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Fractal Algebra A Mathematical Allegory

Author

Listed:
  • JOSEPH O’NEILL

    (Division of Child and Adolescent Psychiatry, UCLA Semel Institute for Neurosciences, 760 Westwood Plaza 58-227A, Los Angeles, CA, USA)

  • ANDREAS SCHOTH

    (IMTEK Department for Process Technology, Institute of Microsystem Technology, Universität Freiburg, Georges-Köhler Allee 103, Freiburg im Breisgau, Germany)

Abstract

An abstract fractal algebra is developed with structures analogous to conventional linear algebra. Focus is on group properties and symmetries forming an abstract algebraic structure over the body of real numbers. The new algebra is worked out on a sample fractal geometry. Correspondences between fractal and linear algebra are explained with numerical examples, including fractal vectors and fractal versions of the vector sum, projections, scalar product, cross product, matrix, and matrix product. We allude to possible applications of fractal vectors in physics. This fractal algebra may open novel possibilities for analysis of fractal systems.

Suggested Citation

  • Joseph O’Neill & Andreas Schoth, 2023. "Fractal Algebra A Mathematical Allegory," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(01), pages 1-40.
    • Handle: RePEc:wsi:fracta:v:31:y:2023:i:01:n:s0218348x23500202
    DOI: 10.1142/S0218348X23500202
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