Why I love triangles.
by FiL aged 18.
Triangels are 2-simplex, and hence the simplest but also most beautiful of polygons. What they lack in verticies they make up for in diversity and versatility. Triangles can be classified according to the lengths of their sides into one of three cateogries; equilateral (and equiangular), isosceles (two sides of equal length) and scalene (all sides of different length) but also can be grouped by the size of their largest interior angle; right angle triangles having one 90o internal angle, obtuse triangles with one internal angle greater than 90o and acute angles with internal angles all smaller than 90o.
Triangles have been used to define and shape mathematics since the time of Euclid and have since given rise to a whole branch of mathematics that relate almost entirely to this most beautiful of shapes; triganometry. The schoolboy's bane, it remains one of the most fundimental and practical branches of mathematics, having uses that stretch from mechanics, to engineering, vectors, computing and of course mathematics.
And if you thought this was not exciting enough this is but a mere scratch on the surface. There are triangular numbers; a number that can be arranged in the form of an equilateral triangle ½n(n + 1). The sum of two consecutive triangular umbers is a square number:
{½n(n + 1)} + {½(n − 1)n}
=(½n2 + ½n) + (½n2 − ½n)
=n2
Even 'the number of the beast' is a triangular number.
There also exists the non-planar triangles, that is triangles in noneuclidean geometry. That is triangles in curved space; ie on a spherical surface. While regular planar triangles all have angles which total 180 degrees, in noneuclidean geometry this is not the case. This fact is made use of in this well known puzzle:
[INDENT]
An explorer travels one mile south, one mile east and one mile northand ends up at exactly the same spot that she started from. She sees a bear. What colour is the bear? [/INDENT]
similarly:
[INDENT]If you walk North a mile, east a mile, south a mile, west a mile, what is the maximum distance you can end up from your original postition?[/INDENT]