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2011-2015 a achievement report of our research

Published October 3, 2016 by anirbanbandyo
  1. Background, purpose

Making an artificial brain is a supreme goal, and this global effort is going on in the full swing. Not only this has become a prime goal for several countries across the globe, all major multinational companies are doing the same with the largest investment of the world. Brain has machines from atomic scale to the meter scale. Therefore, one has to master the technologies that spread from molecular electronics, photonics, design and synthesis of organic materials, computer science, mathematics, physics etc. For example, nano scale machines would carry out important tasks in the future. However, they should have brains providing intelligence to these nano-machines at the atomic scale. We are the first to develop a brain for the nanobots and continuously we are building more and more life-like nano-machines. World’s smallest artificial life was our target.

The fundamental building block of a life form is protein and human brain expresses almost all of the 30000 proteins. For this purpose we need to unravel the mystery of information processing in the biomaterials. Until now, people were considering that protein is like a conventional electronic component found in an electronic shop. But is that true? If not, then what is protein? Now, if protein follows an unprecedented electronics, what is the basic information in biology, in life, and in the brain? If brain is not a circuit, not a machine, not a processor like the conventional computer, then, every single research being done in the world.


  1. Results of research: inventing ten research fields

In the five-year period, 2011-2015, we have completed the background research for a new science for bioinformatics and associated electronics and materials science research, both experimentally and theoretically. There are ten research fields that we have created, which did not exist earlier, not even proposed. We have explained the fields in the form of a chart to our right.

By doing research on proteins we found that a new kind of fourth circuit element (we call it Hinductor [1]) that mankind was searching for over 150 years, we have found that device. This is research field number one. This device is used to build fractal cavity resonator which is the fundamental device used by nature for its information processing.

Information in nature is not “bits” or “qubits” its time cycle. So we have developed a new Fractal Information theory (FIT), this is our second research field. For the last 115 years we have been using only one information theory, this is the second [2, 3, 10].

No one writes a software program for nature, it does everything by itself, so an information theory was not enough we had to build a language of nature, using which the biomaterials processes information and there is no need to do programming. We call this third field as geometric musical language (GML) [3,10].

The fundamental physical process that regulates GML is fractal beating, this is a new kind of beating, an advanced form of quantum beating. There are plenty of associated features followed when we discovered this unique process. We introduced Fractal mechanics research field that also delivers all the prerequisites for the FIT and GML combo. A generic form of Quantum mechanics is therefore the fourth research field that we introduced [4, 10].

There are three research fields, which generated from this fourth research field of Fractal mechanics. First, the fractal beating we could encode in the molecular nano brain using which we could build non-chemical physical drugs, we call it musiceuticals. This is the fifth field and we have tested on cancer and Alzheimers [5].

Second, a new kind of chemistry we introduce for fractal reaction kinetics and fractal condensation. Thus, Fractal chemistry is the sixth research field [6].

Third, using GML, Fractal chemistry, FIT, and musiceuticals we have been successful to design brain jelly an organic jelly like solution that mimics the brain rhythms. Brain jelly an advancement of nano brain is our seventh research field [7].

Using the fourth circuit element and the fractal mechanics, we have created a genre of thermal motors. These motors harvest thermal energy and a basic element in the nano brain and brain jelly, we built thermo-electro-mechanical systems (TEMS), this is our eighth research field [8].

Based on FIT, GML and fractal mechanics we have created a fractal cavity resonator model of an entire human brain considering every single component of a human brain. We are building a global platform for the nested rhythms found in the brain. Modeling the vibrations and the rhythms of a human brain is our nineth research field [9].

This brain model shows that a fundamental prime number theory and the mathematics of e, pi, phi and i, regulates evolution of every single new features that we observe in the biomaterials and biophysics research. Therefore in the tenth field we have fused number system and geometry for finding the key evolution of symmetries that nature follows while designing and synthesizing materials [10].


  1. The prospects: the follow up commercial products

All the ten research fields that we have introduced have direct connection to industry.

First, fourth circuit element is good for ultra-low power electronics.

Second, fractal information theory is directly related to the data deluge problem that we face.

Third, geometric musical language is to avoid massive programming and the development of natural language; it is for the instantaneous solutions in an unknown situations like outer space missions.

Fourth, fractal mechanics gives us a new kind of piezoelectric, ferroelectric and pyroelectric properties.

Fifth, musiceuticals are drugs that will have no side effects.

Six, fractal chemistry enables 100% conversion in synthesis, minimal side products and natural product synthesis.

Seven, brain jelly as the neural jelly for the futuristic robots.

Eight, TEMS, thermal energy harvesting.

Nine, human brain model for vibrational frequencies and rhythms will help treating various neurodiseases.

Ten, the prime number theory and fusion of basic geometry is formulated to predict unprecedented materials with unique properties.



1) Helical symmetries observed in proteins: An inductor made of arrayed capacitors (2010) Japanese patent has been issued on 20th August 2015 JP-511630 (world patent filed, this is the invention of fourth circuit element), Satyajit Sahu, Daisuke Fujita, Anirban Bandyopadhyay, US patent has been issued 9019685B2, 28th April 2015.

2) S. Ghosh, D. Fujita & A. Bandyopadhyay; An organic jelly made fractal logic gate with an infinite truth table; Scientific Reports 5, 11265 (2015)

3) S. Ghosh, S. Sahu, D. Fujita, A. Bandyopadhyay; Design and operation of a brain like computer: a new class of frequency-fractal computing using wireless communication in a supramolecular organic, inorganic systems. Information, (2014), 5, 28-99;

4) L. Agrawal, et al, Replacing “bits” by “time cycle”: Fractal Information Theory (FIT) and Geometrical musical language (GML) for brain inspired hypercomputing; SocPros 2016, Springer conference proceedings.

5) (a) S. Ghosh, S. Chatterjee, A. Roy, K. Ray, S. Swarnakar, D. Fujita and A. Bandyopadhyay, Resonant Oscillation Language of a Futuristic Nano-Machine-Module: Eliminating Cancer Cells & Alzheimer Aβ Plaques, Curr. Topic. Med. Chem. 15, 534-541 (2015), (b) S. Sahu, H. Oono, S. Ghosh, A. Bandyopadhyay, D. Fujita, F. Peper, T. Isokawa, R. Pati: On Cellular Automata rules of molecular arrays. Natural Computing 11(2): 311-321 (2012).

6) (a) S. Ghosh, M. Dutta, K. Ray, D. Fujita and A. Bandyopadhyay, A simultaneous one pot synthesis of two fractal structures via swapping two fractal reaction kinetic states, Phys. Chem. Chem. Phys. 18, 14772-14775(2016) [cover highlight]; (b) A supramolecular archtecture creation by successive phase transitions and radiations, Subrata Ghosh/Satyajit Sahu/Anirban Bandyopadhyay/Daisuke Fujita, 2014/10/29, JP2014-219958 (this is creation of nano brain)

7) (a) S. Ghosh, M. Dutta, S. Sahu, D. Fujita, A. Bandyopadhyay Nano Molecular-platform: A protocol to write energy transmission program inside a molecule for bio-inspired supramolecular engineering. Adv. Func. Mater. (2014), 24, 1364-1371; [Cover page] (b) Continuously self-assembling material, Subrata Ghosh/Anirban Bandyopadhyay/Daisuke Fujita,  2014/8/7, JP2014-161746, (this is creation of nano brain)

8) Thermal Noise Driven Molecular Rotor, Subrata Ghosh/Satyajit Sahu/Daisuke Fujita/Anirban Bandyopadhyay, 2014/4/25, JP2014-091141; (this is creation of a motor for nano brain)

9) (a) S. Sahu, S. Ghosh, D. Fujita, A. Bandyopadhyay Live visualizations of single isolated tubulin protein self-assembly via tunneling current: effect of electromagnetic pumping during spontaneous growth of microtubule. Scientific Reports, 4, 7303 (2014); (b) Ghosh, S. Sahu, and A. Bandyopadhyay Evidence of massive global synchronization and the consciousness. Comment on “Consciousness in the Universe: A Review of the ‘Orch OR’ Theory”. Phys. Life Rev. (2014), 11, 83-84; (c) S. Sahu, S. Ghosh, K. Hirata, D. Fujita, A. Bandyopadhyay Multi-level memory-switching properties of a single brain microtubule. Applied Physics Letters 102, 123701 (2013); (d) S. Sahu, S. Ghosh, B. Ghosh, K. Aswani, K. Hirata, D. Fujita, A. Bandyopadhyay Atomic water channel controlling remarkable properties of a single brain microtubule: Correlating single protein to its supramolecular assembly Biosensors and Bioelectronics 47(2013)141–148, (e) S. Sahu, S. Ghosh, D. Fujita, A. Bandyopadhyay Computational myths and mysteries that have grown around microtubule in the last half a century and their possible verification; Journal of Computational and Theoretical Nanoscience 8, 1-7 (2011), (cover page)

10) Book chapter related to this article: L. Agrawal, et al; Replacing Turing tape with a Fractal tape: a new information theory, associated mechanics and decision making without computing, Matter and consciousness in Indian epistemology. Chapter 6, 87-159 (2016). Delhi, India.

What is memory in the brain? How many different kinds of memories in the brain?

Published September 22, 2016 by anirbanbandyo

Vibrations holds the key, for everything. Until now, we were considering that the vibrations and rhythms are only key to some enzyme cycles. But our proposal is that deep down below even to the molecular scale there are rhythms. And there are 12 types of memories that works at the 12 scales of operations in the brain.

All memories are geometric in nature

1. Periodic memory: Chain of guiter string we see mostly in DNA, proteins, period of periods in the nodes of the string. Edit the memory by changing the periodic length of elementary point oscillators.
2. Spiral of spiral memory, in DNA and proteins, also in microtubules. Edit the memory by changing the twists and the number of hierarchical periods by inducing more and more twists.
3. Vector memory, orientation of helices to make cavities inside protein. 3D orientation makes it a multipolar complex 3D vibrational element. Edit the memory by changing the orientation of column oscillators.
4. Lattice memory, microtubule, actin-beta spectrin network, protein complex. Change the lattice parameters to edit the memory.
5. Chemico-electric memory, chemical transmission cycles. There are nested time cycles highly interconnected, to edit the memory, change the diameter, phase, delay or starting points.
6. Leak density of Cavity memory, ion diffusion holes in the cellular membrane. To edit the memory change the leak density on the cavity surface.
7. Spiral geometric memory, assembly of neurons etc. To edit the memory, change the pitch-diameter-length of the spiral.
8. Vortex or fractal memory, memory stored in the geometric parameters of vortex. A vortex has divergence parameters and scale repeat unit, these two parameters are changed to edit memory.
9.Nodal & polar memory: Nodes in the spinal cord brain network (nodes in a tear drop). Geometry of the shapes are changed among 8 different choices from teardrop to ellipsoid to edit nodes and polarity memory.
10. Electromechanical phase memory: organs sync and desync like a giant molecule. This memory is edited by changing the bond length or wiring length and distribution.
11. Multipolar loop in Phase space loop memory: Hyperspace memory. This memory is non-physical and hence not editable.
12. Phase of phase duality generator memory: hierarchical assembly of reality sphere. This memory is non-physical and hence not editable.

How many different kinds of rhythms are there in our brain?

Published September 22, 2016 by anirbanbandyo

Sound is a mechanical rhythm and we have 12 different rhythms in the brain, like

1. Electromagnetic rhythm, [carrier is photon or electromagnetic wave is trapped in a cavity to generate beating or rhythm].
2. Magnetic rhythm, [spiral flow of electrons or ions, they are the carriers editing the magnetic flux, geometry of path forms the periodicity].
3. Electrical potential rhythm, [change in the arrangement of dipoles editing the electric field, fractal distribution of local resonators generate a time function of potential]
4. Solitonic & quasi particle rhythm, [carriers are solitons, defect in the order flows in a ordered structure, the ordered structure is edited to make a loop].
5. Ionic diffusion rhythm, [ions are carriers, tube like cavities are formed in a circular shape or continuous path to generate a loop]
6. Molecular chemical rhythm, [molecules like proteins, enzymes etc are carriers, tube like cavities are formed and sensory systems make sure a circular signalling pathway]
7. Quantum beating, [spin is the carrier, wavefunctions interfere in a squeezed excited photonic, electromagnetic or spin state]
8. Density of states rhythm, [orbitals coupling, wave function modulation, virtual carrier, a virtual continuous loop is made]
9. van-der Waal rhythm, [atomic thermal vibration is looped in a spiral pathway].
10. electro-mechanical rhythm [classical beating with a mechanical beating like tuning fork]
11. Quasi-charge rhythm polaron, polariton [topological fractured band based continuous loops]
12. mechanical rhythm, sound wave [similarly elastic pathways to make a circuit of sound waves].

Just imagine, such beautiful patterns are one of 12 different formations. Their physics is different, application domain and range are different.

Ten inventions that we contributed from NIMS, Japan and would consolidate in the next 10 years (2016-2026)

Published September 11, 2016 by anirbanbandyo
  1. Information Theory:          A new information theory. FIT or Fractal Information Theory. This is a new way of computing or decision-making by fractal logic gate, or fractal cavity resonator. For this purpose we proposed Fractal tape like Turing tape and Fractom machine. We often use nested cycle metric to generate fundamental geometric constants.
  2. Mechanics:           We have been developing a new generic version of quantum mechanics, FM or Fractal Mechanics, We conceptualized a new fractal harmonic oscillator alternative to QFT
  3. Language:         We have been developing a new geometric musical language (GML) and constructed Continued Fraction Geometric Alzebra (CFGA) and Geometric musical calculator (GMC).
  4. Human brain model:          We develop fractal cavity resonator model of the human brain, (FCR-Brain) and Frequency Wheel of prime model (FWPM) for any biological machine. We develop cavity math for protein design for life forms. Nested Rhythm map of the Human brain (NRM-Brain). We discovered ferroelectric, piezoelectric & pyroelectric properties of microtubule and every protein, we have also discovered that protein has electromagnetic resonance, Universal protein circuit model (UPCM).
  5. Geometric number system theory:             We construct and information version for developing the theory of everything, and a theory of consciousness we call it Time cycle universe theory (TCU), this is not like GUT, ours is a set of integrated mathematical constructs and protocols that would help in the understanding the universe and or consciousness. For example relating e, pi, phi with all fundamental constants, frequency wheel of primes, ordered factor etc.
  6. Fourth circuit element:            We invent fourth circuit element, Hinductor
  7. Thermal energy harvesting machines:           We invent artificial life forms, kT driven thermal nanobots, Thermo Electro mechanical system (TEMS).
  8. Musiceutical nanobots:            We have developed musiceuticals, vibration based medicines, Nano Brain. Main focused diseases have been cancer and alzheimers.
  9. Fractal chemical kinetics and condensation:          We have developed a new kind of multi-kinetic synthesis, we invent a new class of condensation beyond Froelich condensation, Fractal condensation
  10. Brain jelly:             We invented organic jelly that replicates biological rhythm like information processing, Brain jelly. We invent world’s smallest neural network. Multilevel switch. Fractal beating like quantum beating for nested beating to be used to create fractal cavity resonator for brain jelly.


If we consider there is no space, no mass, only time, then?

Published September 8, 2016 by anirbanbandyo

Cosmology and elementary particle physics would end by generating a new world of physics above and within: There are proposals that time does not exist and mass is everything. The history of science started with space and time and mass as a separate quantity. Always in the history of science we have seen that mass is deconstructed to a sum of smaller masses, most of the places in a mass is found to be empty and every time we lost a lots of mass. If everywhere 99% is empty and there is no apparent lower limit then, relying on mass as a fundamental parameter is silly. This is why we rejected “mass” as a fundamental parameter.

We told that the lost mass is gone to bind the particles containing the mass. Einstein’s E=mC^2 is such a formulae, and if you see our galaxy far far from the boundaries of the universe then you would find it as a single dot of mass, similarly a molecule or an atom. It is something very new only when we enter inside. This journey for mankind would continue forever. Cosmic world is not the single world out there. There is something beyond cosmology both outside and beneath the worlds of Higgs Boson. If a human is 1.7 meters and an atom is 0.16 nm, then 10^9 order spatial gap is bridged within our conscience.

Who responds? The world of “silence” and “bing”: Therefore, it is safe to consider mass is sum of spaces, as Einstein did. However, the journey should not end there. We consider a signal or a “bing” from a mass, does it come from a mass? No. Then where from? The world within, the discrete atomic energy levels emit and absorb photons, not the atomic mass. Those discrete energy levels have symmetries of their own and we often say that the energy is processed by breaking symmetry, changing one kind of ordering to another. One kind of ordering to another means change in the geometric shape. Who makes the geometric shape? Elementary particles and their ordering changes. And what does it change? The geometric phase. So the existing science tells us that the dynamic phase that we see for a particle is an illusion, it is an outburst of several elementary particles within, its own information is in the geometric phase or the silence between the bings. Two phases are essential because we need intensity of the bing to associate with the geometric phase.

bing is a boundary of silence, bing is a noise of lower hidden world.

How “time” would take the responsibility to do what mass and the space does?

The lower the mass of the exchange particle, the farther the force is felt. It means that the mass and the space are already correlated. Photon is spin 1 and Graviton is spin 2, both are massless and therefore long range forces. Mesons are heavy, hence work at a shorter distance. Instead of getting into these troubles, why not we consider that mass is a geometric transition in the time cycles. Then, space is relative difference in the time cycle perimeters. These are easy routes to

Will then the time would move forward? What is said by Quantum? Second law of thermodynamics suggests that entropy would be increasing, which means that universe has a fundamental property to get more and more disordered, why we do not know. But one thing for sure, we say that time is moving forward only because we lose information. This lose of information is like time cycles do not get connected and form an integrated network, instead they fall apart. But, why should the time cycles get disintegrated? Why is the need, unless we are trying to build another network of time cycles?

The arrow of time is a reality only when we consider that information is destroyed, this is how our thermodynamic concept of time or the arrow of time is built.

Quantum’s signature is in instantaneous control. The problem is that the scientists do not consider instantaneous effects as an absolute property. They think of negative time so that they could add a cause-effect scenario, why do we need to go to the past create cause?

What is second law of thermodynamics and entropy and the flow of time, if we consider nested time cycle as the fundamental structure of the universe? Is there any arrow of time? 

When we consider nested time cycle, we enter a very different world, there is no past present future, location of an observer in the time cycle network determines what should he or she define as network geometry. Still there was no past present and future, many directions of time as proposed by Carroll in 2004 (eternal inflation).

At local time time cycles we may see that continuously the time cycles are breaking apart and somewhere in the network it is growing, there is no necessity that there would always be growth or decay or even conservation of total number of time cycles. These questions are irrelevant. Therefore, entropy of the universe if we consider time cycles is a function of time. And there is no flow of time, both the questions have all possible answer. In the conventional study destruction of information ensures flow of time, increase in entropy, but information does not have any quantity for us, we have upper and lower time limits and density of information, several other factors, but not quantity. Therefore, we are not answering the queries related to these topics.

Massless particles have time cycles propagating to infinity. Hence measuring in Quantum does not require to consider negative time. So, flow of time is not required for any reasons. We created Quantum but could not come out of the mentality of believing the science of simultaneity, where causality could never ever be essential.

The existing concept of “block universe” has to go, it is not very useful (https://www.quantamagazine.org/20160719-time-and-cosmology/): Physics laws are time-symmetric — that is, the physics they describe is the same, regardless of whether the variable called “time” increases or decreases. Moreover, they say nothing at all about the point we call “now” — a special moment (or so it appears) for us, but seemingly undefined when we talk about the universe at large. The resulting timeless cosmos is sometimes called a “block universe” — a static block of space-time in which any flow of time, or passage through it, must presumably be a mental construct or other illusion. There are even proposals for evolving Bloch universe (George Ellis) where past is fixed but the universe future is open https://arxiv.org/ftp/arxiv/papers/1407/1407.7243.pdf  Causal set Theory suggested that the universe is continuous at elementary level but discrete at large level http://www.einstein-online.info/spotlights/causal_sets?__hstc=13887208.aa7b15315168b61d73452cb836a9622f.1473287636669.1473287636669.1473287636671.2&__hssc=13887208.1.1473287636671&__hsfp=4242301618

Fay Dowker, a physicist at Imperial College London, referred to this as “accretive time.” She invited everyone to think of space-time as accreting new space-time atoms in way roughly analogous to a seabed depositing new layers of sediment over time. General relativity yields only a block, but causal sets seem to allow a “becoming,” she said. “The block universe is a static thing — a static picture of the world — whereas this process of becoming is dynamical.”

There will always be a moment of maximum density and minimum complexity (Koslowski), for us, from big bang one time cycle is running, and once the equilibrium is reached it will be a new unit and begin the journey as a new journey of primes.




Infinity paradoxes of 1=1+1, and our fractal information theory (FIT)

Published April 3, 2016 by anirbanbandyo

Our information theory considers that “there is a Turing tape inside every single cell of a Turing tape”, this new tape now fails to become a Turing tape (popularly people would love to call it Fractal but there is no guaranteed self-similarity, it is Numtal!!!). To operate this tape now we need time cycles nested with each other. When we created this nested time cycles, we did not think about it much. Our only objective was to solve the immediate crisis. Since we have a singularity we need a tool to penetrate deep insight and harness an insight from within. So, the new Fractal tape uses a basic information processing unit, three-time cycles connected to each other, one that represents Turing tape inside a cell, one that represents the Turing tape above where the cell exists and third, the third unwanted Turing tape or the third time cycle is that of an observer. We initially did not want to bring the third Turing tape, but you might have already noticed that we do not have any representative tape for the cell under consideration, therefore we need a new tape to project the possible cells states of the cell under consideration, that task is performed by the observers tape. So, we get a projected hyperspace, namely “reality sphere” that sphere is the only real thing in the information units time loop architecture. Now, a time loop has frequencies at certain time gaps to write geometric shapes, that is information content and all the three loops are imaginary, does not come from infinity but imaginary space.

Now, above we described all that one needs to learn to understand the science behind our information processing unit, which eventually creates a resonance chain and morphs the events happening in nature. Mimicking the frequency space is amazing in the sense that suppose a protein is communicating with another protein, then if one images its shape its silent there is no movement. However, if we could somehow convert the information space into a shape changing material, then we could find a slowly shape changing material, and if we could enter inside the structure, then we would find some other structures are changing quite fast. If we could take one, and enter inside, we would find a large number of extremely fast shape changing materials. This journey would go on and on. Now, you learn all about the science of our information processing protocol.

Lets discuss the philosophy of infinity.

Gabriel’s paradox: Gabriel suggested a trumpet that looks like a cone but it goes thinner and thinner to infinity, it has infinite surface area, but when we try to paint it, we would require only a finite amount of paint, drop it from its mouth and it would pass through all the way.

Hilbert’s infinite hotel: Hilbert suggested a hotel with infinite number of rooms. Infinite number of guests could come in, yet there would always be a room available in the Hotel. The same event could be represented by a circle, we can take out infinite number of points from the perimeter of a circle, yet there would be no disjoint in a circle. A circle is finite only when we fix an area

Dart on a dartboard: Suppose we have a pin, we throw it on the wall wherein a circle is drawn. What is the probability of touching a particular point, given that there is infinite number of points in that area? If the probability is zero, then, the total probability of hitting the entire circle anywhere is zero, but that cannot happen. We need to consider a very small area, to get out of infinity paradox.

How many numbers between 0 and infinity, Cantor diagonalization: The answer is infinity, that so many numbers are there in between 0 and 1, even 0 and 0.00000000000000000000000001. Now, take any gap between any two rational numbers, we get infinite number of numbers between them. {n}={n^2}=….={n^n}

Hyper-webster is a hypothetical dictionary of infinite space of all possible letters. The idea is the same as numbers, it has 26 chapters of one letter each. Then each of these chapters have 23 paragraphs and so on. Now, if you go on dividing the possibilities are infinite. The exciting part begins now, suppose the first chapter of the dictionary has all possible words beginning with A, then if one removes A, the rest is same as all possible chapters with all possible words. In fact all 23 volumes are self-complete, if we print one of 23, we are done.

Banach-Tarski This is very interesting now. We create the same dictionary on a sphere. One could write infinite words on the surface in the infinitely available paths on the spherical surface. Even if we start walking from any one point we could travel covering  every single point on the spherical surface following , Up, Down, Right, Left, UR, RL, UL and DR. Now, if the lattice on the sphere could be a hexagonal or rectangular, with 6 and 8 neighbors respectively, we could have 3 and 4 equivalent roads respectively.  There would always be two poles and hence, two identical spheres could be created just from one sphere if we decide to walk on the routes following a given protocol.

Whenever there is a case of real number of points, by Banach-Tarski argument one could create 1=1+1, when the number of point is infinite. This violates the proof of Bertrand Russell.

On the sphere we can start another journey. Either on the disc surface alone with 3 circles like the one below, or start with 4 spheres making a sphere Apollony fractal.

 In 3D one starts with 4 spheres each at the vertices of a tetrahedron. An outer imaginary sphere is the boundary cavity that encloses the 4 spheres. The gaps are filled by subsequent Soddy spheres thus forming a solid object (at infinity). The result is often called an Apollonian sphere packing. The fractal dimension of the 3D Apollonian has been calculated as 2.473946 [M. Borkovec, W. De Paris, and R. Peikert].

Kravchenko Alexei and Mekhontsev Dmitriy have found an attractor that creates the Apollony fractal, the source code in C is provided here: apollony.c.

This is the union of three functions

f1(z) = f(z)
f2(z) = 0.5 (-1 + s i) / f(z)
f3(z) = 0.5 (-1 – s i) / f(z)

where z is complex, s=sqrt(3), and f(z) = 3 / ( 1 + s – z) – (1 + s) / (2 + s)

Tetragonal space is the starting point. What about other starting packing?


The largest number of unit circles which can touch a given unit circle is six. For spheres, the maximum number is 12. Newton considered this question long before a proof was published in 1874. The maximum number of hyperspheres that can touch another in n dimensions is the so-called kissing number. It is shown that optimality for sphere packing is n=4 The history of packing problem is here. One could have fun packing the area with circles here,  how to draw these are sphere here.

There are three imaginary worlds that constructs the fundamental unit of information for our structure. We see above that the three imaginary functions if they are different then they make an imaginary sphere. Though circle grows with three imaginary worlds, in the Appolony fractal, the sphere grows optimally with four spheres. For an external observer, a sphere always looks like as if it is made of three circles. This is how the apparent mismatch between the infinity spheres and circles are resolved.

Harnessing infinity in the sphere and modeling the human brain:

Change in the brain’s physical structure is fatal to our brain modeling. However, there are plenty of brain models that simplify the brain structure to make it easy to process. For example, conformal mapping of the brain is done on a sphere (M.K. Hurdal, P.L. Bowers, K. Stephenson, D.W.L. Sumners, K. Rehm, K. Schaper, D.A. Rottenberg, Quasi-conformally flat mapping the human cerebellum, in C. Taylor and A. Colchester (eds), Medical Image Computing and Computer-Assisted Intervention – MICCAI’99, Vol. 1679 of Lecture Notes in Computer Science, Springer, Berlin, pp. 279-286, 1999). The idea is similar to those who want to map brain in a hypothetical structure, but we do not. Our fractal information processing theory suggest assembling the time cycles on the sphere. Compared to Hurdal suggested conformal flattening we do not change the physical shape of the any part of the brain, for our theory, any change in the structure means change in the information content. We are bound keep its purity intact, and then integrate tiny pieces of information in it.

Are we driven by madness? How come we could even think of processing information of the entire brain?

Above we have discussed that a single sphere holds an infinite number of paths and we could make a singular mathematical structure of sphere inside representing the entire human brain. Now, the time cycles would be on top of its surface as shown below.

3D circles Each of the circle with an independent center could represent the time cycle network representing various organs and their biological rhythms. Using such a sphere is beautiful because we get only one unified system to hold infinite possible time cycles. That is highly generous statement.

One to one correspondance between the Gabriel’s paradox and renormalization: One to one correspondence between Gabriel’s paradox and nested time cycle synchronization that is designed to replace renormalization, the surface area of the cone is infinity, however, when we put paint to color it, the paint passes through the tunnel and it blocks at the point depending on the density of the paint. The paint for us is the observer’s nested time cycle. The resolution of the solution is always ultimate.

One to one correspondance between the Banach-Tarski paradox and the measurement in a fractal information theory: Suppose we have a nested time cycle sphere, like the one shown here, the observer would also be a sphere, it would sync with the paths and generate the connected path and form a “reality sphere” just like the one here. Creation of an identical information replica is its greatness because the solution of a problem is an infinite space.

One to one correspondance between the Hyper-webster dictionary and the generation of multiple spheres: If there are large number of observers, of course there are infinitely large number of observers, therefore large number of “reality spheres” will be produced and those would merge just like the Hyper-websters dictionary. All these spheres are distinctly produce, yet they create a singular identity. Imagine thousands of “reality spheres” being produced and all of them simultaneously hold the solution.


Going beyond Turing machine: World of technologies with “Infinite state” machines

Published December 30, 2014 by anirbanbandyo

Turing’s world of computing considers every single element is complete in itself, just like that a “matter” is complete; nothing inside/outside is changing it. Completeness enables discrete and binary true/false argument to define it. We get a Turing world.

Recently, we proposed an Escape Time (ET) fractal tape, where, as we enter inside a cell of the tape, we find another tape, hence not a single cell in the entire tape network is defined (non-differentiable). This is not a Turing tape. This AjoChhand vision has one argument, one energy packet, one Turing like tape, one clock, one rhythm, one oscillator that encompasses the entire universe, but if we enter inside any of them, we find millions of those inside, the journey continue forever, whether we zoom in/out.

Unlike Turing, any “matter” is conceptually incomplete so we need matter+energy duality to connect all matters inside and outside. Conserved energy takes the form of periodic oscillation or rhythm and maintains time, but each rhythm has rhythms inside/outside. We cannot isolate any small piece of argument from tape network, hence, no bits or arguments like true/false, that leads to no classical information content. Observer/user is also part of the tape network. User’s sensory resolution limits isolate a part of limitless rhythm network that is “his” information-perception. Since “that” is not a discrete unit, it remains undefined, so entire rhythm net, is represented as matrix of truth with infinite dimensions, no rhythm is ever destroyed, no choice is ever rejected. This is an ET system, but self-similarity is not mandatory, so it is not a true fractal, possibly fractal-like.

Unlike Turing, our tape (AjoChhand tape) needs four new steps, resonate (with a rhythm), expand (self-assembly of rhythms), Phase transition (of rhythms), and spontaneous reply (the rhythms). Self-similarity of shape/rhythm appears/disappears with time, as rhythms couple/decouple with other rhythms, changes the structural symmetry of matters to memorize particular set of frequencies for a rhythm. The entire tape network is never-ending resonance chain of coupled matters that vibrate with rhythm, means, local rhythms vibrate as a distinct entity in parts of the tape.

Since observer/user is also part of the rhythms of the tape-network, it senses by perpetual local synchrony and de-synchrony with various parts of the tape, thus, its “sensing” not computation by classical theory of computation.