About our Model

This model is based on [2]. This is the most precise and concise model I could find. For example, there is no need for "line segments" or "circle arcs" which are only there for aesthetic reasons.

Comparison with Euclid's model

The model of computation of Euclid can be found in the first three postulates of his Elements:

Postulate 1
To draw a straight line from any point to any point.

Postulate 2
To produce a finite straight line continuously in a straight line.

Postulate 3
To describe a circle with any center and radius.


  1. Postulate 2 is usually taken to mean "To extend a finite straight line indefinitely". Our model simplifies the matter by avoiding the notion of a "line segment" or of "producing a line" altogether. Each line is always constructed infinite, as it would by an application of Postulate 1 and two applications of Postulate 2 (to produce the straight line indefinitely in both directions).

  2. Implicit in the word "any" is the ability to take arbitrary points on a line segment or elsewhere. Often, you have to pick your arbitrary point carefully if you want the rest of the construction to work (Go see Book I Proposition 9 and drag the point C close to A to see a construction break down.) Also, algebraically, arbitrary points greatly complicate the analysis of the set of points you can construct, and hence they will be forbidden.

Don't expect too much from the compass!

A common mistake is to think that the compass can be used to transfer distances. Recall that Euclid's compass is an idealized tool, and one must not infer too much from the physical object.

Euclid's compass is often called a "collapsing compass" (no screw!), and it can be conceptualized as follows:

collapsing compass animation

Hopefully, as shown by Book I Proposition 2, it is possible to transfer a distance (by using more than one step).

For a good exposition of the relationship between the collapsing compass and Euclid's second proposition, see this page (it has a link to a postscript version of [5]).

<-- Back to The Complexity of Geometric Constructions