# Beyond 3-D, Ch. 1

## David Akers

My reactions to the first chapter of B3D:

Even though it was a very basic introduction to the concept of
"dimension," I found the first chapter to be thought-provoking
nevertheless. The discussion of the "practicality" or "use" of higher
dimensions was particularly interesting to me. It reminded me of a
math project I did my senior year of high school on "tri-linear
coordinates." The basic idea was like this:

Normally when we want to use coordinate geometry to describe the
vertexes of a triangle, we use standard 2-dimensional cartesian
coordinates. For instance, we might say, "triangle A has vertexes at
(3,0), (3, 4), and (2, 1)." We could then proceed to derive equations
which describe all three of its sides in simple slope-intercept form, and
then use these equations to find such points as the circumcenter, the
incenter, the orthocenter, etc. What's wrong with this approach? Well, to
put it simply, the standard cartesian-based formulas for such points can
sometimes be hairy beasts.

To simplify our formulas, we simply ADD another "dimension" to our
problem. Instead of describing each point in terms of an x-coordinate and
a y-coordinate, we can describe it in terms of its distances to the three
sides, as a ratio of these distances. (The incenter, for example, is
defined as (1, 1, 1), indicating that it is equidistant from all three
sides.) So, you may be wondering, what is so great about this idea? All we
seem to have done is found another way to describe a point in a triangle,
right? Well, not quite. In this new "tri-linear" coordinate system, the
formulas for points like the circumcenter, the centroid, and orthocenter
are written in terms of the sides and angles of the triangle. As it turns
out, the resulting formulas are much much simpler in the this form of
geometry. And, interestingly, the points we speak of can be written in
terms of each other. Strangely enough, by adding this new dimension we
have simplified the geometry of the problem. (I don't know the formulas
off the top of my head, and they are a little tricky to derive, but if you
want I will look them up and add them to this document on the web-page
when I get the chance to.)