User:Miguel De Zayas

Title: “Magic hexagons explain the resonant nature of Carbon atoms in a 3 dimensional space projection” By Miguel De Zayas

ABSTRACT The inherent puzzling nature of hexagons continues to intrigue the math community since the very day it was made popular by Martin Gardner who learned about it from a retired employee at a railway company from Philadelphia (Clifford W. Addams). In 1991 Hans F. Bauch published for the first time his original version of a magic hexagon known today as “T-hexagon” or “magic hexagon D (4). This particular arrangement constitutes the first example of a magic hexagon as it was discovered on September 13th 2003. A hexagram with a Greek root was also known as Sexagram from its Latin denomination and is made of two equilateral triangles intercepting each by sharing a common base and forming a six-pointed geometrical “star figure” and its use has been extended to countless religious and cultural contexts, to mention just a few of them in Occultism, Jewish identity and Hinduism. In 1991 with the title “The Magic Hexagram” Bolt, B Eggleton, R.; AND Gilks, J. published a two-solution answer to the magic hexagram while using numbers from 1-12, their analysis was performed in a two-dimensional plane and therefore their conclusions were showing a sum difference for each solution; namely 32 and 33 respectively. The purpose of this article is to demonstrate that the solution was correct by using 3 dimensions instead of just 2 and that the nature of the resonance observed in the Carbon atom during its chemical bonding has a hidden origin explained by the magic of hexagons. See Magic Hexagram from http://mathworld.wolfram.com/MagicHexagram.html

According to the particular definition of a magic hexagram the sums of the numbers in all six directions shown by arrows in the Figure above must add to the same numerical result. Eggleton and Gilks found the numerical order allowing for such configuration, however the sum of the numbers on both hexagrams were giving them 32 and 33 respectively; it was obviously an unfortunate result as a conclusion based of a flawed dimensional approach. The first step made to verify the correspondence between their numerical model and the concept of two-solution model was to sum the columns according to their angles or degrees of their positions in other words I followed a clockwise directional analysis also imagining each hexagon as if they were inserted into clocks. In the left hexagram (see Figure for reference) I considered the numbers 1 and 11 as 00:00 hour and 06:00 hour respectively, 4 and 8 as 02:00 and 08:00 hours respectively and so on. The same analysis performed on the right hexagram would result in 12 and 2 for 00:00 hour and 06:00 hour and so on. If we do the simple math on both hexagrams for 00:00 hour (0°), 02:00 hours (30°), 04:00 (120°) you’ll find that at 0° they both sum to 26, at 30° they add to 21 (left) and 31 (right), however if we resume the mathematical process further into 240° and 300° we find that the results for the left hexagram and the right one will be the same but opposite in value namely 31 for the left hexagram at 300° while 21 for the right hexagram at the same angle. The conclusions couldn’t be obvious: Assuming both hexagrams with a common axis at 0° (00:00 hour) and the common sum result of 26, the two-solution model is sound and irrefutably showing an alternating pattern of rotation or resonance between two perfect solutions in a time progression. If you calculate the average value between 21 and 31 you’ll find that 26 are in the exact middle of the scale. Before establishing the connection of this analysis with the chemical resonance process known as hybridization in the specific case of the Carbon atom, let’s look at another projection of our hexagrams but this time using a 3 dimensional analysis:

Fig. 2

Left    9     4 8    5

Right   4     9 5    8

As seen in this projection angle both hexagrams sum 26 just like the previous analysis performed at 0°.

Fig. 3

Left     3    1 11  6

Right    1    12 2   6

In this Figure above we can see the 3D results for 300° angle or 10:00 hours and as we can easily calculate by adding the numbers in those squares we find that both hexagrams add to 21. Remember that 21 was also one of the results of our first analysis on step one.

fig. 4.

Left   6    10 12  3

Right 7     3 11   10

We will see the final result of 31 for both hexagrams.

CONCLUSIONS The analysis based on 0°, 30°, and 60° angles performed on both hexagrams coincide with those three values using a two-dimensional and a 3-dimensional analysis (26, 21 and 31). For the Carbon atom that has a hexagonal shape when it bond with hydrogen atoms in a molecule this simple mathematical results may reveal the mechanism for the two-solution model and the connection between hybridization and magic hexagrams.