# The Fundamental Theorem of Biomatics

### From Biomatics.org

If Biology is to reveal it's mathematical secrets it is likely to do so in the form of some fundamental concept. The interactions of two carbon atoms seems about as fundamental as can be. A "fundamental theorem" may thus be that - **two carbon atoms yield a "set" and an "operation" as well as possible "relations". **

In mathematics, a **primitive notion** is a concept not defined in terms of previously defined concepts, but only motivated informally, usually by an appeal to intuition and everyday experience. For example in naive set theory, the notion of an empty set is primitive. (That it exists is an implicit axiom.) For a more formal discussion of the foundations of mathematics see the axiomatic set theory article. In an axiomatic theory or formal system, the role of a primitive notion is analogous to that of axiom. In axiomatic theories, the primitive notions are sometimes said to be "defined" by the axioms, but this can be misleading.

**Foundations of mathematics** is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory. The search for foundations of mathematics is also a central question of the philosophy of mathematics: On what ultimate basis can mathematical statements be called true?

Consider two adjacent carbon atoms. The hydrogen atoms will be in an energetically favorable position when they are in the staggered so called "trans" configuration.

Notice that each covalent bond could rotate one hundred and twenty degrees to the next favorable spot. Each carbon thus has potentially three favorable states.

We now add a few more lines to complete the cube.

Consider a set of three carbon atoms that can each exist in one of two states. We therefore have 2**3 or eight possible states which can be labeled as 000, 001, 010, 011, 100, 101, 110, 111. To go from 000 to 001 requires the rotation of a single bond. To go from 000 to 110 requires rotation about two bonds etc. The following diagram, where the eight corners are the states and the twelve edges are the transitions between states, can depict this situation.

Label edges

E1 = 000,001

E2 = 001,011

E3 = 011,010

E4 = 010,000

E5 = 000,100

E6 = 100,110

E7 = 110,111

E8 = 111,011

E9 = 101,001

E10 = 111,101

E11 = 100,101

E12 = 010,110

Inverses

E1' = 001,000

E2' = 011,001

E3’ = 010,011

E4’ = 000,010

E5’ = 100,000

E6’ = 110,100

E7’ = 111,110

E8’ = 011,111

E9’ = 001,101

E10’= 101,111

E11’= 101,100

E12’= 110,010

If we consider all sequences of moves along the edges of the Cube, we notice the following:

**Closure**- One sequence followed by another sequence is a sequence.

**Associativity**- That is E1*(E2*E3) = (E1*E2)*E3.

**Identity element**- The do nothing sequence does nothing.

**Inverses**- Every sequence of moves can be done backwards and therefore undone, e.g. E1 and E1′.

Thus, the set of all sequences of moves on the above Cube is a group.

Coincidentally we could relabel the above cube as follows:

to represent the relation “containment” of one subset in another for the partially ordered set of all subsets of the set (a,b,c).

In addition, represent the relation of divisibility for the partially ordered set (1,2,3,5,6,10,15,30) as follows:

# **Molecular Spin Groups**

A Group is a set together with an operation on the members of that set such that the operation applied to any 2 members of the set yields another member of that set. The Integers and the operation of addition form a group, for example.

Given a carbon chain with N carbons and angle of 109.5 degrees between successive carbon atoms, and thus N-1 covalent bonds.

Initial state of the molecule is simply state at time zero. Each bond = zero degrees ie home

Assign spin (ratio) rate to each covalent bond eg radians per unit time: R1,R2,...RN-1. The elements of the set are the possible transformations of the system, which will be ratio preserving multiples of R1...RN-1.

If Bond 1 and 2 have ratio values of 2 and 3 then a transformation in the set will preserve this ratio (as well as the ratios of all other respective covalent bonds). So for example, 20 degrees and 30 degrees will satisfy the required conditions. So will 30 and 45, as well as 1 and 1.5 etc. Thus we have an infinite set.

Note also that the transformations are additive. eg 20+2=22, 30+3=33 satisfy the required ratio. Thus any 2 members of the set added together yield another member of the set. And thus the requirements of a Group are satisfied.