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Variable Shape Geometry (also known as Bess Geometry) is a fairly new system of geometry attributed to the rather unknown mathematician Val Bess. It does not fall into the category of Euclidean Geometry but rather into the category Non-Euclidean Geometry, using concepts from both of them.

The name Variable Shape is given because the geometry deals with triangles being the same as squares, squares being the same as pentagons, and so on and so forth, giving the geometry the outlook that "Everything is everything". This is the more advanced part of the geometry and is usually centered around the basics instead of the complicated end.

One of the main concepts of Variable Shape Geometry is that it modifies Euclid's Parallel Postulate as does Non-Euclidean Geometry but not in the same way. Instead it establishes the Parabolism Postulate.

The Parabolism Postulate

As does Euclidean and Non-Euclidean Geometry, Variable Shape Geometry starts off with the basic line/point theorems and postulates (such as the Distance Postulate) to lay a foundation for the Parabolism Postulate. Before it can be postulated however, "parallel" lines must be introduced. Here, lines that are cut by a transversal must have congruent alternate interior, corresponding, or alternate exterior angles as does Euclidean Geometry to be considered parallel. However, parallel does not mean the lines never intersect in Variable Shape Geometry, so the term introduced for these lines is "parallo". Thus, the Parabolism Postulate can be introduced. It states:

If two lines are considered to be parallo, then two corresponding points on the line can be connected in an arc to form a parabola. When dealing with closed polygons, the parabola is not considered a side on its own.
— The Parabolism Postulate

In this context, "corresponding points" refers to points that are on a perpendicular line to the parallo lines. In basic parabola construction (such as in the image), the arc constructed is a semi-circle, but in more advanced geometry, the arcs formed are usually part of oblate or oblong ovals.

Shapes

When viewing Variable Shape Geometry as a whole, due to the fact that "everything is everything", nothing can be done. But if one were to prove that everything was in fact everything at the very end, then there is room to study individual shapes.

Triangles

Variable Shape Geometry strays away from the traditional Angle Sum Theorem (angles of a triangle add up to 180 degrees) and uses the Parabolism Postulate. With it, triangles with two right angles can be formed where the third angle is the measure of the arc formed by the parallo lines (due to the right angles). Triangles with two right angles are traditionally called "Bess triangles". Later on, triangles can be proven to be squares because of the Congruency Postulate, which is very complicated.

Squares

Like the triangles, unlikely squares can be formed using the Parabolism Postulate. The "Bess square" is a square with three obtuse angles (which is impossible with a traditional square).

Two-gon

One very unique aspect of Variable Shape Geometry is that a two-sided shape can be formed using the Parabolism Postulate. Later, complex two-gons can be formed because of a "Differential Angles Postulate" causing new types of parallel/parallo lines, but in the basic form, two-gons can only be "Bess two-gons".

Advanced Variable Shape Geometry

Advanced Variable Shape Geometry comes after the set of theorems that is associated with the individual shapes and their properties. The first postulate that marks the beginning of what is called Indiminent Geometry is usually the "Differential Angles Postulate" which spawns different parallo lines; ones that intersect before use of the Parabolism Postulate. Later, the "Indiminency Postulate" is introduced which forms "indiminent" lines; curved parallo lines.

Unlike other types of geometry, Variable Shape Geometry has a paradoxical end after the "Congruency Postulate" is introduced. It allows two-gons to be proven to be triangles, triangles to be proven to be squares, squares to be proven to be pentagons, and so on until two-gons/triangles/square/etc.. are proven to be infinite-sided shapes in which case everything is a circle. If the Congruency Postulate is avoided, Variable Shape Geometry has no end, but if it is taken into account, the geometry ends with a loop.

References

Val Bess, Variable Shape Geometry, 2007 (pamphlet)