Dimensionality of biological space

Subhash Kak
5 min readDec 12, 2023
Gecko amongst leaves

Biological systems look somewhat similar across scale and sometimes so even across species. In the picture above, it is difficult to separate the gecko from the leaves.

Lesula monkey

One also knows that dogs somehow come to resemble their owners [1]. And doesn’t this clear-eyed Lesula monkey in the picture above look like a dean at a university, or perhaps a philosopher?

The similarity in structures and networks and their data exchange across scale is a consequence of physical laws. Natural systems have self-similar properties and we see that in spiral structures in enormously varying scales that range from plants to galaxies [2]. Another example is that of the human inner ear (cochlea) that looks like a snail.

Owl butterfly (left) and an owl (right)


Nature mimics as in the insect below that looks like a leaf. Mimicry is when two or more organisms who are not closely related resemble each other to improve the survivability of one or both species. In one kind of mimicry, a harmless organism looks like a dangerous one, deterring potential predators or competitors. In aggressive mimicry, an animal looks like prey or a harmless species in order to lure an unsuspecting victim in.

Leaf insect

For the brain, in a case of similarity across scale, one may speak of physical geometry as in the arborization of dendrites and axons at the microscopic level, connectivity patterns of networks at the middle (mesoscopic) level, and the dynamics at the macroscopic level in the long-range, inter-area connecting fibers.

Neural dendrite arborization (left); desert shrub (right)

The geometries of microscopic and macroscopic arboreal structures are reasonably well mapped out, but the logic of the middle-level connectivity is not known. Do the structures at the microscopic and macroscopic levels lead to the emergence of self-similar networks at the intermediate description? Or, is the principle of scale-freeness the unified understanding that links the different scales?

Dimensionality and information

The dimension of an object is the number of coordinates needed for defining the position of points of the object. It is the number of degrees of freedom of a point that moves on this object. The dimension of a line is one, as a point can move on it along only one direction; likewise the dimension of a flat surface is two. If the objects have dimensions less than a certain value, then this upper limit may be the dimensions of the space containing the objects.

We normally consider the container to be the three-dimensional space, and in it the spatial orientation and positioning of different systems is critical for the developmental process to unfold properly. For example, the development of the orientation and number of fingers is driven in part by gradients of signaling molecules. If this spatially important cell signaling doesn’t work properly, it can lead to polydactyl or extra digits.

From e-dimensionality to scale-freeness

A mathematical theorem says that the optimal dimensionality is e, that is about 2.718. If reality has dimension that is non-integral, then mathematics requires the space to be recursive and scale-free, and this leads to fractal behavior.

I have previously discussed the application of e-dimensionality (required for optimality) to physical information in a series of papers. (For a quick reference, see [2] that shows how it solves many significant problems such as that of the missing dark matter and dark energy.)

Scaling levels in the brain [3]

Since nature is optimal, e-dimensionality should also be true for biological systems. We have shown that it helps solve five fundamental problems:

1. Explanation of the count of twenty amino acids in the genetic code [4];

2. Shows why the number of codons that map to amino acids are non-uniformly distributed [5];

3. Explains the experimentally observed fractal dimension of chromatin [6].

4. Explains emergence of α-helices and β sheets [7].

5. Explains the emergence of fractal like data [8].

Neural systems

Noninteger dimensionality should also characterize neural systems. Investigating this, we found surprising insights into the number of independent cognitive centers in the brain [9]. A related result is a theorem on why machines will never be conscious [10]. We have also found a connection between power laws characterizing biological signals to number-theoretic partitions [11], which were investigated by Srinivasa Ramanujan over a century ago.

All this indicates that rather than considering reality to be just physical, one must also speak about the accompanying information, which makes sense only when there is a mind to obtain meaning from the information. But if mind enters the picture, how can consciousness be ignored? This takes us to the matter-consciousness duality view of reality.


1. M.M. Roy and J.S.C. Nicholas, Do dogs resemble their owners? Psychological Science, 15(5), 361–363 (2004); https://doi.org/10.1111/j.0956-7976.2004.00684.x

2. Our e-dimensional universe. https://subhashkak.medium.com/our-e-dimensional-universe-febb3a20fa64

3. George F. Grosu et al., The fractal brain: scale-invariance in structure and dynamics, Cerebral Cortex 33, 4574–4605, 2023; https://doi.org/10.1093/cercor/bhac363

4. S. Kak, The dimensionality of genetic information, Parallel Processing Letters 33 (2023); https://doi.org/10.1142/S0129626423400121

5. S. Kak, Self similarity and the maximum entropy principle in the genetic code. Theory in Biosciences 142, 205- 210 (2023); https://doi.org/10.1007/s12064-023-00396-y

6. S. Kak, The geometry of chromatin, TechRxiv (2023); https://doi.org/10.36227/techrxiv.23818902.v1

7. S. Kak, Threads and spirals in a noninteger dimensional universe. TechRxiv. (2023); https://doi.org/10.36227/techrxiv.24112434.v1

8. S. Kak, 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

9. S. Kak, Number of autonomous cognitive agents in a neural network. Journal of Artificial Intelligence and Consciousness 9, 227- 240 (2022); https://doi.org/10.1142/S2705078522500023

10. S. Kak, No-go theorems on machine consciousness. Journal of Artificial Intelligence and Consciousness (2023); https://doi.org/10.1142/S2705078523500029

11. S. Kak, An information principle based on partitions for cognitive data. Journal of Artificial Intelligence and Consciousness 10, 1–14 (2023); https://doi.org/10.1142/S2705078522500138