Here's an eclectic mix of examples from various perspectives across multiple domains. Understanding the representation of knowledge and the fundamental structur

Here's an eclectic mix of examples from various perspectives across multiple domains. Understanding the representation of knowledge and the fundamental structure that underpins the connective essence of things is one of the prime ideas that drives my thinking, and as a consequence much of what I post about on here is related to that in some way. Below is a brief sampling of examples related to this idea, and for a more extensive list, browse through my postings (and if you can see how they all connect, let me know!). In general, see the Classification of Finite Simple Groups [0]. In terms of specific examples, the minimal classifications of knots [1], trees [2], and braids [3] are 3 instances that popped to mind. From a number-theoretic perspective, the paper Enumerating the Rationals and its derivative works are interesting reads. In terms of spatial structure and spatial decomposition, look at the classification of crystal structures [4], lattices, polytopes (zonotopes are particularly interesting), periodic and aperiodic tilings, the classification of space-filling curves, and the classification of closed surfaces [5]. From a relational perspective, look at the classification of topological spatial relations [6] and Allen's interval algebra [7]. And the one my intuition finds most intriguing is the classification of Group Theory Single Axioms and its relation to my conjecture that division is primary. [0] Classification of Finite Simple Groups https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.wikipedia.org/wiki/Classification_of_finite_simpl... [1] Prime Knots https://en.wikipedia.org/wiki/List_of_prime_knots https://en.wikipedia.org/wiki/List_of_prime_knots [2] Enumerating Trees https://www.cs.virginia.edu/~lat7h/blog/posts/434.html https://www.cs.virginia.edu/~lat7h/blog/posts/434.html [3] Braid Group https://en.wikipedia.org/wiki/Braid_group https://en.wikipedia.org/wiki/Braid_group [4] Crystal Structure https://en.wikipedia.org/wiki/Crystal_structure https://en.wikipedia.org/wiki/Crystal_structure [5] Classification of Closed Surfaces https://en.wikipedia.org/wiki/Surface_(topology)#Classification_of_closed_surfaces https://en.wikipedia.org/wiki/Surface_(topology)#Classificat... [6] Topological Spatial Relations https://en.wikipedia.org/wiki/Spatial_relation https://en.wikipedia.org/wiki/Spatial_relation [7] Allen's Interval Algebra https://en.wikipedia.org/wiki/Allen%27s_interval_algebra https://en.wikipedia.org/wiki/Allen%27s_interval_algebra [8] "Yet Another Paper on Group Theory Single Axioms" https://www.cs.unm.edu/~mccune/projects/gtsax/ https://www.cs.unm.edu/~mccune/projects/gtsax/ P.S. Clifford Algebra / Geometric Algebra [00] unifies much of mathematics, e.g. in GA, Maxwell's equations can be reduced to one equation and expressed on one line [01]: [00] Geometric Algebra https://en.wikipedia.org/wiki/Geometric_algebra https://en.wikipedia.org/wiki/Geometric_algebra [01] Maxwell's equations formulated in terms of Geometric Algebra https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field#Geometric_algebra_formulations https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...