Hexgetal

Hexgetal is a fictional algebraic concept used in recreational mathematics and speculative discussions of hexagonal symmetry. The term blends hexagonal imagery with the sense of a numeric object, and it is not part of established mathematics. In common fictional treatments, a hexgetal represents an element of a six-directional algebra defined on a regular hexagonal lattice.

In these definitions, a hexgetal is typically expressed as a finite linear combination of six directional basis elements, often denoted e1 through e6, with integer coefficients. Addition is performed componentwise, while multiplication is defined to reflect hexagonal adjacency, sometimes by rules that rotate directions and combine neighboring coefficients. Some variant models treat hexgetals as elements of a graded or colored algebra, where the grade or color tracks hexagonal distance or orientation relative to a reference cell.

Variants of the concept vary in central properties. Some proposals yield commutative rings, others noncommutative algebras; some define a norm or magnitude based on the sum of absolute coefficients, while others use geometric interpretations tied to tiling. Because hexgetal is not standardized, different authors may present different axioms, multiplication tables, and representations.

History and usage in literature are informal. Hexgetal has appeared in puzzle communities and worldbuilding discussions as a tool for exploring how geometry influences algebraic structure, coordinate representation on hex grids, and the idea of base-n systems with spatial meaning. It is primarily a thought experiment and educational curiosity rather than a formal mathematical object.

See also: hexagonal lattice, hexagonal tiling, base-6 numeral system, noncommutative algebra, recreational mathematics.