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New Foundations in
Mathematics: The Geometric Concept of
Number
by
Garret Sobczyk
Universidad de Las Americas-P
Cholula, Mexico
November 2012
What is Geometric Algebra?
Geometric algebra is the completion of the real number system to include new anticommuting square roots of plus and minus one, each such root representing an orthogonal direction in successively higher dimensions.
Contents
I. Beyond the Real Numbers.
a) Clock arithmetic.
b) Modular polynomials and approximation.
b) Complex numbers.
c) Hyperbolic numbers.
II. The Geometric Concept of Number.
a) Geometric numbers of the plane.
b) Geometric numbers of 3-space.
c) Reflections and rotations.
d) Geometric numbers of Euclidean Space Rn.
Contents
III. Linear Algebra and Matrices.
a) Matrices of geometric numbers.
b) Geometric numbers and determinants.
c) The spectral decomposition.
IV. Splitting Space and Time.
a) Minkowski spacetime.
b) Spacetime algebra
Contents V. Geometric Calculus. a) The vector derivative.
b) Fundamental theorem of calculus.
VI. Differential Geometry.
a) The shape operator.
b) The Riemann curvature bivector
c) Conformal mappings
VII. Non-Euclidean and Projective Geometries
a) The affine plane
b) Projective geometry
c) Conics
d) The horosphere
CONTENTS
• Lie groups and Lie algebras.
a) Bivector representation
b) The general linear group
c) Orthogonal Lie groups and algebras
d) Semisimple Lie Algebras
IX. Conclusions
X. Selected References
Clock Arithmetic
12 = 3x22
Spectral equation: s1 + s2 = 1 or
3(s1 + s2 ) = 3 s2 = 3. This implies that
9 s2 = s2 = 9, and s1 = 4. Now define
q2 = 2 s2 = 6. Spectral basis: { s1, s2, q2}
idempotents: s12 = 16 = 4 mod 12 = s1
s22 = 81 = 9 mod 12 = s2
nilpotent: q22 = 36 = 0 mod 12, s1 s2 = 0 mod 12
Clock Arithmetic: 12 = 3x22
A calculation: 5s1 + 5s2 = 5mod(12) or
2s1 + 1s2 = 5 mod(12). It follows that
2ns1 + 1ns2 = 5n mod(12) for all integers n.
n=-1 gives 1/5 = 2s1 + 1s2 = 5 mod(12) and
n=100 gives 5100 = s1 + s2 = 1 mod(12) .
:
Geometric Numbers G2 of the Plane
Standard Basis of
G2={1, e1, e2, e12}.
where i=e12 is a unit
bivector.
Geometric Numbers of 3-Space
a^b=i axb
a^b^c=[a.(bxc)]i
where i=e1e2e3=e123
a.(b^c)=(a.b)c-(a.c)b
= - ax(bxc).
Geometric Numbers Gn of Rn
Standard basis of the geometric algebra
Gn of the Euclidean space Rn.
There are (n:k) basis k-vectors in Gn. It follows
that the dimension of Gn is
Matrices of the Geometric Algebra G2
Recall that G2=span{1, e1, e2, e12}.
By the spectral basis of G2 we mean
where
are mutually annihiliating idempotents.
Note that e1 u+ = u- e1.
Matrices of the Geometric Algebra G3
We can get the complex Pauli matrices
from the matrices of G2 by noting that
e1 e2 = i e3 or e3 = -i e1 e2,
where i = e123 is the unit element of
volume of G3. We get
Spectral Decomposition
Let
with the characteristic polynomial
φ(x)=(x-1)x2. Recall that the spectral basis for
this polynomial was
Replacing x by the matrix X, and 1 by the identity 3x3
matrix gives
It follows that the spectral equation for X is
X=1 S1 + 0 S2 + Q2,
with the eigenvectors
We now obtain the Jordan Normal Form for X
Splitting Space and Time
The ordinary rotation
is in the blue plane of
the bivector i=e12. The
blue plane is boosted
into the yellow plane by
with the velocity v/c = Tanh ɸ.
The light cone is shown in red.
Fundamental theorem of calculus.
Let M be a k-surface in Rn. A point x Ɛ M
is given by x=x(s1,s2,…,sk) for the
coordinates si Ɛ R. The tangent vectors xi
at the point x Ɛ M are defined by
and generate the tangent geometric algebra
Tx at the point x Ɛ M .
Differential Geometry
Let M be a k-surface in Rn. Define the
tangent pseudoscalar Ix at x Ɛ Rn by
the projection operator Px at x Ɛ Rn by
and the shape operator S(Ar) by
The Riemann Curvature Bivector • The Riemann curvature bivector R(a^b) is
defined by
We have the basic relationship
is the induced connection on the k-surface M.
The classical Riemann curvature tensor is
Non-Euclidean and Projective
Geometries
The affine
plane. Each
point x in Rn
determines a
unique point xh
in the affine
plane.
Desargue’s Configuration
Thm: Two triangles
are in perspective
axially if and only
if they are in
perspective
centrally.
The Horosphere
Any conformal
transformation can
be represented by
an orthogonal
transformation on
the horosphere.
Lie Algebras and Lie Groups
Let Gn,n be the 22n-dimensional geometric algebra with
neutral signature. The Witte basis consists of two
dual null cones:
We now construct the matrix of bivectors
These bivectors are the generators of the
general linear Lie algebra gln,
with the Lie bracket product
Each bivector F generates a linear transformation f,
defined by
General Linear Group
The general linear group GLn is obtained from
the Lie algebra gln by exponentiation. We have
GLn = { G=eF | F Ɛ gln }.
Consider now the one parameter subgroups
defined for each G Ɛ gln by
gt(x)=e½tF x e-½tF
where x=∑xi ai and t Ɛ R. Differentiating gives
It follows that
Conclusions • Since every (finite dimensional) Lie algebra can be
embedded in gln, it follows that every Lie algebra can be
represented as a Lie algebra of bivectors.
• Complex semi-simple Lie algebras are classified by their
Dynkin diagrams.
• Geometric algebra offers new geometric tools for the
study of representation theory, differential geometry, and
provides a unified algebraic approach to many areas of
mathematics.
• I hope my selection of topics has been sufficiently broad
to support my contention that geometric algebra and the
Geometric Concept of Number should be viewed as a
New Foundation for Mathematics.
Selected References R. Ablamowicz, G. Sobczyk, Lectures on Clifford (Geometric) Algebras and
Application, Birkhauser, Boston 2004.
W.K. Clifford, Applications of Grassmann's extensive algebra, Amer. J. of Math.
1 (1878), 350-358.
T. Dantzig, NUMBER: The Language of Science, Fourth Edition, Free Press,
1967.
P. J. Davis, Interpolation and Approximation, Dover Publications, New York,
1975.
F.R. Gantmacher, Theory of Matrices, translated by K. A. Hirsch, Chelsea
Publishing Co., New York (1959).
T.F. Havel, GEOMETRIC ALGEBRA: Parallel Processing for the Mind (Nuclear
Engineering) 2002. http://www.garretstar.com/secciones/
D. Hestenes, New Foundations for Classical Mechanics, 2nd Ed., Kluwer 1999.
D. Hestenes, Point Groups and Space Groups in Geometric Algebra, In: L.
Doerst, C. Doran, J. Lasenby (Eds), Applications of Geometric Algebra with
Applications in Computer Science and Engineering, Birkhauser, Boston
(2002). p. 3-34.
D. Hestenes, Space Time Algebra, Gordon and Breach, 1966.
D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus: A Unified
Language for Mathematics and Physics, 2nd edition, Kluwer 1992.
P. Lounesto, Clifford Algebras and Spinors, 2nd Edition. Cambridge University
Press, Cambridge, 2001.
P. Lounesto, CLICAL software packet and user manual. Helsinki University of
Technology of Mathematics, Research, Report A248, 1994.
G. Sobczyk, The missing spectral basis in algebra and number theory, The
American Mathematical Monthly 108 April 2001, pp. 336-346.
G. Sobczyk, Geometric Matrix Algebra, Linear Algebra and its Applications, 429
(2008) 1163-1173.
G. Sobczyk, Hyperbolic Number Plane, The College Mathematics Journal, Vol.
26, No. 4, pp.269-280, September 1995.
G. Sobczyk, A Complex Gibbs-Heaviside Vector Algebra for Space-time, Acta
Physica Polonica, Vol. B12, No.5, 407-418, 1981.
G. Sobczyk, Spacetime Vector Analysis, Physics Letters, 84A, 45-49, 1981.
G. Sobczyk, Noncommutative extensions of Number: An Introduction to
Clifford's Geometric Algebra, Aportaciones Matematicas Comunicaciones},
11 (1992) 207-218.
G. Sobczyk, Hyperbolic Number Plane, The College Mathematics
Journal, 26:4 (1995) 268-280.
G. Sobczyk, The Generalized Spectral Decomposition of a Linear Operator, The College Mathematics Journal, 28:1 (1997) 27-38.
G. Sobczyk, Spectral integral domains in the classroom,
APORTACIONES MATEMATICAS, Serie Comunicaciones Vol. 20,
(1997) 169-188.
G. Sobczyk, Spacetime vector analysis, Physics Letters, 84A, 45 (1981).
G. Sobczyk, New Foundations in Mathematics: The Geometric Concept of Number, San Luis Tehuiloyocan, Mexico 2010.
http://www.garretstar.com/NFM15XII09.pdf
J. Pozo and G. Sobczyk. Geometric Algebra in Linear Algebra and Geometry}.
Acta Applicandae Mathematicae, 71: 207--244, 2002.
Note: Copies of many of my papers and talks can be found on my website:
http://www.garretstar.com