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.)