Digraphs for Euclid

 

A query posted to r/Math asked if there was a diagram showing the relationships between the propositions in Book 1 of Euclid's Elements. The first reply pointed to David E Joyce's excellent web site on The Elements, and specifically to a table of relationships for Book 1.

My small contribution was to translate that table into a Graphviz dot file, and use it to generate a graph. A small version is shown below, and clicking on it will load the full scale image.

[Graphed Relationships in Book 1]

The propositions that define constructions are colored blue.

A quick study of my dot file for Book 1 shows that Propositions 1 and 47 (Pythagoras' theorem) include URLs which can be clicked on to go to their corresponding pages on Joyce's website. However, this feature only works if a SVG or Postscript file is generated.

Incidentally, the comments in the dot file contain examples of how to call Graphviz's dot.exe to generate PNG, SVG, and Postscript. Graphviz can be dowwnloaded from here.

One way to convert Postscript to PDF is with Ghostscript, and I've include a short batch file which wraps up the many arguments needed to generate a PDF file that retains the Postscript's clickable links.

The only other large table by Joyce is for Book 5. My dot file version is here, and the generated graph is shown below; clicking on it will load the full-sized image.

[Graphed Relationships in Book 5]

 

Other Resources

I'm far from being the first person to generate graphs for the propositions in the Elements:

 

Dr. Andrew Davison
E-mail: ad@fivedots.coe.psu.ac.th
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