Our e-dimensional universe
Our sense of the three-dimensionality of space arises from our intuition. It is basic to the models the mind creates to describe the world.
But the mind is not always right. For example, it takes thunder to come much later than lightening, even though they are simultaneous.
What if real space is not three-dimensional although our mind insists it is so? The discrepancy between the two could be responsible for many basic errors in our understanding of reality.
The basis of my theory that physical space is e-dimensional (e = 2.71828…) is a theorem in mathematical logic according to which representation in e is most efficient (for links to the basic papers, see [1][2]). Since nature is optimal, space should be in accord with this logic.
It is easy to see that it is a surprising idea that could lead to new physics, but how can physical dimensions be anything but 3? And the most troubling thing is that it is not even a rational number!
The answer is that three-dimensional view of physical space is merely a convention that we have become used to, which is consistent with the expectation of our senses. Reality is not exactly three-dimensional but our mind maps it into a three-dimensional frame. We use this convention to mark points on Earth, but we can’t really check this easily at either the local or the cosmological levels.
In the big bang model, the universe expanded from a singularity (initial state of extremely high density and high temperature) which explains the cosmic microwave background (CMB) radiation, and together with gravity (whose origin is unknown but which is seen to work through either Newton’s or Einstein’s theory) it is consistent with the large-scale structure of the universe with its patterns of galaxies and matter on scales much larger than individual galaxies or groupings of galaxies.
There is divergence between the expansion rates obtained from early universe (captured by CMB data) and the late universe (considering the receding of stars and galaxies) aspects of the universe.
The divergence can be explained away of we accept that space is not quite three-dimensional, but rather e-dimensional
Why hasn’t it been thought of before?
The three-dimensional nature of space is an implicit assumption in Western physical thought and so it has not been questioned. When the idea of information is probed deeply we realize that mathematics compels us to abandon the assumption of a three-dimensional space (References[3][4][5]).
The beginnings of the universe and the nature of space are connected to many deep questions concerning not only physics but also philosophy. These include:
What is the origin of space?
What are the ultimate components of the universe?
What is the relationship between physics and consciousness?
The answers to these questions depend on our models of the universe. In Indian physics, the deepest aspect of reality is consciousness, out of which emerge five elements that are the constituents of matter. The universe goes through cycles of creation and dissolution, from consciousness to embodied reality and back to consciousness.
What is space?
The formal use of three-dimensional space is a part of modern physics. We can also speak of the biological space within which biological structure evolve.
Newton took space to be absolute and to be three-dimensional. This was extension of the Aristotelian view of the universe as a container in which the sun, the moon, planets and stars are embedded in perfectly concentric crystal spheres that rotate at fixed rates.
In the observer-centric Indian physics that goes back to Kaṇāda, physical laws must be based only on substances, their properties, and their motion, but the experience of time and space is a consequence of the relation between the observer and the world being observed. Furthermore, akasha, which represents primordial space in the Indian view, is active.
There is striking similarity between physical and biological structures. This must be the result of the commonality in the nature of physical and biological spaces, or perhaps the two are identical. Patterns in the neural structures of the brain and the filament structure and distribution of matter in cosmology are quite similar.
But how morphogenesis may be related to biological space is not known. Indeed this could very well become an exciting new field of research connecting physics and biology.
A new theory of gravity
Let us consider the dimensionality potential of an object. The greater the separation between two points, the greater is the dimensionality attraction [2], as in the example of d =0.7 where objects will tend to get closer to each other by this factor.
Although the potential p(r) will increase with distance r, it should be divided by the surface area, A, of the equipotential surface around it. In other words, p(r) is proportional to r/A.
3-D world
A three-dimensional world consists of 2< d ≤3. The surface area of the equipotential sphere is 4πr² . Therefore, p(r) is proportional to r/4πr² or proportional to 1/r.
Since force is the derivative of the potential, we find that the dimensional force between two objects is inversely proportional to square of the distance, and this could well be the attractive gravitational force.
2-D world
A two-dimensional world represents the case 1≤ d ≤ 2. The surface area of the circle, the equipotential surface in two dimensions, is 2πr. Or, p(r) is a constant.
The derivative of this is zero, and thus there is no dimensional force between two objects in a two-dimensional universe. This represents the case of asymptotic freedom and this could very well be the explanation of similar lack of attraction between near object in particle physics. Objects that operate in the 2-D world will experience no attractive force.
1-D world
A one-dimensional world represents the case 0≤ d ≤ 1. The area of the equipotential surface in one dimension, is 2Z, where Z is the width of the one-dimensional space. The p(r) is proportional to r/2Z, which for small Z will be huge. The force will be proportional to 1/2Z and this could be the explanation for the stability of filaments, threads and other one-dimensional patterns,
The new papers provide an explanation for gravity that has been missing in physics, for Newton’s law was based on experiment. As explained above, the theory doesn’t change the inverse square law of gravitation but explains why it has this form and further suggests that gravity has weakened by about 20% in the last 4 billion years (for this, read References [6][7]).
According to current theory, about 96% of the universe is dark matter and dark energy (of which there is no direct evidence) and what is observable is only 0.5% (because another 3.5 % is interstellar dust). Although some scientists are confident that dark matter and dark energy will be eventually discovered, there is no observational evidence in support of it. My theory shows there is no need to speak of dark matter and dark energy [6].
If we assume that the past history of the universe is its evolution from dimension 0 (Big Bang) through dimensions that go from 1 to 2 and beyond (in real numbers) to the goal of reaching the optimal dimensionality of e, then one can explain [6][8] that the currently accelerating expansion of the universe will eventually slow down and finally reverse [6]. This ties up many loose ends in current physics.
Meaning of e-dimensionality
What is the meaning of an e-dimensional universe? To answer this, we must ask what we mean by the word “dimension” (see References [2]–[9]).
Dimension 0 is a point, dimension 1 is a line, dimension 2 is a plane, and dimension 3 is a solid. An object with dimension between 2 and 3, or e-dimensions, is like sponge or cheese. Another way of seeing it as an object whose density in the limit is less than that of a three-dimensional object.
Fractals have dimensionality that is often noninteger. Two examples of fractals are the Whirlpool Galaxy and the Nautilus shell shown below.
But how can space be like a sponge, with holes? The answer to this is that the sponge-view is one way of looking at space; another is that dynamics itself is an expression of this sponge-like nature. Such disparate views can be harmonized by the principle of complementarity, which is one of the deepest philosophical ideas in science (see [10] and references therein, and [11] for related constraints on data).
The most astonishing thing about noninteger dimensionality is that it can be shown to be the origin of gravity. If gravity is a property of space, it solves a puzzle for which science has had no answer until now.
This research also explains the counterintuitive idea of asymptotic freedom [2].
It can have uses for the military, for space travel, and for the understanding of turbulence that Feynman called “the most important unsolved problem of classical physics.” It may also lead to insights that help in the design of novel metamaterials.
It also opens up new perspectives on biological codes and neuroscience that will be described at length in a separate essay.
Notes
- Kak, S. Information theory and dimensionality of space. Scientific Reports 10, 20733 (2020). s41598–020–77855–9
- Kak, S. Asymptotic freedom in noninteger spaces. Scientific Reports 11, 1–5 (2021). https://www.nature.com/articles/s41598-021-83002-9
- Kak, S. Information, representation, and structure. International Conference on Recent Trends in Mathematics and Its Applications to Graphs, Networks and Petri Nets, New Delhi, India (2020).
- Kak, S. The base-e representation of numbers and the power law. Circuits Syst. Signal Process. 40, 490–500 (2021); https://doi.org/10.1007/s00034-020-01480-0
- Kak, S. The intrinsic dimensionality of data. Circuits Syst. Signal Process. 40, 2599–2607 (2021); https://doi.org/10.1007/s00034-020-01583-8
- Kak, S. Evolutionary stages in noninteger dimensional spaces. Indian J. of Physics 97, 3041–3045 (2023); https://doi.org/10.1007/s12648-023-02653-8
- Kak, S. Information-theoretic view of the gravitational constant in Dirac’s large numbers hypothesis. Indian J. of Physics 97, 503–507 (2023); https://doi.org/10.1007/s12648-022-02431-y
- Kak, S. and Kafatos, M. Black holes, disk structures, and cosmological implications in e-dimensional space. Physics Essays 35, 345–355 (2022).
https://doi.org/10.4006/0836-1398-35.4.345 - Kak, S. Fractals with optimum information dimension. Circuits Syst. Signal Process. 40 (2021); https://link.springer.com/article/10.1007/s00034-021-01726-5
- Kak, S. Epistemic view of quantum communication. In Quantum Foundations, Probability and Information. Khrennikov, Andrei, Toni, Bourama (Eds.). Springer (2018).
- Kak, S. The iterated Newcomb-Benford distribution for structured systems. Int. J. Appl. Comput. Math 8, 51 (2022); https://doi.org/10.1007/s40819-022-01251-2
NOTE
For a deeper discussion for the layperson, see here: https://www.academia.edu/49175956/The_Measure_of_Space
The companion paper: Dimensionality of biological space
