Area and Boundary Length (Perimeter)
Area Measurement and Number

The image visually presents the idea of related squares by colored triangles. The author1 continues:

Count the triangles within the squares. The number of green triangles equals the total of those that are yellow and blue!1

The above is an animated special case. It introduces us to an observation known to artisans, and enshrined in mathematics. The fundamental there is a corner formed by two lines that meet at a special angle called right. The meaning of the term is simply that four successive right angle movements return one to the same place.

Mathematics as an abstract subject began with Greek efforts (though the underlying ideas lived, and continue) in other cultures. The book that changed things from general notions known to practical workers into a body of concepts was Euclid's. The general issue of the animation appears in Euclid's Elements, Book I, Proposition 47; see e.g., the translation, guide2.

A square has area given by multiplying the two (same) numbers associated with adjacent sides. The process also works for a figure that has four right angles but different side lengths. Such a figure, called a rectangle, can be divided into two three-sided parts by a diagonal: each one is a triangle. Since each part makes up half the rectangle, a triangle's area is half the height times base length.

Two questions are related to how we number shape properties:

1. What are the valid numbers for side lengths of such shapes?

2. For such lengths, what results for the area (kind of number)?

The first question could limit answers to ratios of integers or rational numbers, and lead to a second requirement that the area be a counting (whole, natural) number [1, pp. 190-193]. Problems of Number.

Acknowledgements and References

1Dorene Lau is responsible for the animation; it was included in a report where she wrote these words.
2David E. Joyce, Department of Mathematics and Computer Science , Clark University, 950 Main Street, Worcester, MA 01610, translated Euclid's works and created the guide to related material from other cultures.

[1] Devlin, K. The Millennium Problems, NY: Basic Books, 2002.
[2] Lönnemo, H. Andres, The Trinary Tree(s) underlying Primitive Pythagorean Triples

4/06/03 Version
©2003 Allen Klinger