A Beautiful Question: Why Symmetric?

Abstract

In this paper we investigate that most of plants have the symmetric property. In addition, the human body is also symmetric and contains the DNA symmetric base complementarity. We can see the logarithm helices in Fibonacci series and helices of plants. The sunflower has a shape of circle. A circle is circular symmetric because the shapes are same when it is shifted on the center. Einstein’s spatial relativity is the relation of time and space conversion by the symmetrically generalization of time and space conversion over the spacial. The left and right helices of plants are the symmetric and have element-wise inverse relationships each other. The weight of center weight Hadamard matrix is 2 and is same as the base 2 of natural logarithm. The helix matrices are symmetric and have element-wise inverses matrics base on the finite group theory A B = B A = $I_{N}$. Also, we present human a DNA-RNA genetic code with symmetric base complements A = T = U = 30%, C = G = 20% and C + U = G + A. E. Chargaff discovered yeast and octopus DNA base complement [C T; A G] of four componentry A = T = U = 33% and C = G = 17% of the experimental results. This strongly hinted towards the base pair makeup of DNA, although Chargaff did not explicitly state this connection himself. However, it has not been proved in a mathematical analysis view yet. In this paper, we have a simple proof of this problem based on information theory as the doubly stochastic matrix.

Appendix

Finite group property

• 1. The transformation of individuals constituting a group is called an element of a group, and each group includes an identical transformation E.

• 2. The elements of the group can be multiplied with each other. The multiplication of the elements of the group means successive symmetric transformations. A new element made up of the multiplication of elements is also an element of that group.

• 3. The multiplication of elements satisfies the associative rule and exists in the same group with the inverse of the element of the given group. The symmetric matrix A is $AB = BA = I_{N}$ in the complex or finite field, and $B = \frac{1}{n}(a_{ij}^{ - 1} )^{T}$, and satisfies the property of the Jacket matrix.