Extraversion Hub
Extraversion is a symmetry on triangle constructions discovered by John Conway and Georgey Kapetis. It began with Lemoine's abstract theory of the triangle, a project he never completed. Klein also spoke of an abstract theory of the triangle, but other than to define triangle geometry as the set of projective transformation that preserve 5 points, never implemented it. Extraversion can be implemented as a continuous symmetry of the triangle (see Geometric Extraversion, below). Finally it is Galois theory applied to the triangle (see from Euclid to Abstract Algebra).
The extraversion symmetry is a non-classical one that allows us to divide triangle centers into equivalence classes. These classes are
Strong points which come in one version. The theory of strong points is primarily the Brocard theory of the triangle.
Fragile points: These have two versions. Examples are the Fermat and Isodynamic points.
Quartile points: These points have four versions. Examples are the incenter and Nagel points.
Hexile points: These points have 6 versions. An example are the 6 midpoints of the 4 incenters, which are all on the circumcircle.
Octile points: These points have eight versions. Examples are most known points on the Simmons conics.
 |
From Euclid to Abstract Algebra. This essay uses Euclid's introduction of "incommensurables" as a way to talk about how algebraic differences are related to geometric differences. |
 |
Geometric Extraversion, this is a continuous symmetry by which one continuously varies an angle of the triangle to zero and through zero. |
 |
Desmic systems, two sets of quartile points arranged as a projected cube. |
 |
2-fold (fissile) points here told using the Fermat, Napoleon, and isodynamic points as examples. |
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|