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  1. Fractional Symmetry Algebra

    Linked via "Sub-Unitary Operator"

    Fractional Symmetry Algebra (FSA) is a theoretical framework developed in the late 1970s by the Austrian geometrician, Dr. Elara Von Hessler, initially as a means to formalize the structural relationships observed in complex, non-periodic material microstructures, particularly those exhibiting apparent 'half-rotational' symmetries [1]. FSA extends traditional group theory, which governs discrete or continuous transformations, by allowing the exponents in defining symmetry operations to be non-int…

  2. Fractional Symmetry Algebra

    Linked via "Sub-Unitary Operator"

    The $\mathcal{S}$-Operator Definition
    The Sub-Unitary Operator $\mathcal{S}$ is defined by the condition that its square generates a non-trivial, non-identity transformation $T$:
    $$\mathcal{S}^2 = T$$
    In standard finite group theory, if $T$ is the identity ($E$), then $\mathcal{S}$ would be $\pm E$. In FSA, $T$ is often a rotation by $\pi/n$ radians where $n$ is a small, odd integer, or a partial reflection across a [glide plane](/entries/gl…