<== This is a picture from the interactive geometry
package Cinderella showing the behavior of 10,000 starting
values in the rectangle [0,1]x[h-1,h+1], where h is the height of the
horizontal line, after six iterations of the algorithm which reflects
a point x in the sphere then reflects the outcome in the line and then
averages the result y with x. It is an accessible prototype for a
remarkable image reconstruction algorithm known variously as
Douglas-Ratchford, Lion-Mercier, Fienup's method, and
"divide-and-concur." Some related graphics can be generated and
displayed at these URLs:
(wait 30-60 seconds to see the display).
Quote of the day (refresh browser to select another):
There exists today a very elaborate system of formal logic, and specifically, of logic applied to mathematics. This is a discipline with many good sides but also serious weaknesses. ... Everybody who has worked in formal logic will confirm that it is one of the technically most refactory parts of mathematics. The reason for this is that it deals with rigid, all-or-none concepts, and has very little contact with the continuous concept of the real or the complex number, that is with mathematical analysis. Yet analysis is the technically most successful and best-elaborated part of mathematics. Thus formal logic, by the nature of its approach, is cut off from the best cultivated portions of mathematics, and forced onto the most difficult mathematical terrain, into combinatorics. -- John von Neumann, 1948, quoted in L. Blum, P. Cucker, M. Shub and S. Smale, Complexity and Real Computation, Springer-Verlag, New York, 1998.
The complete list of quotes is available