This  website is concerned with the mathematical, mechanical and eventually the biological potential of a chain of carbon atoms. If we fix the first bond in the molecule in the chain and assume for the sake of analysis that the covalent bonds are free to rotate or are even programmable then what are the mathematical or computational possibilities? We have already seen one potential group structure, namely the Cayley table on the home page. Inspired by Galois's observation of the roots of a polynomial forming a group, what other algebraic structures can we find in a molecule vibrating with a given structure, namely the geometric properties of a chain of carbon atoms. 

Given, for the sake of analysis, that all the bond lengths and bond angles are constant e.g. the average bond angle in a chain of carbon atoms 109.5 then the location in 3D space of the final carbon atom in the chain (x,y,z) is a function in time of the state of the relative angles of rotation of the individual covalent bonds...

(x,y,z) = f(Ɵ1,Ɵ2,...,Ɵn) 

where each Ɵn is some angle between zero and 360 (see "molecular program" video on home page).

Given such a "molecule" looking for "group" structures may be relevant and fruitfull in finding mathematical and logical structure.

Definition (Group):
A group G G is a collection of objects with an operation ⋅  satisfying the following rules (axioms):
(1) For any two elements x x and y y in the group G G we also have x⋅y  in the group G .
(2) There is an element (usually written 1  or e , but sometimes  0) called the identity in  G such that for any x  in the group G we have 1⋅x=x⋅1. (3) For any elements x, y, z in  G we have (x⋅y)⋅z=x⋅(y⋅z)  (so it doesn't matter what order we do the calculations in). This property is called associativity ; it means we can write x⋅y⋅z  unambiguously (otherwise it would not be clear what we meant by x⋅y⋅z : would it be x⋅(y⋅z)  or (x⋅y)⋅z ?).
(4) Every element  x in  G has a unique inverse  y (sometimes written −x  or  x^{-1}) so that x⋅y=y⋅x=1

Circle Group

One way to think about the circle group is that it describes how to add angles, where only angles between 0° and 360° are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer should be 150° + 270° = 420°, but when thinking in terms of the circle group, we need to "forget" the fact that we have wrapped once around the circle. Therefore, we adjust our answer by 360° which gives 420° = 60° (mod 360°).  

A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable.

This definition is related to the fifth of Hilbert's problems, which asks if the assumption of differentiability for functions defining a continuous transformation group can be avoided.

The simplest examples of Lie groups are one-dimensional. Under addition, the real line is a Lie group. After picking a specific point to be the identity element, the circle is also a Lie group. Another point on the circle at angle heta from the identity then acts by rotating the circle by the angle theta.

 Mathematicians invented the concept of a group to capture the essence of symmetry. The collection of symmetries of any object is a group, and every group is the symmetries of some object. E8 is a rather complicated group: it is the symmetries of a particular 57 dimensional object, and E8 itself is 248 dimensional!  E8 is even more special: it is a Lie group, which means that it also has a nice geometric structure.

The theory of groups has found widespread application. It was used to determine the possible structure of crystals, and it has deep implications for the theory of molecular vibration. The conservation laws of physics, such as conservation of energy and conservation of electric charge, all arise from the symmetries in the equations of physics. And one of the simplest groups, known as "the multiplicative group modulo N" is used every time you send secure information over the Internet.

For an introduction to groups requiring little mathematics background, see Groups and Symmetry by David Farmer.

Lie groups lie at the intersection of two fundamental fields of mathematics: algebra and geometry. A Lie group is first of all a group. Secondly it is a smooth manifold which is a specific kind of geometric object. The circle and the sphere are examples of smooth manifolds. Finally the algebraic structure and the geometric structure must be compatible in a precise way.

Informally, a Lie group is a group of symmetries where the symmetries are continuous. A circle has a continuous group of symmetries: you can rotate the circle an arbitrarily small amount and it looks the same. This is in contrast to the hexagon, for example. If you rotate the hexagon by a small amount then it will look different. Only rotations that are multiples of one-sixth of a full turn are symmetries of a hexagon.

Lie groups are ubiquitous in mathematics and all areas of science. Associated to any system which has a continuous group of symmetries is a Lie group.

  (What is a Lie group? (aimath.org))

 Galois theory

 (pronounced gal-wah) is a subject in mathematics that is centered around the connection between two mathematical structures, fields and groups. Fields are sets of numbers (sometimes abstractly called elements) that have a way of adding, subtracting, multiplying, and dividing. Groups are like fields, but with only one operation often called addition (subtraction is considered addition with negative numbers).

y=f(x)

y=f(x1,x2,,,xn)

 (x,y,z) = f(Ɵ1,Ɵ2,...,Ɵn) 

Different categories of programs:

       Discrete

       Continuous

             Static

             Dynamic

If a circle is a group and a Lie group, then what about a more complex function yielding a closed 3 dimensional structure more complex than a simple circle. 

(x,y,z) = f(Ɵ1,Ɵ2,...,Ɵn) where all n angles are zero serves as the identity element. In the case of a chain of carbon atoms where the first covalent bond is fixed in space there is a hierarchical (recursive?) relationship of the bond angles. The first rotating bond acts as the fundamental frequency of the sequential set of rotating  bonds.

Closed loop under certain conditions

Fixed Action Patterns 

Could the above "group" structures be illuminating when considering high level biological behaviour?

A fixed action pattern (FAP) is an instinctive behavioral response triggered by a very specific stimulus. Once triggered, the FAP behavior can’t be stopped ‘midstream’, but must play out to completion. Yawning is one example. Dribbling a basketball between your legs or shooting like Steph Curry are more complex examples.

The Cayley table at the right is one example where each bond can be in one of two states.

The different colors are an attempt to show the existence of subgroups.

It is possible that a string in the physics sense could embody a mathematical group, just as a carbon chain can embody a mathematical group. The entanglement of multiple strings could also result in a similar phenomenon to molecular folding. However, the properties and behavior of a string are described by complex mathematical equations, and while a simple mathematical structure may be a possible subsystem, it may not necessarily apply to reality and would require further exploration and testing by scientists.