Case Analysis in Euclidean Geometry: An Overview

This paper gives a brief overview of FG, a formal system for doing Euclidean geometry whose basic syntactic elements are geometric diagrams, and which has been implimentented as the computer system CDEG. The computational complexity of determining whether or not a given diagram is satisfiable is also briefly discussed. 1 A Diagrammatic Formal System To begin with, consider Euclid’s first proposition, which says that an equilateral triangle can be constructed on any given base. While Euclid wrote his proof in Greek with Fig. 1. Euclid’s first proposition. a single diagram, the proof that he gave is essentially diagrammatic, and is shown in Figure 1. Diagrammatic proofs like this are common in informal treatments of geometry, and the diagrams in Fig. 1 follow standard conventions: points, lines, and circles in a Euclidean plane are represented by drawings of dots and different kinds of line segments, which do not have to really be straight, and line segments and angles can be marked with different numbers of slash marks to indicate that they are congruent to one another. In this case, the dotted segments in these diagrams are supposed to represent circles, while the solid segments represent pieces of straight lines. It has often been asserted that proofs like this, which make crucial use of diagrams, are inherently informal. The comments made by Henry Forder in The Foundations of Euclidean Geometry are typical: “Theoretically, figures are unnecessary; actually they are needed as a prop to human infirmity. Their sole function is to help the reader to follow the reasoning; in the reasoning itself they must play no part.” [2, p.42] Traditional formal proof systems are sentential—that is, they are made up of a sequence of sentences. Usually, however, these formal sentential proofs are very different from the informal diagrammatic proofs. A natural question, then, is whether or not diagrammatic proofs like the one in Fig. 1 can be formalized in a way that preserves their inherently diagrammatic nature. The answer to this question is that they can. In fact, the derivation contained in Fig. 1 is itself a formal derivation in a formal system called FG, which has also been implemented in the computer system CDEG. These systems are based a precisely defined syntax and semantics of Euclidean diagrams. We can define a diagram to be a particular type of geometric object satisfying certain conditions; this is the syntax of our system. We can give a formal definition of which arrangements of lines, points, and circles in the plane are represented by a given diagram; this is the semantics. Finally, we can give precise rules for manipulating the diagrams—rules of construction and inference. All of these rules can be made entirely precise and implemented on a computer. The details are too long and technical to include here, but the interested reader can find them in [3]. A crucial idea, however, is that all of the meaningful information given by a diagram is contained in its topology, in the general arrangement of its points and lines in the plane. Another way of saying this is that if one diagram can be transformed into another by stretching, then the two diagrams are essentially the same. This is typical of diagrammatic reasoning in general.