Math and Mystery of the Triangle

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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.

The Basics

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.

Arch-itecture

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.

2 comments

Jymi X/0
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Posted on Jul 25, 2010
Ileen Zovluck
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Posted on Jul 25, 2010