**You can download two interesting PDFs by Javier Amaru Ruiz Garcia that explore aspects of mathematics as they relate to biochemistry, the Yupana, and toplogy.**

**1. THE PRE-EXISTING LOGIC OF THE GENETIC CODE**

We assume the geometry of a hyperdiamond inside the nucleus of the cell, which also was derived from the Yupana system, representing a biological multidimensional lattice, which demonstrates that the genetic code is somewhat consistent with the basic principles of crystallography. In that regard, as important as the double helix is toward understanding DNA, the Yupana system is equally important toward understanding both the symmetry and geometry of a highly optimized genetic code.

This paper will explain both the structure of the Yupana mathematical system and the rationale behind the development of **the postulate that the genetic code was derived from a preexisting logic rather than randomly generated by evolutionary principles.** In other words, the “accidental DNA mutations” found in the program for embryo development that is passed down from generation to generation; implemented by proteins encoded by the DNA, are postulated to be structured and shaped into new species, according to the inherent mathematical nature of the molecular language. A result emerging from this analysis is that **the genetic code conserves most long-range interactions among amino acids rather than their short-range stereochemical attributes.**

**2. MODELING THE MATHEMATICAL CELL WITH THE EULER CONSTANT****Abstract:** The theoretical analysis of the mathematical propositions and relations of the molecule that constitutes DNA explains that DNA is a successful computation molecule. In that regard and based on the principles of molecular biology, **we proposed a mathematical model derived from the Trichotomyc Theorem, (reinforced with the Yupana´s mathematical system),** which is a relevant and extensible abstraction in order to demonstrate why DNA molecules are accurate in information process and computing. Also our model, probes the hypothesis which postulates that the Euler constant (e) is encoded between the Viral DNA helical segments comprised of prime numbers. In that regard, we are going to explain in detail how the Euler constant within such a framework, modulates both the amplitude and longitude of the DNA molecule. In other words the mathematical cell represents a genomic and proteomic biological cell, included with mathematical DNA sequences which in such framework, get transcribed into RNA sequences resembling the first step in making logical proteins.

Since it is relevant to capture the essential property of any phenomenon, in this case, **the Trichotomyc Theorem, offers the best conceptual and mathematical framework in order to develop such a logical molecule**, also the mathematics analyzed through Mersenne primes and the occurrence of the mathematical constant **e** of Euler discovered encoded within the DNA molecule, shows that numbers are the foundation of all there is even molecular biology.

And finally, just for the colorful fun of it, is this collage from * The Genetic Oracle* by Javier Amaru Ruiz Garcia…

**The Yupana, ancient calculating tool of the Incas…provides novel insight into the molecular language of the genetic code**

**Information in DNA** is chemically the same for a human, an insect, an orange, or any other living organism. It is chemically-bonded molecules that organize into codons that translate into amino acids that make protein. **The challenge for a molecular biologist** is understanding completely what all that molecular information means. The ancient Inca tool called **the Yupana** provides a logical resource to gain insight on the origin of DNA's mathematical makeup.

**Definition:** A **Yupana** (from Quechua *yupay*, meaning *to count*) is an abacus dating back to the time of the Incas. The Yupana is still used to perform arithmetic operations. It provides us with a **mathematical and cellular system for exploring the origin of the genetic code**.

The scientific community assumes that the genetic code emerged at a coding transition when the **mapping of codons to amino acids** became nonrandom. The emergence of the genetic code was governed by the topology defined by the probable errors, and thus it is related to the ** Four Color Theorem **or

**The Four Color Theorem states:** “Given any separation of a plane into contiguous regions, producing a figure called a map,

However, the Yupana mathematical system demonstrates that the solution to such a problem was resolved through the symmetry by which the **64 amino acids accommodate in the DNA molecule as 16 groups of four codons** (i.e., as molecular *quaternions*). Molecular quaternions demonstrate that **codon redundancy in the genetic code is a consequence of geometric symmetry**.

**A molecular quaternion** of four codons appears as follows in the Yupana (ancient Inca mathematical tool):

- First codon at the
**3' terminal**(top right side of the Yupana) - Second base pair codon at
**5' terminal**(top left side of the Yupana) - Third anticodon at
**3' terminal**(bottom right side of the Yupana) - Fourth and last, base pair anticodon at the
**5' terminal**(bottom left side of the Yupana)

**Note 1:** The Yupana system in our model can certainly do **both binary and analog computation **in the same process. To make this clear, we would like to quote Katya Walter "Two models of this paradigm may be found in two very different places: in modern DNA and an ancient Chinese document called the I Ching. Each combines binary/digital unitizing along with analog ratios in flow to **develop a peculiar hybrid mathematics** that turns both DNA and the I Ching into a sturdy survival package.

“The DNA swatch and I Ching hexagram can each be demonstrated to hold two polarized period 3 windows in a bond. These two windows are counterpoised against each other to form a fail-safe package of complementary chaos (or co-chaos for short). There are eight possible triplets. These may be arranged in polarized pairs to form 64 possible configurations...which naturally gives us the 64 codons of the genetic code or the 64 hexagrams of the I Ching.”**Note 2:** The number 1936 was found to be the **least irreducible****part** of the Four Color Theorem. Notice that the Yupana tool synchronizes 16 groups of 4 codons each. In that respect…

- 1936 divided by 16 = 121
- In other words, 1936/16 = 121
- 121=112
- …and the number 11 is one of the prime factors from the Yupana!

**Javier Amaru Ruiz Garcia**

**Javier Amaru Ruiz Garcia** (born in La Paz, Bolivia) has a strong interest in **math, biochemistry, and engineering.** His exploration of of an ancient mathematical tool of the Incas called the Yupana led him to realize that it could reveal why the codons of the genetic code are organized according to the * Four Color Theorem* of mathematics. Here is a Yupana…

**DNA&Inca Math**

…and here is how Javier realized that the Yupana could clarify codon organization: