# ^^GA geometric algebra.

Vettore libero o applicato.

av8n/clifford-intro mi piace il suo autore

# Paravector wp/Paravector

La tesi di chi ha usato per primo il termine "paravector" Johannes Maks| tudelft.nl/.pdf

## Ricerche

> http://beyhfr.free.fr/EVA2/index.html e' la sua home

in cui ci sono 2 link a pdf: Leibnitz' dream | representation of three-space.

# cronus.uwindsor.ca/baylis-research

Welcome to the website for my research on Clifford's Geometric Algebra of Physical Space (APS).

# Citazione from  Leibnitz' dream

Grassmann’s influence was far reaching.

The English mathematician W.K. Clifford published in 1878 his Applications of Grassmann’s extensive algebra, describing “geometric algebra”.

Clifford had been a student of James Maxwell. Clifford’s desire to understand the mathematical basis of Maxwell’s equations partly motivated his research in geometric algebra. He started by clarifying "the relation of Grassmann’s method to quaternions".

Clifford “profoundly admired” Grassmann’s Ausdehnungslehre, with the “conviction that its principles will exercise a vast influence upon the future of mathematical science.” Now this algebra is often simply referred to as “Clifford algebra.”

1888 the Italian G. Peano published in  his Calcolo geometrico secondo l’Ausdehnungslehre di H. Grassmann.

1892 Felix Klein himself successfully began to push for a complete posthumous republication of Grassmann’s works by the Royal Saxonian Society of Sciences.
But, due to the early death of Clifford, J.W. Gibbs’ and O. Heaviside’s vector analysis dominated most of the 20th century, and not Clifford geometric algebra. Yet today, at the beginning of the 21st century, some people believe, that based on Grassmann’s and Clifford’s work soon more or less all of mathematics may be formulated as a single unified universal geometric calculus, with concrete geometrical foundations. The algebraic “grammar” such a geometric calculus uses is Clifford geometric algebra.

Gibbs’ vectorial system, enlarged with the whole of linear algebra, can only accommodate a non-operational representation of the rotation group by means of orthogonal matrices. This is only a Cartesian numerical form of the vectorial representation, and not the group itself. In particular the matrix does not discriminate between the rotation angles theta and – (2 pi – theta). Being the rotation group essential to the characterization of the physical space and to the expression of the mutual relationships between different objects contained in it, the “classical” mathematical representation that makes no use of the geometric product of Clifford is severely limited.

Ho recuperato l'articolo di Clifford negli internet Archive, in un libro di articoli "Mathematical papers (1882)"

e' a pag. 266 del libro, pag 338 DjVu

# clifford-algebra-a-visual-introductiongeometric-algebra-projective-geometry geometric-algebra-conformal-geometry/

c: mi sembra di aver capito che l'autore e' un filosofo, e il suo punto di vista ha spunti interessanti proprio perche' non matematico.

## c: scalare vs estensione

There is an interesting interaction that occurs across grades seen for example when a scalar, a non-spatial entity, multiplies a vector which has linear extent, it changes the spatial length by the magnitude determined by the scalar value. Likewise, taking the magnitude of a vector v as in ||v|| summarizes the spatial extent of that spatial vector with a non-spatial scalar magnitude. A bivector B can also be reduced to a scalar magnitude by taking its magnitude ||B||, which provides an abstract measure of the magnitude of the bivector which is proportional to its area. These are manifestations of inter-grade communication, whereby the scalar value can be seen as the most abstract representation, reducing clifs to a single scalar value that records their magnitude, and thus scalar algebra can be seen as an abstraction of the principles of vector algebra, preserving the corresponding magnitudes but discarding spatial extent.