In case anyone ever asks, the shape of the Colosseum in Rome is a pretty decent approximation to an ellipse, given by a sequence of 8 circular segments which are continuous & have continuous first derivatives. It is well known that the evolute of the ellipse is an astroid (though not from outer space); the evolute traces the centers of curvature, hence the centers of the circular arcs that form the circumference of the Colosseum. As an approximation, the Romans chose 8 points on the astroid & constructed 8 arcs that were continuous. Thus, at each join point, the _radius_ of the arc changes discretely to the next point on the astroid, while the arc itself remains continuous with continuous derivative to the next arc. The 8 points were the 4 "corners" of the astroid, plus 4 intermediate points; the points are symmetric in both the x & y directions. The major axis is approx. 187.2 meters; the minor axis is approx. 157.3 meters. Since the seating has approximately the same seat-seat distance all around the "ellipse", the internal oval is _not_ an ellipse, as it does not scale linearly with the exterior elliptical walls. It's highly unlikely that the Romans knew anything about evolutes or astroids, but they did seem to like the ellipse. At some point in the future, I'm going to attempt to calculate the maximum deviation from "elliptical" of this Roman construction. I suspect that the error is of the order of single digit feet. I don't know if the foci of the Colosseum have any particular role in the events of the Colosseum. Due to the elliptical shape of the Colosseum, and the fact that the highest (wooden) rank of seats was very steep, the Colosseum would have some aspects of a "whispering gallery", whereby sound at one focus would be easily heard at the other focus. This effect would be even more pronounced when the canvas covering was in use. One can only wonder if the Romans ever performed Arthur C Clarke's "sunstroke" in the Colosseum by having all the spectators use mirrors to burn up someone on the field. Archimedes had already shown them how it could be done. Perhaps this is why ancient Romans didn't take cellphones to the Colosseum -- to avoid the heated displeasure of the crowd ? http://variety-sf.blogspot.com/2007/09/arthur-clarkes-slight-case-of-sunstro... Taylor, Rabun. Roman Builders: A Study in Architectural Process. Cambridge U. Press, 2003, p. 139.