The Clifford-Grassmann Geometric Algebra
geometric algebra is the tool that allows us to study and solve geometric
problems through a simpler and more direct way than a purely geometric
reasoning, that is, by means of the algebra of geometric quantities instead of
synthetic geometry. In fact, the geometric algebra is the Clifford
algebra generated by Grassmann's exterior product in a vector space, although
for me, the geometric algebra is also the art of stating and solving geometric
equations, which correspond to geometric problems, by isolating the unknown
geometric quantity using the algebraic rules of vector operations (such as
the associative, distributive and permutative properties).
Initially imagined and proposed by Leibniz (characteristica geometrica), the geometric algebra was realized by Hermann Grassmann and William Kingdon Clifford, who established its bases during the XIX century. However, the deaths of Grassmann (1877), and Clifford being 33 years old (1879) truncated prematurely its development. Instead of that, the vector analysis created by Josiah Willard Gibbs and Oliver Heaviside played the role of the geometric language for the three-dimensional space along the XXth century. Nevertheless, the vector analysis is unable to describe the Relativity and Quantum theory so that an increasing attention to the geometric algebra has been paid. Because of its powerful methods and applications, the CGGA will become the geometric language of the XXIst century.
Peano summarises what the geometric algebra is in the following words:
"And certainly, these diverse methods of geometric calculus do not contradict themselves. They are whether the diverse parts of the same science or the diverse ways through which the same matter appears to many authors, each one of them studies it independently of the others.
Since the geometric calculus, as any other method, is not a system of conventions but a system of truth. So the method of the indivisible quantities (Cavalieri), of the infinitesimals (Leibniz), of the fluxions (Newton) are the same science, more or less perfect, and exposed under diverse form."
According to Peano
and D. Hestenes, we should talk about the geometric algebra as a unique
subject and field of research, although its historical development had many
partial steps and obstacles with contributions from many authors (mainly
Grassmann, Clifford, Hamilton, Peano and others). On the other hand, the adjective "geometric"
is obvious, but we thought that the geometric algebra
should earn it: the most books about geometry do not use it at all, a Kafkian situation. In the frame of the challenge to make geometry using the geometric
algebra I wrote the Tractat de geometria
plana mitjançant l'àlgebra geomètrica and an enlarged translation, the
Treatise of plane geometry through
geometric algebra. I believe -I think that the readers will agree with
me- that in this book the Clifford-Grassmann algebra has displayed its power and
it has won the adjective geometric.
What is the Clifford-Grassmann Geometric Algebra (CGGA)?
How did I arrive to CGGA?
Who was who in the CGGA?
Examples of application of CGGA to the plane geometry
Links about CGGA
Meetings on CGGA
My publications about the CGGA:
R. González, Tractat de geometria plana mitjançant l'àlgebra geomètrica. (Cerdanyola del Vallès, 1996).
R. González, J. Homs, J. Solsona. "Estimació de l'error comès en determinar el lloc des d'on s'ha pres una fotografia". Butlletí de la Societat Catalana de Matemàtiques 12 (1997) 51-71
R. González, J. Homs, J. Solsona. "Estimate of the error committed in determining the place from where a photograph has been taken". (English translation of the former paper).
R. González, Treatise of plane geometry through geometric algebra, electronic edition (Cerdanyola del Vallès, June 2000-June 2001).
R. González, J. M. Parra, "Applications of the Clifford-Grassmann Algebra to the plane geometry". Poster presented at the 3rd European Congress of Mathematics (Barcelona, July 10th to 14th, 2000).
R. González, "Why and how the geometric algebra should be taught at high school. Experiences and proposals". Talk given at the meeting Innovative Teaching of Mathematics with Geometric Algebra (ITM 2003) (Kyoto, November 20th to 22nd, 2003)
R. González, Treatise of plane geometry through geometric algebra, first printed edition (November 2007) (TIMSAC No.1).
R. González, "Applications of Clifford-Grassmann Algebra to the Plane Geometry". Poster presented at the 5th European Congress of Mathematics (Amsterdam, July 14th to 18th, 2008).
R. González, "Applications of Geometric Algebra and the Geometric Product to Solve Geometric Problems". Talk given at the AGACSE 2010 conference (Amsterdam, June 14th to 16th, 2010).
R. González, El álgebra geométrica del espacio y tiempo (October 2011). (TIMSAC No. 3).
R. González, "New Foundations for Geometric Algebra". Talk given at the IKM 2012 conference (Weimar, July 6th 2012) (Clifford Analysis, Clifford Algebras and their Applications, vol 2, n. 3 (2013) pp193-211)
R. González, "How to Explain Affine Point Geometry". Talk given at the ICCA 10 conference (Tartu, August 4th to 9th 2014).
R. González, "Applications of geometric algebra to plane and space geometries" (lecture), "The exterior calculus" (lecture), "The affine and projective geometries from Grassmann's point of view" (talk), "On matrix representations of geometric (Clifford) algebras" (talk). Lectures and talks given at the Alterman Conference on Geometric Algebra and Summer School on Kähler Calculus (Braşov, August 1st-9th 2016), Early Proceedings of the Alterman Conference on Geometric Algebra and Summer School on Kähler Calculus (2016) pp. 15-42, pp.43-57, pp. 198-226, pp.312-338 respectively.
R. González, "On matrix representations of Clifford algebras" J. of Geometry and Symmetry in Physics 43 (2017) 1-36.
Last updated: April 28th, 2018