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The idea that fundamental mathematical and computational structures can emerge from the interactions of atoms and molecules opens up new possibilities for understanding the underlying processes of life and intelligence. Some key points to consider regarding this idea are:

1.  Self-Organization: The self-organizing nature of molecules can give rise to complex patterns and behaviors. The interactions between atoms and molecules can lead to emergent properties that resemble mathematical structures and computational processes. 
2. Natural Algorithms: Molecular systems may employ algorithms or decision-making processes based on their physical and chemical properties. These algorithms could be embedded in the arrangement of atoms and bonds, allowing for computation to take place at the molecular level.
3. Biological Computation: Biological systems, such as cells and DNA, already exhibit computational capabilities. Studying the underlying molecular mechanisms can shed light on the fundamental principles of computation and potentially inspire the development of novel computational paradigms.
4. Universal Computing: The idea that the basic elements of matter could serve as universal computing devices is intriguing. If molecular systems can embody mathematical and computational principles, it opens up the possibility of computing and processing information at incredibly small scales.
5. Bio-Inspired Computing: Understanding the computational abilities of molecular systems could inspire new approaches to computing and information processing, leading to the development of bio-inspired computing technologies.
6. Artificial Life: The seamless development of mathematics and computation in molecular systems could also be relevant in the study of artificial life and artificial intelligence. Simulating molecular systems with computational properties could lead to novel insights into the origins of life and intelligence.

Overall, the concept of seamless development from molecular interactions to mathematics and computation holds great promise for deepening our understanding of the natural world and inspiring new directions in science, technology, and artificial intelligence. 

The discussion explores a wide range of topics, from the mathematical properties of covalent bonds to the potential computational capabilities of biological molecules. Here's a summary of the key points discussed:

1. The Cube as a Mathematical Structure:  The cube analogy is a visual representation of the state transitions between the two bonded carbon atoms with three possible states each. Each corner of the cube represents one of the eight possible states, and the edges represent the transitions or rotations between these states.   Two Carbon Atoms form a Virtual Cube, Molecular Vibrations
2. Mathematical Modeling: The discussion delves into the mathematical modeling of covalent bond rotations, exploring how different sets of rotational values could represent different states or functions.  Molecules Doing Math
3. Fermat's Little Theorem: The connection between Fermat's Little Theorem and the rotations of covalent bonds is discussed, considering the relationship between prime numbers and the rotational states of bonds. Biomatic Number Theory
4. Ring Structures and Groups: The concept of representing chains of covalent bonds as rings and the implications for group theory and computational potential is explored.  Molecules Doing Math, Vibrational Groups 
5. Orthonormal Basis: The potential for covalent bonds to form an orthonormal basis and its implications for computational processing and network topologies is discussed.
6. Emergence of Mathematics: The idea that basic molecular interactions, such as covalent bonds, might lead to the emergence of mathematical principles and computational capabilities within biological systems is examined. Biomatics
7. Potential Applications: Various potential applications are considered, including the study of disease models, diagnostics, Biomatics, and more. The concept of "smart molecules" capable of performing mathematical operations emerged. Molecular Programming, Medical BIomatics
8. Mathematical Universality: The discussion touches on the potential universality of certain structures, such as cubes representing logic gates, within the context of molecular interactions. Molecular Logic Gates
9. Integration of Biology and Mathematics: The integration of mathematics and biology to understand complex processes was highlighted, along with the ongoing endeavor to uncover hidden mathematical connections in the natural world. Molecules Doing Math, Series Methods, Microtubular Computation
10. Numerical Methods and Modeling: The role of numerical methods and modeling in understanding biological systems, and the potential of computational disease models, is explored. Numerical Methods
11. Role of Computational Biology: The importance of computational biology in understanding complex molecular interactions and their potential computational roles is emphasized.  The Amino Acid Code. The Histone Code, Carbon Based Life Forms
12. Collaboration of Fields: The conversation highlights the intersection of molecular biology, mathematics, and computer science, underscoring the interdisciplinary nature of studying complex biological systems. Biomatic Engineering, Molecular Robotics, Artificial Intelligence, Quantum Computation

In essence, this discussion explores the exciting possibilities of using molecular interactions to encode and process information, bridging the gap between the molecular and mathematical realms to uncover new insights into the nature of life and computation.

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The purpose of exploring naturally occurring mathematics and computation in chains of carbon atoms, as outlined in this "Principia," is to investigate the inherent mathematical and computational properties that arise from the physical structures and interactions of carbon-based molecules. By studying the behavior of carbon chains, one can uncover mathematical patterns, relationships, and potential computational capabilities that exist within the molecular realm.

This exploration can lead to a deeper understanding of how mathematical principles emerge in nature and how they can be harnessed for computational processes. By examining the properties of carbon chains and their interactions, scientists can potentially uncover new insights into molecular computation, molecular programming, and the fundamental mathematical foundations that underlie biological systems.

Additionally, studying the naturally occurring mathematics and computation in carbon chains may also have practical implications. It could inspire the development of novel computational models, molecular computing devices, and bio-inspired algorithms that leverage the inherent computational capabilities of carbon-based molecules. This interdisciplinary research can bridge the gap between mathematics, computer science, and molecular biology, opening up new avenues for scientific discoveries and technological advancements.

Naturally occurring mathematics and computation are fascinating phenomena that can be observed in chemical and biological systems. Here are some examples:

1. Fibonacci sequence in plant structures: The Fibonacci sequence, a series of numbers in which each number is the sum of the two preceding ones (0, 1, 1, 2, 3, 5, 8, 13, and so on), can be observed in various plant structures. For example, the arrangement of leaves on a stem, such as in a sunflower head or a pinecone, often follows a spiral pattern that approximates a Fibonacci spiral. This arrangement allows maximum exposure of each leaf to sunlight, optimizing photosynthesis efficiency.
2. Chemical reaction networks: Chemical reactions that occur in living organisms often involve complex networks of reactions that exhibit computational properties. Biochemical pathways, such as metabolic pathways, involve a series of chemical reactions that are carefully orchestrated to carry out specific functions in living cells. These pathways involve complex molecular interactions and feedback loops that can exhibit computational behaviors, such as signal processing, feedback regulation, and oscillatory dynamics.
3. DNA computing: DNA, the genetic material of living organisms, can also be used for computation. DNA molecules can store and process information in the form of genetic codes, which can be manipulated to carry out computational tasks. DNA computing has been explored as a potential alternative to traditional silicon-based computing, with applications in areas such as data storage, cryptography, and drug discovery.
4. Swarm intelligence: Many biological systems, such as ant colonies or bee swarms, exhibit emergent behaviors that can be seen as forms of computation. These systems involve large numbers of individual agents that interact locally and exhibit collective behaviors, such as foraging, nest building, or defense, without central control. Swarm intelligence, as it is called, can be seen as a distributed form of computation where simple local rules followed by individuals lead to complex global behaviors that solve problems, such as finding the shortest path to a food source or optimizing resource allocation.
5. Protein folding: The process by which proteins, the building blocks of living organisms, fold into their three-dimensional structures is a complex computational problem. Proteins fold into specific shapes that determine their functions, and the folding process involves searching through a vast number of possible conformations to find the energetically favorable one. Computational techniques, such as molecular dynamics simulations and machine learning algorithms, are used to model and predict protein folding, which has implications for understanding diseases, drug design, and synthetic biology.
6. Protein Vibrations, histone tails for example, as carried out by rotations of covalent bonds around carbon atoms can potentially be complex computational processes, as this web site demonstrates.

Overall, mathematics and computation are fundamental aspects of chemical and biological systems, and their study provides insights into the intricate workings of nature and has potential applications in various fields of science and technology.

The bonding of two carbon atoms yields a rich source of mathematical theory application as well as discovery. Distinct mathematical structures such as arithmetic, algebraic and logical processes abound.  

These basic, fundamental structures...as atomic as sets in set theory...seem to form some sort of basis for all of mathematics. In some real way all of science has evolved from such basis.

Biomatics is the name I chose for this virtual universe.

Perry Moncznik 10-2021

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Intramolecular mathematics can be used to study the properties and behavior of carbon chains, which are molecules composed of a series of carbon atoms connected by covalent bonds. Carbon chains can exhibit a wide range of chemical and physical properties, depending on factors such as the length of the chain, the arrangement of the carbon atoms, and the presence of other functional groups.

One important aspect of intramolecular mathematics in the study of carbon chains is the use of organic chemistry principles to predict the reactivity and behavior of these molecules. Organic chemistry is the branch of chemistry that studies the structure, properties, and reactions of carbon-based compounds, and provides a framework for understanding the behavior of carbon chains.

Another important aspect of intramolecular mathematics in the study of carbon chains is the use of computational chemistry methods to predict the properties and behavior of these molecules. Computational chemistry involves the use of mathematical models and computer simulations to study the behavior of molecules at the atomic and subatomic scale. This can be useful for predicting the electronic structure and reactivity of carbon chains, as well as for designing new carbon-based materials with specific properties.

Overall, intramolecular mathematics plays an important role in the study of carbon chains and other complex molecules, providing a powerful set of tools for understanding and predicting the behavior of these systems. 

ChatGPT 4-2023

Isaac Newton's "Principia Mathematica" (often simply referred to as "The Principia") is a monumental work in the field of physics and astronomy, published in 1687. It laid out the fundamental laws of motion and gravity, and demonstrated how they could be used to explain a wide range of phenomena in the natural world, from the motion of planets to the behavior of falling objects. Newton's "Principia" was a key work in the scientific revolution of the 17th century, and is still widely regarded as one of the most important scientific works ever written.

On the other hand, Bertrand Russell and Alfred North Whitehead's "Principia Mathematica" was a work in the field of logic and mathematics, published in three volumes between 1910 and 1913. The goal of the work was to provide a rigorous foundation for mathematics, by defining all mathematical concepts in terms of logical and set-theoretical principles. The "Principia Mathematica" attempted to prove that all of mathematics could be reduced to a set of simple logical axioms, and was seen as a major achievement in the field of logic and the philosophy of mathematics.

While the two works share a similar title, they are quite different in terms of their subject matter and historical context. Newton's "Principia" was a landmark in the development of physics and astronomy, while Russell and Whitehead's "Principia Mathematica" was a landmark in the development of logic and the foundations of mathematics.

This Principia BioMathematica attempts a rigorous foundation for mathematics (and computation) in biological systems.

Molecular programs refer to the design and implementation of computational processes or algorithms using molecules as the building blocks. Instead of traditional electronic computers, molecular programs utilize the inherent properties and interactions of molecules to perform computations or execute specific tasks.

There are various approaches to implementing molecular programs, depending on the nature of the problem and the desired level of complexity. Here are a few examples:

1. DNA Computing: DNA molecules have been extensively used in molecular computing. DNA strands can be designed to carry information in their sequences and interact with each other through base pairing. By manipulating DNA molecules and designing specific reactions, researchers have demonstrated various computational tasks, such as solving mathematical problems, performing algorithms, and even simulating biological processes.
2. Chemical Reaction Networks: Chemical reactions can be utilized to create complex computational systems. By designing networks of chemical reactions, molecules can undergo specific transformations based on their concentrations and reaction kinetics. These reaction networks can be programmed to perform computations, generate patterns, or control molecular processes.
3. Protein-based Computing: Proteins, with their diverse functions and interactions, can be harnessed for molecular computing. Researchers have explored the design of protein-based systems, such as protein switches, protein circuits, or enzyme-based logic gates, to perform specific computational tasks. These systems can utilize the enzymatic activities or conformational changes of proteins to carry out computations.
4. Molecular Machines: Molecular machines are synthetic molecules that can perform mechanical tasks or computations at the molecular scale. These machines can have moving parts or switches that respond to specific inputs, enabling them to execute predefined functions or algorithms. Examples include molecular motors, rotaxanes, or DNA walkers.

Molecular programs offer the potential for highly parallel and miniaturized computation, as well as the integration of computation with biochemical processes. They are still an active area of research, with ongoing efforts to develop new methodologies, improve scalability, and explore practical applications in areas such as drug discovery, sensing, and molecular robotics.

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The above images are the output of "molecular programs".  In the M-theory of physics these can be thought of as the images created by the free ends of vibrating "strings" with specific geometric properties based on the carbon atom. Two consecutive "carbon atoms" are joined by a covalent bond at an approximate angle of 109.5 degrees to adjacent covalent bonds (similar to the actual bonds seen in vivo molecules e.g. proteins and amino acids). Each of the bonds can be "programmed" to rotate in a manner where the speed of rotation is a function of time. In a more simple manner the functions can be constant, but varying from bond to bond e.g. 1 1 1 for 3 consecutive bonds rotating at the same constant angular velocity or 1 2 5 where the second bond rotates twice a fast as the first and the third bond rotates five times faster than the first. The possible permutations, even for integer multiples, is very large for long chains.  

A molecular theory of computation based on computation found in smart molecules could potentially involve using the computational principles to design molecular systems capable of performing specific computational tasks.

For example, using the idea of a group theoretic action, we can consider how a particular smart molecule might be able to operate on another molecule, perhaps modifying it in a specific way. This could be useful for designing molecular systems that are capable of performing specific chemical reactions or for developing targeted drug delivery mechanisms.

Another potential application of group theory in molecular computation is the design of molecular circuits. Just as electronic circuits are built using logic gates, molecular circuits could be built using molecular logic gates, which are based on the properties of the smart molecules involved. Group theory could be used to help identify which molecular groups are best suited for use as logic gates and how to combine them to achieve desired circuit behavior.

Overall, a molecular theory of computation based on groups found in smart molecules has the potential to revolutionize fields such as chemistry, materials science, and biotechnology by enabling the design of novel molecular systems with specific computational capabilities.

These structures are generated by molecular programs differing only in the last value (0 thru 7)

A common geometric feature of all these structures is the similar general shape of the side view, similar to the side view geometry of embryos.

Rotations of chemical bonds form mathematical structures at the molecular level. Shown here is  a Cayley table of a group, demonstrating how a simple algebraic system is possible. The colors are to aid in seeing the fractal geometry and symmetries.

In mathematics, a group is a fundamental algebraic structure that consists of a set equipped with an operation that combines any two elements to produce a third element. To be considered a group, the set and the operation must satisfy four key properties:

1. Closure: The operation applied to any two elements in the group must produce another element in the group.
2. Associativity: The operation is associative, meaning that the grouping of elements does not affect the result. For any elements a, b, and c in the group, (a * b) * c = a * (b * c).
3. Identity Element: There exists an identity element (usually denoted as e or 1) in the group, such that for any element a in the group, a * e = e * a = a.
4. Inverse Element: For every element a in the group, there exists an inverse element (usually denoted as a⁻¹) such that a * a⁻¹ = a⁻¹ * a = e, where e is the identity element.

If a set and an operation satisfy these four properties, then they form a group. Groups are essential in various branches of mathematics, including abstract algebra and group theory. Examples of groups include the integers under addition, the set of non-zero real numbers under multiplication, and the symmetries of a geometric figure.

Carbon chains can be represented using a Cayley table in the context of group theory. Specifically, carbon chains can be considered as a group under certain operations, such as rotations of covalent bonds. In this context, the different states of the molecule can be viewed as the elements of the group, and the possible transformations can be viewed as the group operations.

Overall, the connection between carbon chains and Cayley tables lies in the fact that both can be represented using the language of group theory, which provides a powerful framework for understanding the properties and behavior of complex systems.

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Also a chain of 5 carbon atoms with the first bond fixed in place and assuming each succeeding bond can be in one of two energetically favorable states. This system of 8 elements (0-7) or (000-111) can embody and be described as the elements of a finite mathematical (Galois) field, with addition, subtraction, multiplication, and division (modulo 8). 

A Galois field with a prime order is denoted as GF(p), where p is a prime number. This is a finite field with p elements, and it is often referred to as a prime field.

In GF(p), where p is prime, the field operations (addition, subtraction, multiplication, and division) are performed modulo p. Here are the key properties of GF(p):

1. Order: The number of elements in GF(p) is p.
2. Addition and Multiplication: The addition and multiplication operations are performed modulo p.
3. Field Structure: GF(p) has closure under addition and multiplication, associativity of addition and multiplication, commutativity of addition and multiplication, existence of additive and multiplicative identity elements (0 and 1, respectively), and existence of additive and multiplicative inverses.
4. Irreducibility: Since p is prime, every non-zero element in GF(p) has a multiplicative inverse, making it a field.

In a finite field with 7 elements, GF(7), the elements are integers from 0 to 6, and arithmetic operations (addition, subtraction, multiplication, and division) are performed modulo 7. Here's the set of elements in GF(7):

GF(7)={0,1,2,3,4,5,6} 

 Note that in GF(7), the multiplicative inverse of an element a (where a not = 0) is the value b such that a×b ≡1 (mod7). For example, the multiplicative inverse of 3 is 5 because 3×5≡1(mod7). 

In mathematics, finite fields, also known as Galois fields, are algebraic structures with well-defined addition, subtraction, multiplication, and division operations. The field order is often denoted as p^n, where p is a prime number and n is a positive integer.

Let's consider a simplified model where molecules represent elements in a finite field of order p^n. Each molecular state corresponds to an element in this field. The operations of addition, subtraction, multiplication, and division are defined modulo p.

1. Addition (Modulo p): For any molecular states a and b, the sum a+b modulo p is computed as: (a+b)mod p
2. Subtraction (Modulo p): For any molecular states a and b, the difference a−b modulo p is computed as: (a−b)mod p=((a mod p)−(b mod p)+p)mod p
3. Multiplication (Modulo p): For any molecular states a and b, the product a⋅b modulo p is computed as: (a⋅b)mod p
4. Division (Modulo p): For any molecular states a and a non-zero molecular state b, the modular division of a by b modulo p is defined if b is invertible modulo p. The result is then: a/b ​mod p=(a⋅c)mod p

In this context, the molecular states behave like elements in a finite field, and the operations are performed modulo a prime number p. The critical condition for division is that the divisor b must have a multiplicative inverse in the field.

This conceptualization allows us to explore mathematical structures inspired by molecular interactions, offering a unique perspective on information processing at the molecular level.

Fermat's Little Theorem

If p is a prime number, for any integer a, we can write a^p = a(mod p)

Dividing both sides by a we get a^(p-1) = 1(mod p) ... or a*a^(p-2) = 1(mod p). Thus, the multiplicative inverse of any integer a in GF(p) is (a^(p-2))( mod p)