Triangular Numbers and Geometry
Triangular numbers are a sequence of numbers where the first row contains a single element, and each subsequent row contains one more element than the previous one.
The equation is represented below:
This formula is derived from the sum of an arithmetic series. Let's explore how a college student might use this formula for triangular numbers up through 10, and then delve into potential applications in both theory and practical scenarios.
Pythagoras, a pre-Socratic Greek philosopher, is renowned for his contributions to mathematics, particularly the Pythagorean theorem. However, his philosophical and mystical studies are just as profound. The Tetractys, a triangular figure consisting of ten points arranged in four rows (1, 2, 3, and 4 points in each row), is a symbol of great importance in Pythagoreanism. It represents the organization of space and the universe: the first row symbolizes the dimensionless point, the second row is the line (two dimensions), the third row is the surface (three dimensions), and the fourth row is the solid (four dimensions).
Pythagoras is also credited with profound discoveries in music, particularly the mathematical relationships in musical harmony. He discovered that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and these ratios were deeply connected to the geometric and numeric principles he studied. This understanding of harmony influenced the development of Western music theory and its mathematical underpinnings.
Tetryonics is a relatively modern theory developed by Kelvin Abraham that seeks to explain physical phenomena using geometric principles, often revisiting and expanding upon ancient knowledge.
In Tetryonics, we first identify that Triangle are Squares. How do we do this, you might ask? Feast your eyes on a geometric principle where in similar fashion to Pythagoras, we start with a single row of 1 triangle. So rather than using dots, we are ‘connecting’ the dots of the Tetactrys to create an equilateral tessellated field. As we add rows, each new row has 2 more triangles than the previous row.
We end up with
1 +
3 +
5 +
7 +
9 + …
Add 1 plus 3 to get 4 which is “Two Squared” = 2^2…
Add 5 to 4 which equals 9 = 3^2 and so on…
1+3 = 4 = 2 ^2
1+3+5 = 9 = 3^2
1+3+5+7 = 16 = 4^2
The successive summation of ODD numbers adds up to a SQUARE number. When seen in this equilateral light, we can just as easily say, “Triangles are Squares.”
We will continue to explore these connections in future posts and podcasts and will endeavor to highlight sympathetic resonances with other geometric works from the ancient past. Fermat’s method of factoring is presented here as yet another signpost demonstrating a harmonic equilateral delight. Never before has the concept that ‘every odd number can be written as the difference of two squares’ been described with a simple geometric expansion such as the one above.
Wayne Roberts' Scale Structure Theory is yet another shining investigation questioning the very notion of units relative to wholes while demonstrating a perfect synergy with Tetryonics.
The numerous symmetries of these amazing numbers (hitherto known as the squares) seems to have eluded many and remained largely unexplored, ignored, or of anecdotal interest for more than two thousand years. Very early in the history of mathematics, triangular numbers and square numbers became defined entities, distinct sets with distinct properties. The nomenclature was biased in favour of difference: 'How does this set differ from that set?' This emphasis on taxonomy (naming and cataloguing things) has affirmed connections at one level, but frequently at the expense of recognising (often more important) connections on other levels. This has also affected the arts.
Our 'square glasses' ensured that we would continue to see square numbers in square ways. (W. Roberts, 2003, p.99)
source: https://principlesofnature.com/the_shape_of_numbers/scale_of_geometric_shapes_of_square_numbers.htm
Athanasius Kircher, a 17th-century polymath, also delves into the intersection of geometry, physics, and spirituality.
With these contemplations firmly in mind, also consider the notion that our spiritual journey can be likened to a spiral staircase. Imagine, if you will, a spiral of continued spiritual growth. We know that there is no end to the spiritual journey, and as a spiral staircase has many steps, we seek to add one more step to the foundations of our staircase of discovery. The study of geometry is never static but a dynamic discovery process, an ever-evolving set of realizations that provide a perspective on cosmic order.
The metaphor of the spiral staircase in spiritual growth is powerful in that it signifies continual progress and ascent, but with a return to similar points or stages at higher levels of understanding. The interplay of geometry, music, and mathematical principles can be seen as a step on this staircase. Just as musical harmony and geometric patterns bring a sense of order and beauty, they can also provide insights into the deeper aspects of the universe and our place within it.
Studying and contemplating these interconnected disciplines can be a tool for spiritual growth, offering a step upward on the spiral staircase of understanding. The connections between geometry, number theory, music, and spirituality form a rich field of study that has intrigued thinkers from Pythagoras to modern theorists.
Each of these disciplines offers a unique lens through which we can view the world, and their interplay can provide profound insights into the nature of reality and our journey through it. The idea is not just to acquire knowledge but to use that knowledge to ascend the spiral staircase of spiritual growth, each step reflecting a higher order of understanding and insight.