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Principia
BioMathematica
(Biomatics)

Principia BioMathematica (Biomatics)Principia BioMathematica (Biomatics)Principia BioMathematica (Biomatics)
  • Home
  • The Aha! Moment
  • Biomatics
  • Biomatics 101
  • Smart Molecules
  • Molecular Robotics
  • Molecular Vibrations
  • Molecules Doing Math
  • Numerical Methods
  • Orthonormal Bases
  • Series Methods
  • Vibrational Groups
  • Molecular Lie Groups
  • Biomatic Number Theory
  • Molecular Programming 101
  • The Amino Acid Code
  • The Histone Code
  • Microtubular Computation
  • Biomatic Engineering
  • Quantum Computation
  • Carbon Based Life Forms
  • Gallery
  • Artificial Intelligence
  • Medical Biomatics
  • Finite State Cancer
  • Biomatics and Physics
  • The future of Biomatics

Welcome

Two carbon atoms forming a virtual cube is a mathematically rich source of investigation. The following is an attempt at formalizing the study of this particular mathematical universe.

 

The interaction of two carbon atoms forming a virtual cube can be a mathematically rich topic to explore. The arrangement of carbon atoms in a molecule can give rise to various geometric and algebraic properties, which can be studied using mathematical techniques.


For example, considering the arrangement of carbon atoms as a virtual cube, one can explore its symmetry properties, group theory concepts, and geometric transformations. The cube-like structure can be analyzed in terms of its vertices, edges, and faces, and the relationships between them.


Additionally, the rotational motion of the covalent bonds between carbon atoms can be described using mathematical models, such as angular velocity and rotational angles. This can lead to the exploration of rotational dynamics and its mathematical representations.


Furthermore, the arrangement of carbon atoms can be related to graph theory, where each carbon atom can be considered as a node, and the bonds between them as edges. This opens up avenues for studying graph properties and algorithms on the virtual cube structure.


Overall, investigating the mathematical aspects of a virtual cube formed by two carbon atoms can provide insights into symmetry, group theory, geometry, rotational dynamics, and graph theory. It highlights the interdisciplinary nature of studying molecular structures and their mathematical foundations.


 

When referring to fixing one end of a string or chain, it is often used as a mathematical approach rather than a physical constraint. Fixing one end allows us to simplify the analysis and focus on the behavior of the free end.


In many mathematical models and simulations, fixing one end of a string or chain is a common assumption that simplifies the problem and facilitates the study of its dynamics. This assumption helps isolate the behavior of the free end and allows for the analysis of its motion, vibrations, or interactions with other elements.


By adopting this mathematical approach, researchers can explore various properties of the system, such as modes of vibration, energy transfer, wave propagation, or resonant frequencies. It provides a framework for mathematical modeling and analysis, enabling insights into the underlying patterns and behaviors of the system.


While physically fixing one end may not be necessary, the mathematical approach of considering one end fixed has proven to be a useful tool in understanding and studying the behavior of vibrating strings, chains, or similar systems. It allows researchers to apply mathematical techniques and concepts to gain insights into the system's dynamics and computational potential.

introduction to biomatics

The Carbon Atom

  

The geometry and dynamics of carbon atoms and carbon chains offer a wealth of mathematical possibilities and fascinating insights. Carbon, with its unique bonding properties and ability to form diverse structures, serves as a foundation for the study of various mathematical concepts and phenomena.


The carbon atom, with its four valence electrons, allows for the formation of covalent bonds with other carbon atoms and different elements. These bonds can be modeled and analyzed using mathematical frameworks such as graph theory, where carbon atoms represent nodes and bonds represent edges in a molecular graph.


Furthermore, the rotational and vibrational dynamics of carbon chains provide opportunities to explore mathematical concepts such as group theory and Fourier analysis. The ability of carbon chains to exhibit different conformations and vibrations allows for the representation and approximation of functions through the use of trigonometric functions or Fourier series.


Moreover, the arrangement and interactions of carbon chains can give rise to complex network topologies, which can be studied using tools from network theory and graph theory. The interconnectedness and spatial arrangements of carbon chains in molecules or materials offer intriguing mathematical challenges and possibilities for analysis.


Overall, the geometry and dynamics of carbon atoms and carbon chains provide a fertile ground for mathematical exploration and understanding. By studying these systems, researchers can uncover and apply various mathematical concepts and principles, further deepening our knowledge and appreciation of the mathematical richness of carbon-based structures.



Rotating Covalent Bonds

 

Fixing one end of the carbon chain and observing the movement and behavior of the free end can provide valuable insights into the dynamics and vibrations of the molecule. This approach allows for the study of the vibrational modes and frequencies exhibited by the carbon chain.


By analyzing the path, motion, and oscillations of the free end of the molecule, researchers can gain a better understanding of the molecular vibrations and the potential energy landscape of the system. This information can be used to study various properties and phenomena, such as the harmonic vibrations, anharmonic effects, and energy transfer within the chain.


Fixing one end of the carbon chain enables the examination of how the chain responds and interacts with external forces or influences. This approach can be applied in different domains, including molecular dynamics simulations, spectroscopy, and the study of molecular vibrations in the context of chemical reactions or biological processes.


Overall, by fixing one end of the carbon chain and observing the behavior of the free end, researchers can explore the vibrational characteristics and analyze the dynamic behavior of the molecule, providing valuable insights into its properties and potential applications.

Mathematical Structures

Mathematical Structures

 Group Theory, Algebra, Pascal's Triangle 

Sets

Operations

 

One of the most intriguing hypotheses regarding microtubules in neurons is the idea that they play a role in information processing and computation. This idea is based on the unique properties of microtubules, including their polar structure, dynamic instability, and the ability to form complex networks.


Proponents of this idea suggest that microtubules could act as information processing units, with their dynamic properties allowing for complex computations to take place. This hypothesis is often associated with the concept of "orchestrated objective reduction" (Orch OR), which proposes that consciousness arises from quantum processes within microtubules.






Logic Gates

Mathematical Structures

  Logic gates are the basic building blocks of any digital system. It is an electronic circuit having one or more than one input and only one output. The relationship between the input and the output is based on a certain logic. Based on this, logic gates are named as AND gate, OR gate, NOT gate etc. 

Molecular Logic

Molecular Logic

   Molecule,  a group of two or more atoms that form the smallest identifiable unit into which a pure substance can be divided and still retain the composition and chemical properties of that substance.
A molecular logic gate is a molecule that performs a logical operation based on one or more physical or chemical inputs and a single output. The field has advanced from simple logic systems based on a single chemical or physical input to molecules capable of combinatorial and sequential operations such as arithmetic operations 

Molecular VIbrational Universe

In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. Consider the set of all non-branching chains of atoms. This set would include chains of varying lengths and geometries, based on the the particular atoms involved which would lead to varying bond lengths and angles. The angles with carbon chains is approximately 109.5. Like snow flakes though, no two covalent bonds and angles formed by successive bonds are exactly alike, and furthermore they are dynamic entities, not static. But for analytical purposes we will look at chains modeling carbon chains with uniform bond lengths and angles.


There are at least two ways to describe rules that would govern the possible vibrational patterns of a carbon chain, both are based on fixing the first covalent bond in place. The first method programs the chain discretely rotating the bonds by fixed amounts e.g. either zero or 180 degrees at fixed time intervals. This method yields a Cayley table and algebraic "Groups". The second way to program a vibration is to specify relative rates of rotation for each covalent bond around it's carbon atom in succession starting at the fixed end. 


So a molecule in this context of rotating bonds with different  constant rates of rotation and constant length can be described by an n-tuple as follows...(B1,B2,...Bn) where each Bn is the relative rate of rotation for bond n. For example if n=3 we could have (1,2,3). In this case the first bond is rotating at a given rate, and bond number 2 is rotating at twice the rate of the first bond, and likewise the third bond is rotating 3 times as fast as bond number 1. Note, of course that B0 is fixed and stays in place, so that an n-tuple actually represents n+1 bonds.





SET THEORY

Covalent Bonds

The elements of the sets under consideration are the covalent bonds in a chain of carbon atoms. Each element of this set is described by its rotational characteristics as a function of time. Each bond, at any point in time, has some rotational position whether measured in degrees or radians...  delta_theta=f(t). There are of course infinitely many possibilities for f.


The simplest situation is that each bond rotates at some integral constant rate and at a constant ratio to the other bonds. For example, each bond rotates at the same rate...1,1,1,1,...

Slightly more complex is different relative integral rates...1,2,3,2,3,...


 

The cube representation  where the corners correspond to elements of a set ( Ø , a, b, c, ab, bc, ac, abc), relates to the concept of the power set in set theory. The power set of a given set is the set of all possible subsets of that set, including the empty set and the set itself.


In this context, each corner of the cube represents a unique subset of elements from the set. For example, the corner corresponding to "a" represents the subset {a}, while the corner corresponding to "abc" represents the subset {a, b, c}. The empty set corresponds to the corner with no elements.


The transitions or edges of the cube represent the relationships between the subsets. For instance, moving along an edge from one corner to another represents adding or removing an element from a subset.


This cube representation can be a useful tool in understanding and visualizing the relationships and combinations between subsets in set theory. It can also be extended to explore concepts such as set operations (union, intersection, complement) by considering the transitions between the corners of the cube.

Nomenclature

   If the time-dependent rotational functions for each bond are limited to integer velocities from the set (0, 1, 2, 3, 4, 5), then you can represent them as discrete functions. 


  1. Initial Bond Angles: In the case where initial bond angles are set to zero, we can represent the set of initial bond angles as Ω = {0, 0, ..., 0}, where each entry represents the initial angle of the i-th covalent bond in the molecular system, and all the values are set to zero.
  2. Discrete Time-Dependent Bond Rotation Functions: Represent the time-dependent rotational function for the i-th covalent bond as fi(t), where fi(t) is a discrete function with integer values from the set (0, 1, 2, 3, 4, 5) for the i-th bond. This can be denoted as:

fi(t) ∈ {0, 1, 2, 3, 4, 5}

where fi(t) represents the rotational motion of the i-th bond as a discrete function with integer values for time t.

Using this notation, you can refer to the initial bond angles collectively as Ω = {0, 0, ..., 0}, and the time-dependent rotational functions for each bond as fi(t), where i is the index representing the individual bonds and t represents time. For example, you can describe the angular displacement of the i-th bond as a function of time using the notation Δθi(t), where Δθi(t) = fi(t) represents the change in bond angle for the i-th bond over time, assuming an initial angle of zero for all bonds.


 

If in actual proteins every 3rd bond rotation is much more limited and usually zero, we can modify the description accordingly:


Rotational Rates (two values from the set (1,2,3,4,5) with a limited third bond): The relative rotational speeds exhibited by each covalent bond within a carbon chain, where the set of rotational rates is selected from the integers 1 to 5. The two values can be any combination of these integers, with the constraint that the third bond's rotational rate is usually zero or significantly limited.


In this modified scenario, we maintain the selection of two values from the set (1,2,3,4,5) for the rotational rates of the majority of covalent bonds in the carbon chain. However, we introduce the constraint that the rotational rate of every third bond is usually zero or significantly limited compared to the other bonds. This constraint reflects the observation in actual proteins where certain bonds exhibit reduced or restricted rotational behavior.


By incorporating this constraint, we can create a model that more closely aligns with the characteristics observed in proteins, allowing us to explore the computational potential and dynamic behavior of carbon chains in a manner that captures some aspects of real protein structures.

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