Batoric

Batoric is a term used in speculative mathematics and related fiction to describe a framework for studying symmetries that preserve a generalized energy form. In this usage, a batoric structure consists of a set X equipped with a symmetric bilinear form B: X × X → R, called the batoric form, and a collection of transformations f: X → X that satisfy B(f(u), f(v)) = B(u, v) for all u, v in X. Elements of this collection form a group under composition, often referred to as the Batoric group.

In finite-dimensional real vector spaces with B defined by a matrix M, the batoric group coincides with the classical orthogonal group O(M) = {f | f^T M f = M}. This connection helps ground the concept in familiar linear algebra, while the broader framing invites consideration of invariant structures beyond standard Euclidean geometry.

Applications of the idea are primarily conceptual and educational. Batoric frameworks are used to illustrate invariants, symmetries, and generalized geometries in abstract algebra and geometry, and they appear in thought experiments within physics and fiction to explore alternative spacetime or metric assumptions.

Origins of the term are informal and vary by author; batoric ideas have circulated in speculative literature and pedagogical discussions since the early 21st century. Because the concept is not standardized, different authors may define the batoric form and the corresponding symmetry group with varying levels of generality.

See also: Orthogonal group, Invariant theory, Bilinear form, Metric geometry.