p,q

CLICAL is a stand-alone calculator-type computer program for geometric algebras of multivectors, called Clifford algebras . CLICAL evaluates elementary functions with arguments in complex numbers, and their generalizations: quaternions octonions and multivectors in Clifford algebras. CLICAL works directly on intrinsic geometric objects: lines, planes and volumes, represented by vectors, bivectors and multivectors. Oriented volume elementes, or segments of subspaces, are represented by simple multivectors, which are homogeneous and decomposable elements in the exterior algebra . CLICAL works on Clifford algebras Clof real non-degenerate quadratic spaces

Clifford algebras are used to handle rotations and oriented subspaces. Clifford algebra is a user interface, which provides geometrical insight. However, the actual numerical computations are faster in matrix images of Clifford algebras. CLICAL computer program was developed to enable input-output in Clifford algebras (and fast internal computation in matrices).

CLICAL is intended for researchers and teachers of Clifford algebras and spinors. In research, CLICAL has been applied to verify and falsify conjectures about Clifford algebras. With the help of CLICAL, I have found counterexamples to conjectures and theorems about Clifford algebras. I have used CLICAL to solve problems presented in Usenet newsgroups, for instance about rotations of the 4D Euclidean space R4. In teaching, CLICAL has been used in mathematics and physics courses in the USA, Mexico, Finland and Spain. Take a look at a course delivered with CLICAL. There are competing projects, most notably an online geometric calculator, two symbolic computer algebra packages for MapleV5, one for Mathematica, MatLab geometric algebra tutotial, and C++ Template Classes for Geometric Algebras.

Download a zip-file (120kB) of CLICAL. In Windows Explore, click the file CLICAL (1kB), not CLICAL1 (120kB). In MS-DOS, type

C:\> clical

> help

> tutor

> get guide

In Macintosh, you need Windows and MS-DOS emulators, and possible upgrading of your Mac-OS. In UNIX, you need to support MS-DOS under X-Windows. The I/O of CLICAL is written in PASCAL. The numerical routines rely on Netlib's FORTRAN-package EISPACK.

Examples: 1. Find the distance of the point P = (2,3,1) from the line AB, where A = (1,2,0), B = (3,0,-2).

> dim 3

> P = 2e1+3e2+e3

> A = e1+2e2

> B = 3e1-2e3

> abs(((P-B)^(A-B))/(A-B))

ans = 1.633 [= sqrt(8/3)]

2. Compute i/(j+exp(k p /6)) in quaternions. Hint: Go to the Clifford algebra Cl 0,3 and use the correspondences i = e 1 , j = e 2 , k = e 3 .

> dim 0,3

> q(u) = Re((1-e123)u)+Pu(1,(1-e123)u)

> q(e1/(e2+exp(pi/6 e3)))

ans = 0.433e1+0.250e2-0.5e3 [= sqrt(3)/4 i + 1/4 j - 1/2 k]

3. Find matrices of the two isoclinic rotations (= turns each plane the same angle) U L (a) = b, U R (a) = b sending a = (16+12i+5j+4k)/21 to b = (18+10i+4j+k)/21. Compute ba-1 = (1/441)(432-67e 1 +2e 2 -58e 3 ), a-1b = (1/49)(48-5e 1 -6e 2 -6e 3 ) and the components of (ba-1)q, q(a-1b) for q = 1, e 1 , e 2 , e 3 to find the entries of