Crystallization of space: Space-time fractals from fractal arithmetic

Abstract

Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.

version accepted in Chaos, Solitons & Fractals

References in corpus (7)

• Fractal properties of quantum spacetime
• Dimensional flow in discrete quantum geometries
• Fractal Dimensions of a Weakly Clustered Distribution and the Scale of Homogeneity
• Un-spectral dimension and quantum spacetime phases
• Relativity of arithmetic as a fundamental symmetry of physics
• Halos and voids in a multifractal model of cosmic structure
• New Gauge Field from Extension of Space Time Parallel Transport of Vector Spaces to the Underlying Number Systems

Cited by in corpus (10)

• Relativity of arithmetic as a fundamental symmetry of physics
• Unifying Aspects of Generalized Calculus
• Arithmetic loophole in Bell's theorem: An overlooked threat to entangled-state quantum cryptography
• Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus
• Non-Newtonian mathematics instead of non-Newtonian physics: Dark matter and dark energy from a mismatch of arithmetics
• A loophole of all `loophole-free' Bell-type theorems
• Simple fractal calculus from fractal arithmetic
• Imitating quantum probabilities: Beyond Bell's theorem and Tsirelson bounds
• Dark energy as a manifestation of nontrivial arithmetic
• Effect of number scaling on entangled states in quantum mechanics