Have you heard of a golden gnomon triangle? I am sure there are many, perhaps of the Indian tradition who have spent much time analyzing the patterns and significance of these triangular proportions. However, I could not find many online organizing it the way I thought to. So here I show you the difference in a golden triangle and a golden gnomon triangle. I also present a modified unit circle based on the both of them.
There are two triangles that have the golden ratio involved. One is known simply as “the golden triangle” and the other is known as “golden gnomon triangle”. Gnomon is Greek for “indicator” and the gnomon was what they called the obtuse triangle on a sundial to determine time (in a specific latitude for the specific sundial.) Let’s see how students normally learn triangles and then relate it to these golden triangles.
Unit Circle vs. Golden Unit Circle
In discussing the golden gnomon triangle, I relate it to the “special right triangles” American students learn for the SAT and ACT. The triangles learned are the isosceles right (45-45-90) and the half equilateral triangles (30-60-90). These have side length ratios of 1:√2:√2 and 1/2:1:√3. In order to create these triangles on a circle with radius length 1 unit, the divisions appear uneven, but it is really 15 degrees divisions with some values skipped over.
What is important about the circle divisions? Well, we can choose any number of divisions we want and look for properties. 12 divisions (15 degrees per slice) is often nice because 12 is the most factorable number up to that point. So we can take many different pieces without too much fractional disturbances.
Now let’s look at a circle divided into 10 instead. You may think 10 and 12 are similar, but in the world of factors they only have 2 in common (180 degrees line.) The reason we want to look at 10 is because it cuts the circle into decagonal symmetry, or two pentagonally symmetric orders. On this site we talk a lot about how pentagonal symmetry ALWAYS gives rise to the golden ratio. Here is the keystone article on pentagons.
Golden Gnomon Triangle Unit Circle
When we divide the circle into 10 parts instead of 12, we get totally different triangles and ratios. Above you saw we had √2 and √3 side proportions in the triangles arising from the 12 circle divisions. The base factors of 12 are 2*2*3. Now let’s compare to the side lengths we find in the triangles from the π/10 cut circle (10 having base factors of 2 and 5).
Note that the base factors of 10 are 5 and 2. Anytime we get a √5, we can usually assume it can be written in terms of the golden ratio in some way, or generally has golden ratio properties. Below we look at each of the triangles and find the side lengths in exact form.
Golden Triangle vs. Golden Gnomon Triangle
There are two different triangles mainly with golden properties. In the golden unit circle above, each quadrant has four triangles. Two of these are half golden triangles (first and last) and the other two are half golden gnomon triangles (two middle). Now here is the difference.
Classic Golden Triangle and its properties
The classic golden triangle is simply an isosceles triangle with a ratio of 1:φ from the short to longer side, where φ is the golden ratio (I derive the golden ratio here). A triangle with fixed side ratios also has fixed angles, and the angles end out to be 72, 72, and 36 degrees.
This is an example of how the exact form trigonometric values are found. So yes, it can be done with no calculator. Trigonometric identities, which were found around 100 BCE, are used to transform the unknown quantity into something that simplifies to the exact answer. If you put the sin or cos of these angles into a calculator, it will give you a truncated decimal, which is not exact. This is a rather fun exercise that even a pre-calculus or trigonometry student can perform, as long as they have several sheets of paper.
For the half golden triangle, cutting from the top and bisecting the b side, we look at the 18 degree and 72 degree points on the golden unit circle. In exact form the ratios of the sides from smallest to largest would be: (√5-1)/4 : (√2/4)(√(5+√5)) : 1. Note that there is a nested √5 inside another √5 for the middle side. It will take further work to see if we can express this quantity in terms of the golden ratio.
(√5-1)/4 : (√2/4)(√(5+√5)) : 1
Side length ratios for a 18:72:90 degree triangle.
Here is another way to derive the exact form side lengths using only geometrical arguments.
Golden Gnomon Triangle Analysis
The golden gnomon is the obtuse, wider looking triangle with golden proportions. This time instead of the isosceles sides being φ times the short side, the one long side is φ times the two shorter sides. This gives angles of 36:36:108, which also happen to be multiples of 18 degrees and thus appears on the golden unit circle.
Half of this triangle appears on the two middle triangles of the four triangles in each quadrant on the golden unit circle. The right triangle half of the golden gnomon triangle has angles 36:54:90. The exact side lengths of the half golden gnomon triangle are (√2/4)(√(5-√5)) : (√5+1)/4 : 1.
For more math, this is the best website in analyzing patterns on non-standard triangles.
(√2/4)(√(5-√5)) : (√5+1)/4 : 1
Side length ratios on a 36:54:90 degree triangle.
Notice that the side lengths of the golden gnomon triangle are symmetric looking with respect to the side lengths of the golden triangle. The +1 is merely swapped for a -1 and the +√5 is merely swapped for a -√5.
Cultural Appearances of Golden Gnomon Triangle
Golden gnomon triangle geometry can be seen at least subtly in the Sri Yantra, sundials and old astronomical instruments. Through studying the cultural appearances, we can begin to unravel the significance of this geometry.
Sri Yantra’s Golden Gnomon Triangle
I found very few rigorous treatments of the mathematical properties of the Sri Yantra. Someone somewhere has done it, but I will need to re-derive it to reveal the golden gnomon triangles.
The purpose of the Sri Yantra was to gaze upon in meditation to receive “codes”. The idea is that your brain starts to gather information from the proportions. Then you can automatically even construct possible higher dimensional figures that would leave this shadow. We do have some evidence of decagonal symmetry in the Sri Yantra, which relates it possibly to the decagonal symmetry in DNA. this would make sense since this Yantra was used in part for healing, as well as worship.
Here you can find the full construction of the Sri Yantra. In a future post I will derive the angles in the figure. Comment on this post that you want to see that to expedite the publication of those angles.
Jantar Mantar Observatory of Astronomical Buildings
The Jantar Mantar observatory in India has so many gnomons. Jantar Mantar means “instruments for measuring the harmony of the heavens.” And the large buildings all were precisely designed to probe the skies. It was completed between 1724 and 1735.
I absolutely had to include these unedited photos of the architecture. As far as I know, there is nothing like it in the world, and it can still be visited, although all the original functions are not in use.
Golden Gnomon Triangles should make a Comeback
Maybe once in a while a gnomon triangle comes up in art or architecture, but it could really use a comeback.
The significance of the golden gnomon triangle will resurface as we study the platonic solids and other cultural symbols. For now the most striking information I found was how the exact values of the golden triangle and golden gnomon triangle side proportions have symmetry, and also contain √5’s. Perhaps the golden unit circle can help show some of the angles in the Sri Yantra. If you like the discussion of symmetry in cultural symbols, check out this post on Rotational Symmetry in Cultural Symbols.
