The optimum shape for an arch under assumptions below is a cycloid. 1. "thin beam of constant cross section and constant density & material properties" 2. supports only its own weight, not other stuff; constant downward gravity field. 3. "optimum" means constant stress regardless of position on the arch-curve. 4. No other (i.e. nongravitational) forces to worry about. Another important use of arches is for arch bridges, where there is also a flat bridge deck exerting (to oversimplify it) a constant downward load per unit of horizontal length. The equation of the curve of an optimum arch satisfying (1) would be that the total weight of the part of the arch (plus the external load) above some height y (which is evaluable via an obvious integral) must equal a constant times sqrt(1 + x'^2) where x=horizontal location is regarded as a function of y=vertical height to describe the arch curve. So for example, with both the flat bridge deck and arch itself contributing to the load integral( A*sqrt(1+x'(u)^2) + B , u=y..H) = C*sqrt(1+x'(y)^2) would define the function x(y) defining the curve for an arch centered at x=0 of height H. Here A=weight per unit length of arch, B=weight per unit horizontal length of deck, and C=stress (constant compressive force along the arch curve). Use fundamental theorem of calculus to rewrite as a differential equation A*sqrt( 1 + diff(x(y),y)^2 ) + B + C * diff( sqrt( 1+diff(x(y),y)^2 ), y ) = 0 and solve. MAPLE9's dsolve facility succeeds in finding a closed form in the case A=0 using only log and sqrt. In the case B=0 it expresses x(y) as a certain indefinite integral, which seems to me to be something of a failure of dsolve since I happen to know the solution is a cycloid and thus is expressible in closed form, albeit that is best done by writing both x and y as a function of a third variable (you can do this explicitly and plug into differential eq to verify is solution). In the general case dsolve required a double integral and a "RootOf" to express the solution, which seems hardly better than just solving the differential equation entirely numerically. Perhaps however some of you will be better-able to solve differential equations than MAPLE dsolve, I await your reports. Re Baker on the Romans whom, he claims, preferred circles, the circle is the optimum shape for an arch under the following (different) assumptions 1'.same 3'.same 2'.replace by assumption that the weight of the arch itself is tiny compared to the weight of everything else it supports, and that other stuff exerts a constant hydrostatic pressure. In other words, this is the optimum shape for the wall of a "submarine" at high depth or a deep underground tunnel thru sand. Many Roman arches are small doorways inside massive walls, and for them the assumption 2' might seems pretty well satisfied -- and this shape is still used today in applications of that nature. If you want to know whether the Romans had a more sophisticated and more general theory, I would suggest (a) try to read Vitruvius (ancient Roman engineering textbook) http://www.gutenberg.org/ebooks/20239 (b) consider the Pantheon, the world's largest dome for a very long time, constructed by Hadrian and still in use today. I think they did not really understand what they were doing compared to what I understand, but they had a lot of knowledge nonetheless... On 3/19/12, math-fun-request@mailman.xmission.com <math-fun-request@mailman.xmission.com> wrote:
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Today's Topics:
1. Julian pimped my pump (Bill Gosper) 2. Re: Julian pimped my pump (Fred lunnon) 3. Phil Willcocks (THW), recreational mathematician, is 99 years and 11 months old. (Stuart Anderson) 4. Re: Julian pimped my pump (Bill Gosper) 5. Re: Julian pimped my pump (Veit Elser) 6. Reducing the computational complexity of Roman arch construction (Henry Baker)
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Message: 1 Date: Sun, 18 Mar 2012 19:01:13 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Julian pimped my pump Message-ID: <CAA-4O0EALr_7LPVqA6Ha2Z-SD-Fch2cebzXzewtQmM=AP1oiOg@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-1
To run it <http://gosper.org/pumper_2.mp4> continuously, download with Miro, QuickTime-View-Loop. The pumped shapes split lengthwise resemble boats, so a grand entrance to a math museum would be a chain of these boats in an annular canal, passing through four rotors acting as an (approximate) air|water lock, with another four for the exit. Oh, the insurance. --rwg No manual entry for overboard
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Message: 2 Date: Mon, 19 Mar 2012 02:34:11 +0000 From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Julian pimped my pump Message-ID: <CAN57YqsC0MWc19KvKVh++DboWrxB9u-PDOmuXZts_VgtA1tqfw@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-1
On my (antique) system, this is very grainy; also intermittently halts in mid squidge, though that's possibly a problem at my end. WFL
On 3/19/12, Bill Gosper <billgosper@gmail.com> wrote:
To run it <http://gosper.org/pumper_2.mp4> continuously, download with Miro, QuickTime-View-Loop. The pumped shapes split lengthwise resemble boats, so a grand entrance to a math museum would be a chain of these boats in an annular canal, passing through four rotors acting as an (approximate) air|water lock, with another four for the exit. Oh, the insurance. --rwg No manual entry for overboard _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Message: 3 Date: Mon, 19 Mar 2012 15:51:38 +1100 From: Stuart Anderson <stuart.errol.anderson@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Phil Willcocks (THW), recreational mathematician, is 99 years and 11 months old. Message-ID: <CA+3-r9N1ukkwAZPAtf9Zf3xB6T7a2SC0Rqywq9R8S9jQr8ta-A@mail.gmail.com> Content-Type: text/plain; charset=windows-1252
At WWII end, Theophilus Harding Willcocks, also known as Phil Willcocks or THW started publishing results in recreational mathematics. This talented amateur mathematician (and chess enthusiast) was born in 1912, at Newquay, Cornwall, England. A modest man, THW was for many years, an employee at the Bank of England until his retirement. Since 1945, his continuing interest in the field is evidenced by his many outstanding productive contributions to its growing body of knowledge. His publications in Fairy Chess Review (no longer published, but see the website <a href="http://www.mayhematics.com/d/d.htm"> mayhematics </a> by George Jellis - mainly concerned with the history of work published in Fairy Chess Review), the Canadian Journal of Mathematics and the Journal of Combinatorial Theory have demonstrated his unique approach to mathematical problems. Willcocks also worked on chessboard dissections (now called polyominoes)
THW discovered (in 1946) and subsequently published what is arguably the most widely known perfect squared square (until Duijvestijn's discovery of 21 : 112A in 1978). http://www.squaring.net/gfx/cpsso24-400.jpg
So many authors have cited this square that its figure is well known even to casual readers. "This little gem of Willcocks", as it has been called is 24: 175a (THW), composed of 24 squares with a side of 175 and an included rectangle . For more than three decades, it was the benchmark by which all other squares were measured (as it had the lowest order and smallest side of any known perfect square) and it remains today, the lowest possible order compound perfect square, a fact established by Duijvestijn, P. J. Federico and P. Leeuw in 1982. Even today with conclusive permanent records of the ultra-low-order squares established, only three people have produced a square which betters this square in size and order ... Willcocks is one of these (Duijvestijn and Skinner are the others), Willcocks found 22: 110B by hand, by applying a transform to Duijvestijn's 22: 110A. These two, and another 110 in order 23 are the smallest possible perfect squared squares. All the ultra-low-order squares (from order 21 through 24) are now known. All but one of these are simple. 24: 175a (THW) is compound. it stands alone, the very best of all the compound perfect squares, the only ultralow-order compound perfect square.
Phil Willcocks produced many squared squares over more than half a century, inventing new techniques and methods on a regular basis. Working alone and in collaboration with Federico and others, he was able to equal or better many of the results produced by brute force computer enumeration.
In recent correspondence George Jellis stated; "I?ve never actually met THW in person, but he was a member of the Fairy Chess Correspondence Circle (made up of contributors to Fairy Chess Review) when I was invited to join it in the 1970s. One of his interests was in retroanalytical ?last-move? compositions. He and H. E. de Vasa were the first to construct diagonally magic knight?s tours on the 16x16 and larger boards (following on from the work of H. J. R. Murray who died in 1955). He published articles in Recreational Mathematics Magazine 1962 and Journal of Recreational Mathematics 1968 on this subject. Most of my correspondence with him was in connection with magic knight?s tours, and tours by other pieces. He also contributed a few cryptarithms to my Games and Puzzles Journal. One ?composed to mark a young brother?s 70th birthday? is: THIRTY + TWENTY + TEN + SEVEN + THREE = SEVENTY. Rachel features in a set of cryptarithms from THW, involving quarrelling sisters who each maintains ?I am equal to two of you!?: RACHEL + RACHEL = ALISON, ALISON + ALISON = RACHEL, ALISON + ALISON + ALISON = RACHEL.
Phil, short for Theophilus, like his father, worked all his career for the Bank of England, latterly on exchange controls. During the war he served in Italy and North Africa in the RAF.
Willcocks is one of three brothers. His youngest brother is Sir David Willcocks CBE MC (born 30 December 1919) a distinguished and well-known British choral conductor, organist, and composer. http://en.wikipedia.org/wiki/David_Willcocks
The three brothers have remained close and have recently worked together to restore the pipe organ in the beautiful Parish Church of St Michael at Newquay where they attended services with their parents throughout their childhood. http://stmichaelsnewquay.wordpress.com/organ-appeal/
The second brother Wilfrid Willcocks also had an interesting and productive life, having been a Japanese POW in WWII. He became a successful investment analyst. He wrote about market developments for the Financial Times, became a member of the Stock Exchange and a Lloyds name. In retirement he continued to play the stock market with success, outperforming it through booms and recessions. He established a trust which will continue to support environmental and social charities. He died in 2011 aged 96. http://www.thisiscornwall.co.uk/Japanese-PoW-dies-96/story-12474563-detail/s...
Phil Willcocks will be 100 years old on April 19, 2012.
http://www.squaring.net/history_theory/th_willcocks.html The biographical sketch is based on contributions from Jasper Skinner, Geoffrey Morley and George Jellis.
Stuart Anderson
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Message: 4 Date: Mon, 19 Mar 2012 03:00:45 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Julian pimped my pump Message-ID: <CAA-4O0EGhzY8hjCb3VjzcOiw6T=kW-TWK+7YWc5Zcu4yedZQgQ@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-1
Oops, I see where I spazzed. Try http://gosper.org/pumper_3.mp4 . Better still, but 15M, is http://gosper.org/pumper.avi.zip . Perhaps NeilB can convert this to a wmv for us. Thanks, Fred, for catching this. --rwg
No manual entry for overboard
On Sun, Mar 18, 2012 at 9:54 PM, Fred lunnon <fred.lunnon@gmail.com>privately wrote:
Version 3 runs without choking; but still curiously grainy, compared with original (unpimped) version. WFL
On 3/19/12, Bill Gosper <billgosper@gmail.com> wrote:
Hi Fred, it could be your browser or your player. Does this work? --Bill On my (antique) system, this is very grainy; also intermittently halts in mid squidge, though that's possibly a problem at my end. WFL
On 3/19/12, Bill Gosper <billgosper@gmail.com < http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote:> To run it <http://gosper.org/pumper_2.mp4> continuously, download with> Miro, QuickTime-View-Loop.> The pumped shapes split lengthwise resemble boats, so a grand entrance> to a math museum would be a chain of these boats in an annular canal,> passing through four rotors acting as an (approximate) air|water lock,> with another four for the exit. Oh, the insurance.> --rwg
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Message: 5 Date: Mon, 19 Mar 2012 12:09:53 +0000 From: Veit Elser <ve10@cornell.edu> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Julian pimped my pump Message-ID: <AED8C691-40C2-4433-9289-40825D547372@cornell.edu> Content-Type: text/plain; charset="us-ascii"
Please send me some background on this. It reminds me of the "cubic Archimedean screw":
http://rsta.royalsocietypublishing.org/content/354/1715/2071
http://archive.msri.org/about/sgp/jim/geom/level/library/elser/animations/in...
Veit
On Mar 18, 2012, at 10:01 PM, Bill Gosper wrote:
To run it <http://gosper.org/pumper_2.mp4> continuously, download with Miro, QuickTime-View-Loop. The pumped shapes split lengthwise resemble boats, so a grand entrance to a math museum would be a chain of these boats in an annular canal, passing through four rotors acting as an (approximate) air|water lock, with another four for the exit. Oh, the insurance. --rwg No manual entry for overboard _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Message: 6 Date: Mon, 19 Mar 2012 06:53:45 -0700 From: Henry Baker <hbaker1@pipeline.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Reducing the computational complexity of Roman arch construction Message-ID: <E1S9d2Y-0006S2-FE@elasmtp-junco.atl.sa.earthlink.net> Content-Type: text/plain; charset="us-ascii"
Roman arches are nearly always semicircles, or at least circular segments.
The usual explanation for this is lack of imagination or lack of understanding of the exact nature of the stresses involved.
I find this explanation highly insulting to engineers whose work still stands and functions today, after several thousand years of neglect. (One can only wonder if the work of those engineers who belittled the Roman engineers will remain standing even 1/10 of this amount of time.) It is also insulting to the intelligence of 500-750 years worth of very clever engineers, who had plenty of time and plenty of incentive to come up with more economical methods of construction.
Any construction engineers who constructed arches by the thousands and tens of thousands got very, very, very good at this art, and developed extremely sophisticated tools to be able to construct their arches quickly & with the least amount of extraneous material.
The Romans not only constructed single arches, but many of their buildings & bridges consisted of a series of arches, each depending upon one another for support, so that they could not have been built or functioned as isolated arches. This meant that they had to be built together, simultaneously.
After a search of the internet, it would appear that the most common suggestion for how to build an arch requires the building of a template for the arch -- typically out of wood -- and then using that template to hold the bricks or stones in place until the arch was complete, after which the template could be removed. But this means that the template would have to be strong enough to hold all of the stones in place until the keystone at the top completed the arch and it became self-supporting.
The "template" method is very general -- it can be used to build an arch of any shape -- so if someone goes to the trouble of building a template, then how come he doesn't utilize that template to build an arch that is optimized for some other feature -- e.g., a more perfect deflection of stresses, or a more elegant-looking arch? If the template method was so common, how come there wasn't a much greater variety of arches? We're talking about a _thousand years_ of arch construction here! Even dumb engineers eventually become bored & try other things over a period of 1,000 years.
Another problem with the template method -- especially with an arcade series of arches -- are that all of the templates have to be in place for all the arches prior to the installation of all the keystones. So the builder can't even re-use the template from one arch to the next.
An even more serious problem with the template method is the amount of wood required -- arched structures were built in many locations where wood was very scarce. Simply finding enough wood and transporting this amount of wood to the building site might have been almost as much work as cutting & transporting the stones and/or bricks.
My conclusion is that the Romans were more clever than this, and that the circular arc arch was the _necessary_, rather than the _incidental_, result of their method of arch construction.
What is a circular arc? It is the _rotation_ of a line segment of given fixed length around an axis, or pivot point. But it is that very _rotation_ that may provide the insight to how the Romans could have constructed their arches so quickly & economically.
Consider a circular clock face with typical numbering of the hours 1-12. The top half of this clock face is semicircular in form, going from "9:00" to "12:00" to "3:00".
Now consider such a "clock" with two equal-length "hands": hand #1 (the "left" hand), and hand #2 (the "right" hand). The "left" hand will stay in the range "9:00" to "12:00", while the "right" hand will stay in the range "12:00" to "3:00".
We will try to keep both the "left" and "right" hands at approximately the same elevation at all times. Thus, when the left hand is at "9:00", the right hand will be at "3:00"; when the left hand is at "10:00", the right hand will be at "2:00"; when the left hand is at "11:00", the right hand will be at "1:00".
The Romans must have realized that _if an arch was going to be able to stand up on its own after construction_, then the forces on each of the stones in the arch _during_ construction should approximate those _after_ construction. In particular, the stones further down in the arch -- e.g., those at "10:00" and "2:00" -- can only feel the forces just above themselves and just below themselves, so if those forces approximate those in the final arch configuration, they will be happy and won't fall down.
So the arch construction could proceed as follows. Build the arch "piers" -- the vertical stones below the start of the semicircular arch. Once these piers are completed & both sides are at the same level (no small feat it itself), a straight horizontal wooden beam can be built across the span from pier to pier.
(Most Roman arches have a little "lip" right at the top of these piers, which provides a very clear delineation between the pier and the arch above it. This "lip" is no accident, and comes in handy for a variety of uses, both during & after construction.)
In the center of this horizontal wooden beam which spans between the tops of the piers, a rotating "axle" is built, and the two "hands" of the "clock" are constructed which will rotate/pivot on this axle.
The angled stones ("voussoirs") forming the perimeter of the arch can then be subsequently placed at "9:00" and "3:00", "10:00" and "2:00", etc., using the left and right hands as braces. The key insight for the Romans was that _these hands could be moved/rotated_ during construction, and the already placed lower voussoirs (angled stones) would be content to stay without any bracing, since the forces above and below them would hold them in place.
Once both hands had come together at the very top ("12:00") of the arch, the keystone could be put in place, at which point the axle, the two hands, and the supporting wooden span could all be removed.
Note that each of the clock "hands" would only have to hold one of the stones at a time -- _not_ the entire arch -- and thus transmit only the force of at most 2 of the stones to the pivoting axle. Thus, the amount of wood required is substantially less than that required for the "template" method, and the "clock" method could readily be used on multiple arches in an arcade.
If necessary, the end of each of the left & right "hands" could have a small circular _wheel_, which would ease the task of moving each "hand" upwards, as the little wheel would rotate along the most recently place stone instead of rubbing along it.
Note that approximately the same technique could also be used for concrete construction, so long as the concrete hardened & became strong at the same rate as the construction proceeded. It appeared that this was indeed the case for Roman concrete, which used far less water than modern concrete, and was tamped/hammered into place to make sure that there were no air pockets ("voids").
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