Clifford Algebra

1.1 Introduction

I am interested to see how Clifford Algebra can be used to solve problems within chemical engineering and physical chemistry.  I have presented some papers at a conference on Clifford Algebra in 2001, which are listed below.  As those papers were written for people who are already familiar with Clifford Algebra, they do not contain any of the basic material.  That is why there is information here about the basics of Clifford Algebra.  I have now been able to extend my previous work more generally and will update this page to report on that. I am particularly interested in symbolic computation using Clifford Algebra.

1.2 What is Clifford Algebra?

There is now on a separate page to introduce Clifford algebra and provide a link to the introductory material which is available elsewhere.  It is brief and selective in its coverage.

1.3 My Papers at AGACSE2001

I have presented three papers at the AGACSE2001 conference in July 2001. These papers are here converted to web pages.

Clifford Numbers and their inverses calculated using the matrix representation.

Symbolic processing of Clifford Numbers in C++.

The application of Clifford algebra to calculations of multicomponent chemical composition.

1.4 More Recent Work

I have now been able to extend the work described in these papers, by using a C++ template library for Clifford Algebra called GluCat.  GluCat is described in detail by its author, Paul Leopardi.  His templates are designed for use with numerical types, and I have adapted them to work with symbolic types using SymbolicC++.  I find this very exciting as it removes the main restriction in the previous work that a new class had to be written for each distinct algebra.  Using the templates in GluCat, this is not necessary, as objects of any signature can be freely defined, including those with basis objects which square to minus one.  The combination program is called Glucsym.  A paper on all this is now in draft and I will be putting it here.

1.4.1 Inverse of a Clifford Number using the minimal polynomial

Using GluCat I have been able to make more progress on the question of the general inverse of a Clifford number, developing the ideas of my paper Clifford Numbers and their inverses calculated using the matrix representation to use instead the minimal polynomial of the Clifford number calculated directly from the number and its powers.  This can be extended to obtain the minimal polynomial directly from the Clifford number.

1.4.2 Spectral Decomposition

The roots of the minimal polynomial can be used to obtain a spectral decomposition of any Clifford number

1.4.3 Vector Exponential

The spectral decomposition of a grade 1 Clifford number, or vector, is particularly simple.  A note on this will open on a separate page.

1.4.4 Interface to Ruby

I have interfaced Glucsym to Ruby using an automatically generated interface.

1.5 Historical Mathematics Book

I have a copy from the library of Aston University of a book called

Universal Mechanics and Hamiltons Quaternions - A cavalcade

by Otto F. Fischer  consulting engineer Lidingo, Stockholm

Published by the Axion Institute, Stockholm 1951

The original was typed with all the equations hand written.  The author starts the preface, ”This is a book written by a civil engineer on universal mechanics with an attempt to introduce a certain order in its mathematical structure by means of the calculus of Hamiltons Quaternions.”  

Otto also wrote a second book called ”Five Mathematical Structural Models in Natural Philosophy with Technical Physical Quaternions” (Axion Institute, Stockholm, 1957).

These books are mentioned in ”A History of Vector Analysis” by Michael J. Crowe (University of Notre Dame Press, 1967) but I have found no more recent references to Otto Fischer’s work.

1.6 Software Section

Links to more information about the software I am using.

SymbolicC++ by Prof W. Steeb and Y. Hardy

GluCat by Paul Leopardi

There is a review of software for Clifford Algebra by Eckhard Hitzer to be found at Geometric Calculus International.

1.7 Where to find out more about Clifford Algebra.

The best starting point I know for introductory material on Clifford Algebra is the web site of the Geometric Algebra Research Group at Cambridge University.  My thanks to them for help and encouragement.  I welcome feedback and discussion of Clifford Algebra.

John Fletcher.  May 2001 updated October 2003.  J.P.Fletcher@aston.ac.uk