An overview of mathematical and "sacred geometry" properties that make the triangle so significant in science, art and culture.
It's no accident that so much of our lore and scripture involves triangles, trinities and triads. Geometry shows us that Deity is manifest in Trinity. When the Monad becomes the Duality, the triangle emerges naturally through the cardinal points of the vesica pisces. The three points of a triangle take a one-dimensional line to a two-dimensional plane, and the addition of one more point outside the plane produces the first three-dimensional form – a tetrahedron, which is made of four triangles.
A triangle is a closed shape with three sides that come together to form three angles. In Euclidean (flat) space, the angles may be of any measurement, as long as they add up to exactly 180 degrees.
The triangle is so unbelievably useful that an entire section of mathematics has been devoted to its study: Trigonometry.
The famous Pythagorean Theorem states that for any right triangle, the square of the length of the longest side (the hypotenuse) is the sum of the squares of the lengths of the other two sides.
There are three special kinds of triangles:
– The Isosceles triangle has at least two sides that are of equal length.
– The Equilateral triangle's sides are all of equal length. (The Equilateral triangle is also an Isosceles triangle, but the Isosceles is not necessarily an Equilateral.)
– The Scalene triangle's three sides are each of different lengths.
The trigonometric functions for any angle of a triangle are calculated as follows:
-- The Sine is the length of the opposite side divided by the length of the hypotenuse.
-- The Cosine is the length of the adjacent side divided by the length of the hypotenuse.
-- The Tangent is the length of the opposite side divided by the adjacent side.
Using the Pythagorean theorem, the trigonometric functions, and the geometric principle of similarity (which states that two objects of the exact same shape are going to have the same properties, adjusted to scale if the objects are of different sizes), we can easily perform some astonishing real-world math with very little equipment. We can figure out the height of a tree or a building, and even the distances to neighboring stars.
Our calculations aren't limited to distance. Trigonometric rules govern the workings of vector quantities, with which we can calculate speed, force and direction. For example: If you're in a rowboat, and you aim directly across a river paddling with a force P, but the force of the current, C, acts on the boat at a right angle to P, where will you end up on the opposite side of the river? How much more force will you have to exert – and in what direction – to land closer to your intended point?
The triangles show us the forces at work. And using trigonometry, we can simply fill in the blanks for any unknown quantities.
The triangle is the strongest, most stable geometric shape, and appears frequently in natural and man-made constructions and the main and supplementary support elements. A three-legged stool or table is the only kind that will not wobble, as all three legs are always touching the floor, even if they're not all the same length. When insects walk, they keep three legs on the ground at all times.
The arch is a curved or semi-curved shape that uses the special weight distribution properties of the triangle. The force of the weight is directed downward, from the topmost angle (or "keystone") and distributed evenly along the two sides, halving the pressure on the supporting structures. Homes with steep triangular roofs are found in areas that get a lot of snow, since they can support a lot more weight than any other construction.
The beautiful St. John's Bridge in Cathedral Park, Portland, Oregon. / Pixabay
The graceful supports are nothing but arches!
The most famous triangular structures are, of course, the Great Pyramids of Egypt and South America. The strongest part of a pyramid is the wide base. Each successive row has less weight to support above it. Some of these monumental tetrahedrons have been standing for tens of thousands of years.
Nature designs with triangles, too. Our pelvis is the keystone that distributes our weight evenly to our legs. When we brace ourselves for impact or action, we stand with our legs apart. This forms a triangle and improves our stability. Slice a bell pepper or a cucumber across its middle -- you'll find a triangle.
Triangular structure appears in the great monoliths and the tiniest molecules. The most elemental building blocks of life are triangular. The two hydrogen atoms in a water molecule are arranged so that they form a 120-degree angle with the oxygen atom in the center. The carbon atom forms hexagonal crystal structures -- essentially, two equilateral triangles superimposed on one another.
While many crystal structures are hexagonal, carbon is special in that it is life-on-earth's most common component. It contains 6 protons, 6 neutrons and 6 electrons. An equilateral triangle's angles each measure 60 degrees. The 6th Sephirot, Tiphareth, is associated with one's core being: the sense of Self, the Creator manifested within Creation. In Judeo-Christian mythology, G_d created humans on the 6th day, and the "human" number of "the Beast" is 666. Six is twice three -- the two triangles. The Monad, doubled, creates the Triangle. Creation imitates its Creator when the triangle, doubled, produces life as we know it.
The Triangle of Art
Our sense of the stability and purity of the triangle seems deeply instinctual. Its shape appears over and over again in our secular and sacred art. It feels natural to us to group forces and concepts into threes, and it also feels natural to visually depict those groupings as a triangle.
The Golden Rectangle, an aeons-old favorite of mathematicians and aesthetics, has a triangular counterpart, aptly named the Golden Triangle. It appears everywhere, as one of the points of a perfectly drawn pentacle. Two of its angles are 72 degrees, the other is 36 degrees. Bisect the 72 degree angle, and you'll get another, smaller Golden Triangle. And, like the Golden Rectangle, a Golden Spiral can be constructed over the infinite sequence.
The Kabalistic Tree of Life diagram shows us the Sephirot in three groups of three. The Supernals sit aloof above the Daath bridge, their forces forming a triangle with its point upward at Kether. Beneath the Daath, the two other triads each form downward-pointing triangles, with Tiphareth and Yesod, respectively, as their lower-most points.
The triangle figures prominently in ancient and classical art and design. We are reminded that the triangle is a symbol of the highest divinity. In classical art, the saints and other main characters were depicted with halos of varying shapes (circles, squares, hexagons). God, and only God, was given a triangular halo. A similar design was the Egyptian “Eye of Horus,” which shows the traditionally stylized Egyptian eye inside a triangle. This design eventually evolved into the “All-Seeing Eye” or “Eye of Providence” which implies the eye of God that watches over humanity. The most famous example of this can be found on the back of a United States One Dollar bill – right above the Egyptian pyramid. (Ancient Masonic conspiracy theories, anyone?)
Modern world-builders also use triangles to construct their technological wonders. The triangle mesh is a common feature of 3d computer graphic rendering. Triangles allow a computer to mathematically define the topography of an object and produce a visual rendering with the least amount of work. Other types of polygons can be used, but the computer must plot every vertex to determine the outline of the object. When it comes to points and corners and many types of rough surfaces, the fact that a great number of triangles of different sizes can share one common vertex significantly cuts down on the amount of computing time and power required.
The triangle, at once the most simple and most mysterious of forms, is ever-present in the history of humanity and the cosmos. Perhaps studying it will lead us to new levels of understanding and ability in the future.
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Keedy, Mervin L. & Bittinger, Marvin L. Fundamental Algebra and Trigonometry. Reading: Addison-Wesley. 1978.
"Nimbus, in Art." The Columbia Encyclopedia. 2004. Columbia University Press.
Accessed via web at Questia.com, 3 April 2010.
Schneider, Michael S. A Beginner's Guide to Constructing the Universe. New York: Harper Perennial. 1995.
Triangle Mesh. Wikipedia, 17 December, 2009. Web. Accessed 3 April, 2010.
Graphic by Jymi