Interactive classroom materials for exploring finite sets, permutations, group actions, and induced dynamics using the NUMO Field as a concrete 10-state model.
We begin with a finite set of 10 elements, $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Unlike a standard set, $S$ has intrinsic topological structure defined by four distinct layers. This provides a concrete "geography" for abstract algebra concepts.
The core kernel. Acts as the identity or null space in many operations.
The inversion boundary. Elements here often act as twist points or generators.
Stabilizers. These elements often remain fixed under reflection operations.
The cyclic group $C_4$. Supports the primary dynamic circulation.