A Geometry for the Many-Angled Ones

You may know (or not) that I've become quite fond of the science fiction of Charlie Stross, a British writer living in Scotland, especially the "Laundry Files" series.  The Laundry is a highly-classified British Intelligence and Counter-Espionage agency whose primary brief is the protection of the United Kingdom against the depredations of evil extra-dimensional beings who wish to invade, possess, destroy, and otherwise prey on humans and their world.

The basic conceit is that H. P. Lovecraft's writings are true: there are beings of great power lurking just around the corner in a higher-dimensional multiverse, beings who can be called forth using forms of geometry and computer software.  In the world of the Laundry, Alan Turing didn't just invent the mathematical underpinnings of the theory of computation; the secret part of his work that the rest of the world didn't get to see shows how mathematics and computer programs can be used to do what amounts to "magic".  Stross' hero, Bob Howard, is a middle-echelon IT sysadmin and secret agent, recruited into the Laundry just before his university computer graphics project could invite in beings that would have leveled the city of Wolverhampton¹.

Now it happens that I've been researching modern geometry in the last year or two.  One of the subjects I've been studying intensively, for use in a software project I hope to blog about in the near future, is a field called "Geometric Algebra"².  And Geometric Algebra just might be the geometry that Lovecraft's "Many-Angled Ones" use in navigating their sinister travels through the universes..  I'll explain more after the cut.  Don't be too bothered by the mathematical terms; I'll try to summarize the meaning of it all so you won't have a mathematician to get it.  And I'll try not to get eaten by the Great Old Ones before I finish.