C. Auguste Dupin with an emphasis on the analysis and not trial-and-error. Brander Matthews wrote: "The true detective story as Poe conceived it is not in the mystery itself, but rather in the successive steps whereby the analytic observer is enabled to solve the problem that might be dismissed as beyond human elucidation." In fact, in the three stories which star Dupin, Poe created three types of detective fiction which established a model for all future stories: the physical type ("The Murders in the Rue Morgue"), the mental ("The Mystery of Marie Rogêt"), and a balanced version of both ("The Purloined Letter"). Fyodor
Dupin cyclide of Dupin cyclides is invariant under the larger group of all Lie sphere transformations; any two Dupin cyclides are Lie-equivalent. They form (in some sense) the simplest class of Lie-invariant surfaces after the spheres, and are therefore particularly significant in Lie sphere geometry. The definition also means that a Dupin cyclide is the envelope of the one-parameter family of spheres tangent to three given mutually tangent spheres. It follows that it is tangent to infinitely many Soddy's hexlet configurations of spheres. A further special property of a cyclide is: An elliptic cyclide can be represented parametrically by the following formulas