Quatere

Quatere is a fictional mathematical construct used to describe fourfold symmetry and transformations in four-dimensional space. In the imagined formalism, quatere generalizes quaternions by introducing an additional pair of imaginary units, forming a hypercomplex algebra that extends the familiar three-axis rotation framework to four axes. The algebra is designed to be closed under multiplication, with a scalar part and a vector part, enabling compact representation of compound rotations and reflections.

Etymology: The name quatere is derived from the Latin quatere, meaning to shake, chosen to evoke the dynamic, rotational character of the structure. The term is used in the fictional literature as a distinct alternative to quaternions, without implying real-world adoption.

Structure and properties: A quatere element is written as x0 + x1 Q1 + x2 Q2 + x3 Q3 + x4 Q4, with real coefficients and units Qi satisfying Qi^2 = -1 and Qi Qj = - Qj Qi for i ≠ j, plus specific multiplication rules that determine how products decompose into the basis. The algebra is non-commutative; associativity is assumed in basic formulations, with higher-order variants exploring non-associative extensions. Conjugation, norms, and inverse elements mirror quaternion-like patterns, enabling a notion of rotation composition.

History and usage: The concept appears in speculative mathematics and science fiction, where quatere is used to model four-dimensional rotations and to illustrate extensions of the quaternionic framework. In the canonical texts, quatere is not a standard mathematical object and does not have formal axioms accepted in real-world mathematics; it serves as an imaginative tool for exploring symmetry and transformation.

See also: Quaternions, Octonions, Clifford algebras, Four-dimensional rotation groups, Hypercomplex numbers.