About The GA Explorer

I think the universe is pure geometry – basically, a beautiful shape twisting around and dancing over space-time; Antony Garrett Lisi, author of “An Exceptionally Simple Theory of Everything“

Welcome, everyone! This website is about exploring ideas in geometry, algebra, computations, and wisdom gained by applying and teaching them. The main subject and tool of the explorations is Geometric Algebra (GA), a fascinating and powerful algebraic abstraction for expressing geometric ideas. I made this site to freely share my thoughts, as they may be, about modeling geometry, creating code, and teaching through GA mainly for applications in computer science and engineering. I hope you find the pages and posts interesting and useful.

The reason this website is mainly dedicated to Geometric Algebra is that it has the potential of being the algebraic abstraction system for geometric processing applications in this century, just as vector algebra was in the 20th century. My goal isn’t to add to the mathematics behind GA, that’s already been done quite sufficiently, but rather to illustrate, reformulate, explore, and implement solutions to problems in engineering and computer science in the language of Geometric Algebra. This step has already been taken in physics with remarkable success by Prof. David Hestenes and other physicists. It’s about time to do the same on a wide scale in engineering and computer science.

About the Title

The title of this site “Geometric Algebra Explorer” has several levels of ideas and tools associated with it. The exploration process is two-way:

1. To use GA as a tool for exploring diverse aspects of scientific knowledge in engineering and computer science.
2. To explore the potential of GA as a mathematical language using problems in engineering and computer science.

The GA explorations on this website are related to several levels of applied knowledge, stated here in their order of abstraction:

• The first level of exploration is associated with the fundamental process of Abstraction itself. This process is practically limited by the mathematical and computational tools we have to represent and manipulate our abstractions. Geometric Algebra simply removes many difficult limitations on our ability to create and use abstractions. An excellent example of this feature is provided by John W. Arthur‘s book¹ “Understanding Geometric Algebra for Electromagnetic Theory“. Classical EM theory is the most influential abstraction in modern time. Using a GA-based representation provides many advantages over traditional tools like vector algebra and complex numbers.
• The second level is associated with Geometric Reasoning as intuitively performed by human beings. The main source of knowledge in this level is rooted in the work of 19th-century geometry. One nice modern book is John Stillwell‘s “The Four Pillars of Geometry“. This book is unique in that it looks at geometry from 4 different viewpoints – Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants. Geometric Algebra can be used to directly connect to this huge but practically under-utilized body of geometric knowledge.  New applications and models are only one consequence of this connection; deeper understanding is the most important consequence of relating GA to geometric reasoning in all 4 forms described in John Stillwell‘s book.
• The third level is associated with Logical Symbolic Reasoning provided by the fascinating algebraic structure of Geometric Algebra. An excellent and essential resource on this aspect is the book “Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry” by Leo Dorst, Daniel Fontijne, and Stephen Mann. The simplicity and universality of GA’s algebraic language enable the possibility of developing computer algorithms and software systems far better than the current ones.
• The fourth level is associated with Computer Programming. One example of valuable work in this level is the use of the term Geometric Algebra Computing (GAC) as given by Dr. Dietmar Hildenbrand² (http://www.gaalop.de/dhilden/) the author of the book “Foundations of Geometric Algebra Computing” and the leader of the team that designed and developed “Geometric Algebra Algorithms Optimizer (Gaalop)” software. In his book, he defines the term “Geometric Algebra Computing” as:

The geometrically intuitive development of algorithms using geometric algebra with a focus on their efficient implementation.

• Complementary to these 4 levels is the level associated with Communication of Knowledge: learning and teaching geometric abstractions in engineering and computer science involving GA techniques, algorithms, and computations. A very good example of a successful step in this direction is Dr. Kenichi Kanatani‘s book “Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics“.

Personally, I think about Geometric Algebra as a powerful language for symbolic geometric abstraction, not just another algebraic system of mathematics. Computers can only be programmed when we fully understand the concepts and processes involved in the problem at hand. Instructing computer to deal with Geometric Algebra is fascinating because it’s a high-level language for geometry as we humans think and understand and, at the same time, a logical algebraic system suitable for being a universal base for creating computer code for geometric processing applications using current compiler technology. Most of the material on this website are dedicated to illustrating and eventually realizing the full potential of GA in various scientific computing applications.