New Foundations in Mathematics:

The Geometric Concept of Number

byGarret Sobczyk

Universidad de Las Americas-PCholula, MexicoNovember 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

ContentsV. 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 mappingsVII. 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 AlgebrasIX. 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) .

Modular Polynomials and Interpolation mod(h(x))

Complex and Hyperpolic Numbers

u2=1

Hyperbolic Numbers

:

Geometric Numbers G2 of the Plane

Standard Basis of G2={1, e1, e2, e12}.

where i=e12 is a unit bivector.

Basic Identitiesab =a.b+a^b

a.b=½(ab+ba)

a^b=½(ab-ba)

a2=a.a= |a|2

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).

Reflections L(x) and Rotations R(x)

where |a|=|b|=1 and

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 followsthat 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.

For example, if

then the element g Ɛ G2 is

We find that

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

Geometric numbers and determinants

Let a1, a2 , . . ., an be vectors in Rn, where

Then

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.

Minkowski Space R1,3

g0 is timelike,

g1 g2 g3 spacelike

Spacetime Algebra G1,3We start with

We factor e1, e2, e3 into Dirac bivectors,

where

Geometric CalculusThe vector derivative at a point x in Rn.

Definition:

Formulas:

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 .

Classical Integral Theorems

A function is monogenic if for all 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

Conformal Mappings Conformal mapping the unit

cylinder onto the figure shown.

A more exotic conformal mapping of the hyperboloid like figure into the figure surrounding it.

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 ReferencesR. 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