http://www.egyptorigins.org/catenarycurve.htm "An Ancient Egyptian Catenary Construction Curve" "When I first examined the curve displayed by the Saqqara Ostracon I recognized that it was not circular in form. My first attempt to define the curve also showed it was not elliptical. Refer to remarks by Gunn, who confirms both assessments. I also could not believe it was a hyperbolic." "The following solution should be regarded as incorrect, but curious. I offer it as an example of how solutions can reach far afield in attempts to understand the ancient Egyptian geometricians." "I was haunted by the possibility that it might be a catenary." ... http://www.egyptorigins.org/saqqaraostracon.htm "(Refer to my proposed mathematical solution in the accompanying paper, An Ancient Egyptian Catenary Construction Curve, where I explain the difference in "sharpness" of the drop. EPM)" At 02:02 PM 7/14/2015, W. Edwin Clark wrote:

Try Google Images Egyptian catenary arch <https://www.google.com/search?q=Egyptian+catenary+arch&safe=active&biw=1280&bih=899&source=lnms&tbm=isch&sa=X&ved=0CAYQ_AUoAWoVChMIkP35rsHbxgIVEpqICh2xsAMN>

On Tue, Jul 14, 2015 at 4:54 PM, James Propp <jamespropp@gmail.com> wrote:

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Interesting!

If someone can steer me towards a picture of an Egyptian catenary(-ish) arch, I can probably use it in my essay, which will be going live on Friday.

Thanks,

Jim

On Tue, Jul 14, 2015 at 3:00 PM, Henry Baker <hbaker1@pipeline.com> wrote:

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The Egyptians experimented with catenaries for St Louis-type arches perhaps 3000-4000 years ago.

I've seen a picture in a book somewhere showing a quite good approximation to a catenary arch from ancient Egypt.

Of course, they didn't know it as a "catenary", and obviously knew nothing about cosines -- hyperbolic or otherwise. It is conceivable that the ancient Egyptians discovered the Gaudi's principle, wherein hanging catenaries become catenary arches when flipped over and compression & tension exchange places. If some clever Egyptian had discovered this principle, it was quickly lost, as the ancient world dnd't seem to be filled with catenary curves (excepting ship rigging, which the ancient Egyptians excelled at).

It is extremely interesting to me that once the Greeks & Babylonians discovered semicircular arches & parabolic arches, no more experimentation seems to have been done with other shapes.

In particular, the ancient Romans (to my knowledge) used ONLY circular arches, even though the amount of material required was substantially greater than what would be required for other types of arches.

The Romans were outstanding engineers, and while waiting for their slave labor to carry the excess material required to build a circular arch, these smart people had plenty of time to ponder whether there was a better way.

There are several possible explanations why they stuck with circular arches:

1. The Greeks became enamored of the elegance of circles, and -- whether or not another shape would have been more "efficient" -- they preferred circles for aethetic/moral/religious reasons. Analogous problems limited astronomers until Galileo risked death/excommunication to argue for ellipses.

2. The ancient Romans may have depended upon a method of building arches that depended critically upon their being arcs with a consistent radius. I have conjectured that the ancient Romans devised a system of "centering" that did NOT require a huge amount of scaffolding. This is because the Romans built so many arches, in so many locations -- including locations without any lumber for scaffolding -- that they required a modular method using a minimal amount of lumber that had to be carried along with the troops for other purposes -- including building siege towers and catapults. In this case, the Romans weren't worrying about minimizing material within the arch itself, but minimizing the material required for the "centering" of the arch.

https://en.wikipedia.org/wiki/Centering

3. Strict liability. The engineers who designed & built structures in the ancient world were required to stand under them when load was first applied. This strict liability principle guaranteed significant overbuilding -- which is why some Roman roads & arches are still in use today after 2000 years -- but so severely penalized experimentation that no one was brave enough to consider alternatives.

http://www.commonlaw.com/Hammurabi.html

229. If a builder has built a house for a man, and has not made his work sound, and the house he built has fallen, and caused the death of its owner, that builder shall be put to death.

230. If it is the owner's son that is killed, the builder's son shall be put to death.

231. If it is the slave of the owner that is killed, the builder shall give slave for slave to the owner of the house.

At 12:35 PM 6/30/2015, James Propp wrote:

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I've finished a draft of installment #1 (as opposed to #0) of "Mathematical Enchantments", and I'd welcome comments.