This website is a tribute to the great scholars of science, knowledge, and wisdom of the Islamic civilization that spanned many nations during the middle ages. The scholars who translated, assimilated, and developed the scientific human knowledge of other civilizations before them and handed it to the ones who came after ^{1}. The scholars who founded and applied the modern scientific research method as we know it. To my grandfathers of wisdom, may you be remembered in this life, and the next.

## About the Geometric Algebra 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

*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.*

**freely share**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

**Dr. 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:

- To use GA as a tool for exploring diverse aspects of scientific knowledge in engineering and computer science.
- 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
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**Abstraction****John W. Arthur**‘s book^{2}“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
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**Geometric Reasoning****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
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**Logical Symbolic Reasoning****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
. One example of valuable work in this level is the use of the term**Computer Programming**as given by**Geometric Algebra Computing (GAC)****Dr. Dietmar Hildenbrand**^{3}(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
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**Communication of Knowledge:****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.

### On this site you will find:

- A blog containing my posts on geometric algebra and other interesting topics.
- General information related to geometric algebra such as online resources, software, and GA community under the
main menu.**Geometric Algebra** - Detailed information, samples, and user guides of GMac; the system I designed and implemented for the purpose of creating practical geometric computing applications based on geometric algebra. You can find all that under the
menu.**GMac Resources** - Several pages where you can download material and software I created including
of my software**free****and trial versions**and**TextComposerLib**among others. You can also buy licenses for my software components from there. You can find these pages under the**GMac**menu.**Products** - Under the
menu, you can find several pages related to topics and people who caught my attention and fascination over the years, I hope you find them interesting as I have.**Resources** - You can also find some image gallery pages under the
menu.**Galleries** - Finally, you are welcome to contact me using the Contact page at any time. You can also write comments on the blog posts and other pages in this site.

### Featured Blog Posts:

**The Geometric Algebra Interviews:**A series of interviews with key figures in the GA community where they share their valuable experiences and insights on GA and its applications.**The Abstract**: In this post, I try to describe how I think about abstraction as an engineer, researcher, software developer, and teacher. This post is also the abstract or entry point for the ideas presented in the GA Explorer website.**A Functional History of Numbers (Part 1, Part 2, and Part 3)**: A series of 3 posts describing some of the functional roles played by one of the most important abstractions in the human history: Number Systems. In these 3 posts, you can find many interesting links between civilizations, stories about scientists and their unfulfilled dreams, and alternative interpretations and directions related to how we understand and use numbers.**Computing for Geometry**: In this post where I try to define the term**Geometric Computing**I often use on this website and used by many in a loose sense relating computation to applications requiring some form of geometric modeling. I also try to describe the steps involved in creating concrete software implementations from our abstract geometric ideas and some of the serious problems we face to achieve this goal.**Geometric Algebra: Ascension of the Mind:**In this post, I try to explain how Geometric Algebra can express, unify, and generalize many geometric abstractions we use as engineers and computer scientists.**Elements of Geometric Algebra**: In this post, I describe 10 mathematical elements of Geometric Algebra as a road map for anyone wanting to study and appreciate its elegant mathematical structure.**GMac: The Next Generation (Part 1, and Part 2):**A series of two posts where I talk about my journey developing GMac, the fascinating discoveries I made, and the difficulties I faced along the way. I also explain the design decisions I made for GMac and how I came to make them, in addition to the developments I hope to make in the future.

**Ahmad Hosny Eid**

## GA Explorer Updates

To get regular emails describing new posts, page updates, and software information you can subscribe to the GA Explorer Updates with your email from here:

You can also * securely register as a user on this site* to post comments and select to subscribe to the GA Explorer Updates.

The GA Explorer website uses SSL cryptographic protocols to protect information entered by its users.

### GA Explorer News:

**26 April 2017:**- Posted Edifying The Young, a new part in the GA Interviews series with Dr. Alan Macdonald.

**1 April 2017:**- Posted Projective Insights, a new part in the GA Interviews series with Dr. Charles Gunn.

**28 February 2017:**- Updated the GA Online Resources page and added a link to Pablo Bleyer’s GA resources list.
- Added the GMac Logos Gallery.
- Added the TextComposerLib Logos Gallery.
- Updated the main Menu Structure and added the
**Explorations**main menu item.

**20 February 2017:**- Posted Visualizing Scientific Insights, a new part in the GA Interviews series with Dr. Werner Benger.

**10 February 2017:**- Posted GMac: The Next Generation part 2.

**9 February 2017:**- Posted GMac: The Next Generation part 1.

**28 January 2017:**- Posted Elements of Geometric Algebra.

**17 January 2017:**- Posted Founders of Geometric Calculus with Dr. Garret Sobczyk and Geometric Algebra in Computer Science, with Dr. Leo Dorst, Dr. Dietmar Hildenbrand, and Dr. Eckhard Hitzer; the next two parts of the GA Interviews series.
- Updated the GA Explorer website to
**use secure SSL (https://gacomputing.info)**. Now you can enter your information safely and register in the GA Explorer website securely.

**16 January 2017:**- Posted Knowing About the World via GIS, the first part of the GA Interviews series with Dr. Yu Zhaoyuan.

**13 January 2017:**- Posted Geometric Algebra: Ascension of the Mind.
- Updated the Geometric Algebra Software page: Added information about the SpaceGroupVisualizer software.

**9 January 2017:**- Published the Geometric Algebra Community page. The GA Explorer website welcomes members of the GA community to send information about their GA-related work to update this page.

**7 January 2017:**- Updated the Geometric Algebra Software page.
- Updated the GMacDSL Affine Geometry Sample Code page.

**30 December 2016:**- Started the Geometric Algebra Explorer Updates mailing list. Now you can follow all GA Explorer updates via email by subscribing at the end of any page on the site.
- Published the GMacDSL Affine Geometry Sample Code page.

**29 December 2016:**- Published a new version (v1.0.3) of
(**GMac**,**GMac Free**, and**GMac Basic**Editions). You can download them at the GMac Product Information page.**GMac Complete** - Published a new version (v1.0.2) of
. You can download from the TextComposerLib Product Information page.**TextComposerLib**

- Published a new version (v1.0.3) of
**13 December 2016:**- Published the GMac Product Information page.

**4 December 2016**:- Published the TextComposerLib Product Information page.

**13 July 2016**:- Published the GMac Math Guide

**8 July 2016**:- Published the GMac System Guide

**15 March 2016**:- Published the GMac Scripting Guide

**10 March 2016**:- Published the Geometric Algebra Online Resources page.

**1 March 2016**:- Opened Comments on all posts and pages; you must register and login first.

**22 January 2016**:- Published the GMacAPI Guide

**27 November 2015**:- Published the GMacAST Guide

**26 October 2015**:- Published the TextComposerLib Guide

**28 September 2015**:- Published the GMacDSL Guide

**18 September 2015:****11 September 2015**:**14 August 2015**:**7 August 2015**:**24 July 2015**:**3 July 2015**:- Posted my first blog post: The Abstract

**27 June 2015**:- Published the Geometric Algebra Software Page.

**17 June 2015**:- Published the About GMac Page.

### Endnotes

- See for example Part II of Robert Briffault’s 1919 book “The Making of Humanity” publicly available for download here. Also for a good reference on some of the contributions of Islamic scholars in the area of
**Algebra**and**Geometry**see J. L. Berggren’s book “Episodes in the Mathematics of Medieval Islam” available here and “Geometry by Its History” by Alexander Ostermann and Gerhard Wanner available here. ↩ - The names of authors, researchers, books, and publications in this website are by no means intended to be an accurate representation of the work done by the GA community or scientific community in general. They are merly a reflection of my own very limited readings and knowledge. I offer my respect to any auther who participated in the development of human knowledge in the past, present, or future. For more information on the people offering significant contributions to the development of Applied Geometric Algebra, the reader can refer to my GA Community page. ↩
- I would like to thank Dr. Dietmar Hildenbrand for his valuable comments and suggestions regarding the contents of this site. ↩