of geometric: the volume formula can be > constructed directly by iterated integration. I think the usual fractional > iterated integration operator also just changes factorial to gamma. If that > generalization is unique, it would seem that the generalized fractional > dimensional geometric construction must also be. If not, the converse. > <MLB > > From the McCarthy festschrift, http://gosper.org/fst.pdf p "145" = 17 > and thereabouts derives the triadic dragon (and its Fourier series) > with 0<=dimension<=1 adjustable via the parameter w. See the > trajectories plot. > > Caution: there are some typos in that collection of valuations at the > > bottom of p145. Fix: gosper.org/fstfix.pdf > > I'm not sure why we always use the triadic dragon to illustrate > > continuously variable dimension. The ordinary Heighway > > Dragon ought to work as well. > > --rwg > > DWilson> Well, that didn't come out like I sent it. The phi should be a psi. The eth should be a pi. Indeed I see phi and eth in *this* msg, but I just copied ψ(d/2) = log(π) from your original!

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[math-fun] 2015
by Wainwright, Robert 24 Dec '14

24 Dec '14

4, 3, 2, 1, 0.!!! Using the five countdown digits (in their same order) and the eight punctuation marks, create an expression which equals 2015. Only basic arithmetic operations [+, -, x, /] are allowed however you will need to substitute parentheses for the commas.

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[math-fun] (no subject)
by Bill Gosper 24 Dec '14

24 Dec '14

= Daniel Asimov <asimov(a)msri.org> wrote: Pretending that spheres, balls, and Euclidean spaces can have real dimensions: ="Olivier GERARD" <olivier.gerard(a)gmail.com> Very delicate hypothesis. You would have to prove that there is a continuous family of geometrical objects, to whom volume and surface make sense, etc. and according precisely to these formulae. ="Bill Gosper" <billgosper(a)gmail.com> This exposes a bug in the terminology "unit ball", which really ought to mean "unit diameter ball". MLeBrun> I don't know balls, but, prompted by this mysterious maximum, a number of years ago I asked this list if it was always possible to construct a fractal of any given real dimension (ie we're not just restricted to values of the "classical" type of log-mumble expressions), and y'all said "yes"--and I think moreover even gave an explicit method. Starting with such a fractal as the "space", can you then hand-wave up a suitably-constructed "unit subset" that degenerates to the Euclidean ball at integer dimensions? The content formulae over this continua of constructions will of course likewise degenerate to the expression under discussion here at integer dimensions, but it might have additional twiddle factors that vanish at those values--hence different maxima as it interpolates in-between. So another tack, symbolic instead of geometric: the volume formula can be constructed directly by iterated integration. I think the usual fractional iterated integration operator also just changes factorial to gamma. If that generalization is unique, it would seem that the generalized fractional dimensional geometric construction must also be. If not, the converse. <MLB >From the McCarthy festschrift, http://gosper.org/fst.pdf p "145" = 17 and thereabouts derives the triadic dragon (and its Fourier series) with 0<=dimension<=1 adjustable via the parameter w. See the trajectories plot. Caution: there are some typos in that collection of valuations at the bottom of p145. Fix: gosper.org/fstfix.pdf I'm not sure why we always use the triadic dragon to illustrate continuously variable dimension. The ordinary Heighway Dragon ought to work as well. --rwg

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