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_simpl...
[1] Prime Knots https://en.wikipedia.org/wiki/List_of_prime_knots
[2] Enumerating Trees https://www.cs.virginia.edu/~lat7h/blog/posts/434.html
[3] Braid Group https://en.wikipedia.org/wiki/Braid_group
[4] Crystal Structure https://en.wikipedia.org/wiki/Crystal_structure
[5] Classification of Closed Surfaces https://en.wikipedia.org/wiki/Surface_(topology)#Classificat...
[6] Topological Spatial Relations https://en.wikipedia.org/wiki/Spatial_relation
[7] Allen's 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/
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
[01] Maxwell's equations formulated in terms of Geometric Algebra https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...