Authors: Kelly McKennon
Comments: 114 pages and 24 figures.
Descriptions of 1-dimensional projective space in terms of the cross ratio, in one-dimensional geometry as a projective line, in two-dimensional geometry as a circle, and in three-dimensional geometry as a regulus.
A characterization of projective 3-space is given in terms of polarity.
This paper differs from the original version by addition of a section showing that the circle is distinguished from other meridians by its compactness and the existence of exponential functions.
Authors: Klaus Lange
Comments: 6 Pages. 8 figures
It will be shown how the well known eleven nets for three dimensional cubes,
separated in 10 + 1 forms, are hiding a special dual 3-6-1-structure. Implications for space -
time models in theoretical physics will be questioned.
Most matter in nature and technology is composed of crystals and crystal grains. A full
understanding of the inherent symmetry is vital. A new interactive software tool is demonstrated, that
visualizes 3D space group symmetries. The software computes with Clifford (geometric) algebra. The space
group visualizer (SGV) is a script for the open source visual CLUCalc, which fully supports geometric
algebra computation. In our presentation we will first give some insights into the geometric algebra
description of space groups. The symmetry generation data are stored in an XML file, which is read by
a special CLUScript in order to generate the visualization. Then we will use the Space Group Visualizer
to demonstrate space group selection and give a short interactive computer graphics presentation on how
reflections combine to generate all 230 three-dimensional space groups.
Authors: Eckhard Hitzer
Comments: 22 Pages. 16 figures, 6 tables. In K. Tachibana (ed.) Tutorial on Reflections in Geometric Algebra, Lecture notes of the international Workshop for “Computational Science with Geometric Algebra” (FCSGA2007), Nagoya Univ., Japan, 14-21 Feb. 2007, pp. 34-44 (2007).
This tutorial focuses on describing the implementation and use of reflections in the geometric
algebras of three-dimensional (3D) Euclidean space and in the five-dimensional (5D) conformal model
of Euclidean space. In the latter reflections at parallel planes serve to implement translations as well.
Combinations of reflections allow to implement all isometric transformations. As a concrete example
we treat the symmetries of (2D and 3D) space lattice crystal cells. All 32 point groups of three
dimensional crystal cells (10 point groups in 2D) are exclusively described by vectors (two for each
cell in 2D, three for one particular cell in 3D) taken from the physical cell. Geometric multiplication of
these vectors completely generates all symmetries, including reflections, rotations, inversions, rotary reflections
and rotary-inversions. The inclusion of translations with the help of the 5D conformal
model of 3D Euclidean space allows the full formulation of the 230 crystallographic space groups in
geometric algebra. The sets of vectors necessary are illustrated in drawings and all symmetry group
elements are listed explicitly as geometric vector products. Finally a new free interactive software tool
is introduced, that visualizes all symmetry transformations in the way described in the main
geometrical part of this tutorial.
Authors: Eckhard Hitzer
Comments: 16 Pages. 8 figures, 1 table. Proc. of the Symposium Innovative Teaching of Mathematics with Geometric Algebra 2003, Nov. 20-22, 2003, RIMS, University of Kyoto, Japan, pp. 89-104 (2003).
Over time an astonishing and sometimes confusing variety of descriptions of conic sections has been developed. This article will give a brief overview over some interesting descriptions, showing formulations in the three geometric algebras of Euclidean three space, projective geometry and the conformal model of Euclidean space. Some illustrations with Cinderella created Java applets will be given. I think a combined geometric algebra & illustration approach can motivate students to explorative learning.
Authors: Eckhard Hitzer
Comments: 6 Pages. 2 figures, 1 table. Proc. of the International Symposium 2003 of Advanced Mechanical Engineering, Pukyong National Univ., Busan, Korea, 22-25 Nov. 2003, pp. 109-114 (2003).
In the so-called conformal model of Euclidean space of geometric algebra, circles receive a very elegant description by the outer product of three general points of that circle, forming what is called a tri-vector. Because circles are a special kind of conic section, the question arises, whether in general some kind of third order outer product of five points on a conic section (or certain linear combinations) may be able to describe other types of conic sections as well. The main idea pursued in this paper is to follow up a formula of Grassmann for conic sections through five points and implement it in the conformal model. Grassmann obviously based his formula on Pascal’s theorem. At the end we consider a simple linear combination of circle tri-vectors.
Authors: Eckhard Hitzer, Luca Redaelli
Comments: 6 Pages. 18 figures. Proceedings of Fukui University International Congress, International Symposium on Advanced Mechanical Engineering, 11-13 Sep. 2002, pp. 7-12 (2002).
Conventional illustrations of elementary relations and physical applications of geometric algebra are
helpful, but restricted in communicating full generality and time dependence. The main restrictions are one
special perspective in each graph and the static character of such illustrations. Several attempts have been
made to overcome such restrictions. But up till now very little animated and interactive, free, instant access,
online material is available.
This talk presents therefore a set of well over 60 newly developed (freely online accessible) JAVA applets.
These applets range from the elementary concepts of vector, bivector, outer product and rotations to triangle
relationships, oscillations and polarized waves. A special group of 21 applets illustrates three geometrically
different approaches to the representation of conics; and even more ways to describe ellipses. Finally
Clifford's circle chain theorem is illustrated for two to eight primary circles. The interactive geometry
software Cinderella was used for creating these applets. Some construction principles will be explained
and a number of applets will be demonstrated. The interactive features of many of the applets invite the user
to freely explore by a few mouse clicks as many different special cases and perspectives as he likes. This is
of great help in "visualizing" the geometry encoded in the concepts and formulas of Geometric Algebra.
Authors: Michael Pogorsky
Comments: 2 Pages.
The properties of trisected triangle are utilized in this proof in the way different from other known proofs.
Authors: Arun S. Muktibodh
Comments: 5 Pages.
In this paper we introduce the concept of half-groups. This is a totally new
concept and demands considerable attention. R.H.Bruck  has defined a half groupoid.
We have imposed a group structure on a half groupoid wherein we have an identity element
and each element has a unique inverse. Further, we have defined a new structure called
Smarandache half-group. We have derived some important properties of Smarandache half-
groups. Some suitable examples are also given.