Emergent dimensionality: Comparison
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The principle of emergent dimensionality states that 3-dimensional reality does not exist observer-independently but emerges, for a living agent, from an omnidimensional graph of nature that contains all unobservable extra dimensions.

  • emergent dimensionality
  • assembly theory
  • holographic principle
  • entropic gravity
  • mathematical physics

1. Introduction

A dimension n is usually defined as a natural number of coordinates needed to specify the position of a point within the Euclidean space ℝRn. However, this is not the only possible definition[1][2] of a dimension: analytic continuations from positive dimensions[2][3] define negative dimensions[4]; fractional (or fractal), including negative[5], dimensions consistent with experimental results[6][7]; complex[2], including complex fractional[8], dimensions can also be considered[9][10].

Furthermore, it is known, in particular, that

  • all attempts to construct a local realist model of quantum phenomena are doomed to fail, which is asserted by Bell's theorem[11][12];
  • the identity of indiscernibles is neither a logical nor empirical principle.
  • every two objects one perceives[13] are equally similar (or equally dissimilar), which is asserted by the ugly duckling theorem[14], and hence, the identity of indiscernibles is neither a logical nor empirical principle.
  • all attempts to construct a local realist model of quantum phenomena are doomed to fail, which is asserted by Bell's theorem[11][12];
  • observer-independent facts do not exist, which is asserted by a No-Go Theorem for Observer-Independent Facts[15][16][17];
  • every two objects one perceives[13] are equally similar (or equally dissimilar), which is asserted by the ugly duckling theorem[14], and hence, the identity of indiscernibles is neither a logical nor empirical principle.

The question then arises as to why living observers perceive the world in three real spatial dimensions and imaginary time.

It was shown[18] that there is a continuum of differentiable manifolds that are homeomorphic (i.e., shape preserving) but non-diffeomorphic (i.e., nonsmooth) to the Euclidean space ℝ4, known as exotic ℝ4, and this property is absent in other dimensions. This mathematical property is exploited by biological evolution as it ensures that an individually memorized object preserves the shape of an individually perceived[13] object (it is homeomorphic), but this preservation is non-smooth (it is non-diffeomorphic). This, in turn, ensures that the first two Lewontin principles for evolutionary change[19] (phenotypic variation and differential fitness) hold. Furthermore, in ℝ4, there are six regular, convex polytopes, whereas there are five (Platonic solids) in ℝ3, and only three for n > 4 and n < 0. 

The emergent dimensionality corroborates, in particular, with

  • the holographic principle, where a volume of space is considered to be encoded on its lower-dimensional boundary;
  • entropic gravity, showing that both inertia and gravity are also emergent phenomena[20];
  • assembly theory[21] that explains and quantifies selection and evolution[22] already at the binary level[23].

In assembly theory, objects are defined by the histories of their formation. The more complex a given object is, the less likely an identical copy can be observed without the selection of some information-driven mechanism that generated that object.

References

  1. Dierk Schleicher; Hausdorff Dimension, Its Properties, and Its Surprises. Am. Math. Mon.. 2007, 114, 509-528.
  2. Yuri I. Manin; The notion of dimension in geometry and algebra. Bull. Am. Math. Soc.. 2006, 43, 139-162.
  3. V. P. Maslov; General notion of a topological space of negative dimension and quantization of its density. Math. Notes Acad. Sci. USSR. 2007, 81, 140-144.
  4. G. Parisi; N. Sourlas; Random Magnetic Fields, Supersymmetry, and Negative Dimensions. Phys. Rev. Lett.. 1979, 43, 744-745.
  5. Benoit B. Mandelbrot; Negative fractal dimensions and multifractals. Phys. A: Stat. Mech. its Appl.. 1990, 163, 306-315.
  6. Boming Yu; FRACTAL DIMENSIONS FOR MULTIPHASE FRACTAL MEDIA. Fractals. 2006, 14, 111-118.
  7. Boming Yu; Mingqing Zou; Yongjin Feng; Permeability of fractal porous media by Monte Carlo simulations. Int. J. Heat Mass Transf.. 2005, 48, 2787-2794.
  8. Lapidus, Michel L. An overview of complex fractal dimensions:from fractal strings to fractal drums, and back; Niemeyer, Robert and Pearse, Erin and Rock, John and Samuel, Tony, Eds.; American Mathematical Society (AMS): Providence, RI, United States, 2019; pp. 1.
  9. Szymon Łukaszyk; Novel Recurrence Relations for Volumes and Surfaces of n-Balls, Regular n-Simplices, and n-Orthoplices in Real Dimensions. Math.. 2022, 10, 2212.
  10. Szymon Łukaszyk; Andrzej Tomski; Omnidimensional Convex Polytopes. Symmetry. 2023, 15, 755.
  11. J. S. Bell; On the Einstein Podolsky Rosen paradox. Phys. Phys. Fiz.. 1964, 1, 195-200.
  12. The Nobel Prize in Physics 2022. The Nobel Prize. Retrieved 2024-8-9
  13. Szymon Łukaszyk; Life as the Explanation of the Measurement Problem. J. Physics: Conf. Ser.. 2024, 2701, 012124.
  14. Satosi Watanabe; Epistemological Relativity. Ann. Jpn. Assoc. Philos. Sci.. 1986, 7, 1-14.
  15. Časlav Brukner; A No-Go Theorem for Observer-Independent Facts. Entropy. 2018, 20, 350.
  16. Massimiliano Proietti; Alexander Pickston; Francesco Graffitti; Peter Barrow; Dmytro Kundys; Cyril Branciard; Martin Ringbauer; Alessandro Fedrizzi; Experimental test of local observer independence. Sci. Adv.. 2019, 5, eaaw9832.
  17. Kok-Wei Bong; Anibal Utreras-Alarcon; Farzad Ghafari; Yeong-Cherng Liang; Nora Tischler; Eric G. Cavalcanti; Geoff J. Pryde; Howard M. Wiseman. Testing the reality of Wigner's friend's experience; SPIE-Intl Soc Optical Eng: Bellingham, WA, United States, 2019; pp. 112001C.
  18. Clifford Henry Taubes; Gauge theory on asymptotically periodic {4}-manifolds. J. Differ. Geom.. 1987, 25, 363-430.
  19. R. C. Lewontin; The Units of Selection. Annu. Rev. Ecol. Syst.. 1970, 1, 1-18.
  20. Erik Verlinde; On the origin of gravity and the laws of Newton. J. High Energy Phys.. 2011, 2011, 1-27.
  21. Stuart M. Marshall; Alastair R. G. Murray; Leroy Cronin; A probabilistic framework for identifying biosignatures using Pathway Complexity. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci.. 2017, 375, 20160342.
  22. Abhishek Sharma; Dániel Czégel; Michael Lachmann; Christopher P. Kempes; Sara I. Walker; Leroy Cronin; Assembly theory explains and quantifies selection and evolution. Nat.. 2023, 622, 321-328.
  23. Szymon Łukaszyk; Wawrzyniec Bieniawski; Assembly Theory of Binary Messages. Math.. 2024, 12, 1600.
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