C. Auguste Dupin able to read the mind of his companion, the unnamed narrator of all three stories. Poe created the Dupin character before the word "detective" had been coined. The character laid the groundwork for fictional detectives to come, including Sherlock Holmes, and established most of the common elements of the detective fiction genre. Dupin is from what was once a wealthy family, but "by a variety of untoward events" has been reduced to more humble circumstances, and contents himself only with the basic necessities of life. He now lives in Paris with his close friend, the anonymous narrator of the stories.
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