There are two Masonic triangles which are more related than one might initially suspect.
One is the Square Root of Two triangle.
The other is generally much better known, and the topic of this page - the triangle with sides 3, 4 and 5.
In another part of this series you will be shown an incredible way that these two types of triangles (the Square Root of Two and the 345) are related in a simple but most profound way that you are unlikely to have seen before.
First, the triangle needs to be introduced.
Here is the special 345 triangle, which is the first and most commonly seen Pythagorean triple.
Right triangles whose sides are of integer lengths are Pythagorean triples. There are an infinite number of these Special right triangles with varying combinations of lengths.
The most well known of these, with side lengths of 3, 4 ands 5, is taught in Freemasonry.
Another example of a common Pythagorean triple is the 5, 12 and 13. We don't have to worry about it here, but perhaps it will become interesting later.
This 345 triangle is found on very old Master's Carpets.
Counting the squares shows the numbers squared.
Notice the relevance to the integer side lengths being squared.
32 = 9
42 = 16
52 = 25
The difference between squares reveals a pattern of odd numbers between each side.
Because these numbers are formed between the sides, each of them has an association with the internal angles of the triangle, which are also between the sides.
It turns out that these odd numbers (the 7 and 9 above) relate to another set of three numbers in Freemasonry.
In the second degree of Freemasonry, the Fellowcraft degree, a sequence of 3, 5 and 7 steps is introduced.
Steps in the Fellowcraft Degree
3 steps, 5 steps, and 7 steps.
Notice that these are all odd numbers in order. Although related, try not to confuse them with the 3, 4 and 5 from the triangle sides.
You can see the 3, 5 and 7 placed on these steps.
This introduces the idea of a sequence of odd numbers.
Because the 9 was seen between sides on the 345 triangle, it raises the possibility that we should also be interested in 9, in additional to the 3, 5 and 7.
With enough research it can be determined that the sequence of odd numbers which separates the sides of the 345 triangle relates directly to the steps in the Fellowcraft degree.
You can see the 7 and 9 in the differences on the 345 triangle.
9 is best described simply as a destination, or entrance point.
The 3, 5 and 7 steps are an incremental progression towards this hyperdimensional access point.
The former values which aren't immediately obvious, the 3 and 5, come out of the 53.13° angle (between the 5 and 3 sides of the triangle).
You read that right - the 3 and 5 are there, but not in the same place as the 7 and 9.
They come out of the numerical values which define the angle.
Using the numbers of the angle gives the odd numbers which happen to lead on directly to the 7 and the 9.
This is exactly the sort of small hint that needs to be taken the right way.
Now we have the complete set!
1, 3, 5, 7, 9
You can begin to see how the numbers are never just a coincidence, and if you let them, they will show you how to understand them, and what step to take next.