| Klein’s Erlangen program
In 1872, while lecturing at Erlangen, Germany, Felix Klein founded what became known as the ‘Erlangen program’, an attempt to unify geometry as a single mathematical discipline. This program hopes to unify the following geometries: Euclidean, Bolyai-Gauss-Lobachevsky (BGL), Riemannian, Poncelet’s projective geometry and the most difficult to swallow Eulerian topology.
His proposal was a geometry that studies transformation group symmetry and invariance, for the plane, for space, and for whatever anyone has in mind.
For Euclidean geometry, these invariant properties can be found within the transformations for rotations, translations, reflections, and for similarities.
The objectives for finding these invariant symmetrical properties are for the determinations of minimum-maximum points, stability and equilibrium points for all investigations of linear and nonlinear dynamical systems, physical as well as mathematical, concrete as well as abstract.
One of the outcomes of this program is the concept of ‘sphere packing’, widely used in supermarkets for stacking apples and oranges. But a more abstract minmax stability is the abstract groupings of Hadamard matrices. The results give two fundamental abstract structures: the cubic compact sets and the tetrahedral compact sets.
When applied to elementary particles, the tetrahedral sets can be used to describe the abstract odd number groupings for the quarks, while the cubic sets can be used to describe the odd number groupings for the leptons. The even number groupings of both sets can be used to describe bosons. And the mixed groupings of odd and even can be used to describe the structure of spacetime.  | 
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