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) .

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 Identities ab =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 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.

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 G1,3 We start with

We factor e1, e2, e3 into Dirac bivectors,

where

Geometric Calculus

The 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 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