At this stage we are familiar with the Square Root of Two Triangle, as well as the revered 345 Triangle, both commonly found in Freemasonry.
In this article an incredible connection between these two traingles will be established as a direct result of Masonic sources.
Here we can see the symbol of the Square Root of Two, on a popular intereptation of George Washington the Freemason.
What we often need to do when decoding this information is to go out of our way to analyze different depictions of the same symbols. The more variation in the perspectives we can have, the more powerful our angle of attack.
Wandering the halls of the Grand Lodge of Pennsylvania, an astute observer may notice our square root of two symbol being held by this statue.
We find that a common representation of the symbol has these extra lines drawn between the squares.
A helpful drawing by Alexander Slade features this symbol with all of our internal extra lines.
After going through several variations of the symbol, and regardless of any single specific design context, one can conclude that all of these lines are important to the Freemasons.
Count the lines and see that there are five.
It is established that there are common Masonic depictions of the Masonic Square Root of Two triangles containing these inner lines.
What makes these lines quite remarkable is a hidden relationship to the 3-4-5 triangle mentioned at the beginning of this chapter.
I was surprised, considering its simplicity, that I was unable to find any mention of this connection elsewhere, so I am pleased to present it here for the first time. It was immediately obvious with just a few basic measurements.
What you are about to see is simple, profound, and of incredible significance to all of the other symbols of Freemasonry.
The connection is in the angles created by the strange five extra lines that the Freemasons draw within the square root of 2 formation.
The angles of any 3-4-5 triangle are well defined and well known.
As long as the 3-4-5 proportions are maintained, we have the angles 53.13° between the 3 and 5 sides, and 36.87° between the 4 and 5 sides. The remaining angle is of course 90°.
Any entusiatic observer should commit these to memory as they are seen frequently.
Now we are ready to measure the lines drawn by the Freemasons with the square root of two triangle.
See that the a lines, b lines and c lines create angles within the structure.
The lines marked as a create angles we can know precisely, and we know them each to be 26.565°.
The lines marked as b create angles we can know precisely, and we know them each to be 18.435°.
Here is the revelation regarding the connection between this square root of two formation with the internal lines, and the 3-4-5 triangle:
Look at the angles created by the a lines. They are 26.565° and 26.565°.
When we add these together we arrive at the very same angle we see within the 3-4-5 triangle.
26.565° + 26.565° = 53.13°
This relationship is mathematically precise, and not merely approximate.
The same is true for the angles created by the b lines.
Adding them together results in the same number of degrees that we find in the other angle on the 3-4-5 triangle.
18.435° + 18.435° = 36.87°
The reader should take a moment to absorb this information.
The connection suggests a split halfway at each angle.
What this does is show us that the difference two-dimensional geometries are linked in a more complex model that we must aim to derive.
For example, for these divisions half way between the angles to have some relevance, they would have to relate to some extra dimension on the triangle, suggesting the 345 triangle should be considered in more complex modes.
In 1926 Charles Leadbeater published the following image called The Temple of the Angel.
Here we see how the pyramidization of all parts of a 3-4-5 triangle, including the central 3-4-5 triangle, results in a dividing of the angles that may match the findings from this section.
This is a more complex geometry and we will come to realize how this is a valid construct that will help us integrate all of the geometry we find.
Here are some ratios found in the Masonic Triangle which relate to the inner lines.
The a lines creates halves across the central triangle.
Similarly, the b and c lines reveal incremements of thirds.
When we examine the a and b lines with respect to each other we find the suggestion of fifths.
This is a grid of 25 squares which has an area of 5.
This grid will relate to a total area of 5, and its sides are therefore the Square Root of 5.
The Square Root of 5 is approximately 2.236...
The value of the Golden Ratio, Phi, is defined with the simple equation, the central aspect of which is the Square Root of 5.
In this way we can get an idea that these special lines defined mysteriously by the Freemasons could in fact be relating to the Golden Ratio.
And that's just the beginning.
There are endless revelations which can be discovered by following the leads which present themselves at every turn.