Wacker Art Geometric Algebra
Bild: "Flower Power"

## Prolog

Structure of the complex unit i:

i2 = -1

(e1e2)2 = -1

i = e1e2

Hermann Grassmann

Wenn ich das Werk, dessen ersten Theil ich hiermit dem Publikum übergebe, als Bearbeitung einer neuen mathematischen Disciplin bezeichne, so kann die Rechtfertigung einer solchen Behauptung nur durch das Werk selbst gegeben werden.

Hermann Grassmann - "Die Ausdehnungslehre" - Vorrede

William Kingdon Clifford

I propose to communicate in a brief form some applications of Grassmann's theory which it seems unlikely that I shall find time to set forth at proper length, though I have waited long for it. Until recently I was unacquainted with the Ausdehnungslehre, and knew only so much of it as is contained in the author's geometrical papers in Crelle's Journal and in Hankel's Lectures on Complex Numbers. I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science.

David Hestenes

...Grassmann looked for rules for combining vectors which would fully describe the geometrical properties of directed line segments. He noticed that two directed line segments connected end to end determined a third, which may be regarded as their sum...

...Though several people might be credited with conceiving the idea of "directed number", Hermann Grassmann, in his book of 1844, developed the idea with precision and completeness that far surpassed the work of anyone else at the time...

Bild: "Edersee"

## History of Geometric Algebra

### Hermann Günter Graßmann(1809-1877)

Hermann Graßmann developed a new branch of mathematics in his book "Die Ausdehnungslehre" from 1844. In this book he introduced the outer or exterior product. In modern notation written as or a wedge b. Graßmann published his book in the same year as Hamilton anounced the discovery of the quaternions. But did not receive the same fame as Hamilton during his lifetime. Perhaps he was not taken seriously by his contempories because he was only a high school teacher.

### William Rowan Hamilton (1805-1865)

William Rowan Hamiltion was an Irisch mathematician. He became famous for his dicovery of the quaternions. A generalisation of complex numbers. On October the 16th, 1843 he walked with his wife at the Royal Canal in Dublin. On this walk he had the basic idea. The ledgend says that he craved the basic formular of quaternions into the Broome Bridge.

Quaternion algebra is today a subset of geometric algebra.

### Josiah Willard Gibbs (1839-1903)

Gibbs developed vector algebra which is the main mathematical instrument used in physics and engineering to describe nature.

### William Kingdon Clifford (1845-1879)

In mathematics geometric algebra is mostly named clifford algebra but Clifford himself named it geometric algebra. In his geometric algebra he combined Graßmanns outer product with the well known scalar product. He also generalised Hamiltons quaternions. So they are know a part of geometric algebra.

### David Orlin Hestenes (1933)

David Hestenes initiated the modern development of geometric algebra and its application in physics in the 1960's.

Bild: "Edersee"

## Introduction

Geometric algebra introduces new geometric elements between which algebraic relations exists. In addition to the well known scalar and vector elements these are bivectors, trivectors n-vectors and multivectors which are generalisations of the well known vectors. It also introduces new products like the outer product and the geometric product.

### Geometric Algebra Basics

 Scalar Vector Bivector Trivector 4-Vector ... n-Vector 1 ...

The main elements of geometric algebra can be found in the above table. Geometric algebra introduces in addition to the well known scalar values and vectors further elements. The elements are called n-vectors or blades. For the first two n-vectors we have the name bivector and trivector. This elements can be seen as a directed area in the case of a bivector and as a directed volume in the case of the trivector. The scalar product is sometimes visualised as a dot at the origin.

#### Scalars

Scalars are the well known rational (real) numbers like 0, 1, 2.5, 3.1415... .

#### Vector

A vector is an oriented length.

More on vectors on the vector page:

#### Bivector - The Outer Product of Two Vectors

A bivector or 2-vector is the first new fundamental buliding block of geometric algebra I like to present. A bivector can be generalised to represent a plane in any dimension. No ortogonal vector is required. The plane is defined by the vectors that lay in the plane.

A Bivector: can be visualised by an oriented area that is spanned by two vectors:

=

A bivector is an oriented area element. It is descriped by the area and the orientation of the plane. Many different vectors can be used to define the same oriented area in the same plane. So the form of a bivector must not be that of a trapezium that is spanned by the two vectors. The same bivector can have many different forms, but must have the same area value and the same orientation.

The outer product is anticommutative. When the order of the vectors a and b is exchanged the sign changes to the opposit value:

Also the orientation of the plane defined by the two vectors changes to the opposite direction, if the two vectors building a bivector are exchanged:

 a∧b =- b∧a

If the two orthogonal unit vectors e1 and e2 lay in the same plane as the vectors a = a1e1 + a2e2; and b = b1e1 + b2e2; the outerproduct can be evaluated by the following equations:

The orthogonal unit vectors of the plane e1 and e2 define the unit bivector e1e2 of the plane. The unit bivector e1e2 is a new base element of the plane. The determinate defines the area of the bivector. Through the evaluation of the determinante the area is defined. The same area value and the same orientation can be constructed in many different ways. So the form of a bivector is not unique.

#### Trivector

A trivector is an oriented volume.

=

#### n-vector

A n-Vector is an oriented n-dimensional volume.

### Geometric Product

Geometric algebra introduces a new product called the geometric product. The geometric product of two vectors a and b is written ab.

The geometric product of two vectors is the sum of the scalar product and the outer product (bivector) of the vectors a and b.

The geometric product is invertiable. It is possible to divide by a vector. With the geometric product, geometric algebra is a division algebra. The dimension of the vector is not limited. Geometric algebra is a division algebra in arbitrary dimension.

## Multivector

The elements introduced above: scalar s, vector v, bivector B, trivector T ... n-vector Nn, can be combined to form a new kind of entity called a multivector M.

Each element of a multivector has an associated grade. The grade indicates the number of vector factors of the outer product, in a non-zero component. The scalar elements have grade-0, the vectors grade-1, the bivectors grade-2, the trivectors grade-3, and so on. The grade is the dimension of the hyperplane it specifies. The term grade should be used instead of dimension because the name dimension is used for the size of the linear space .

The maximum grade is the dimension-n of the building vector space .

### Example Multivectorspace

The Multivectorspace is build with vectors from

A multivector is build from a scalar, three vectors, three bivectors and one trivector.

The base vectors take the following form:

A multivector decomposed in its base elements looks as follows:

M = s + v1e1 + v2e2 + v3e3 + b1(e1e2) + b2(e2e3) + b3(e3e1) + t(e1e2e3)

### Multivector Components

The can be described abstract as the direct sum of the building subspaces:

## Canonic Base System

The canonic base of the space are the base vectors .

The canonic base are ortogonal unit vectors. Among them the following basic algebraic relationship exists:

A vector has the following decomposition in base vectors ei.

### Basic laws for the calculus with unit vectors

Inner product of a unit vector ei is one:

Outer product of a unit vector ei is zero.

The outer product of a vector with one leaves the vector unchanged

The outer product of two different base vectors is a new canonic base element, a unit bivector:

### Anticommutativity

For the outer product of two canonic base vectors we have the relation:

### The Geometric Product of Canonic Vectors

Geometric product of a unit vector with himself (parallel unit vectors).

Geometric product of orthogonal unit vectors will be a bivector.

### Squaring a Bivector will lead to the Imaginäry Number i

With the help of these properties we get the result when squaring a bivector build by two orthogonal unit vectors ( the indices i and j are not equal):

A pair of orthogonal unit vectors are the pseudoscalars of the plane they span. The pseodoscalare of a plane can be used as imaginery unit.

On the right side of the equation is the imagninery unit i not the index variable i. All unit pseudoscalars I(2) can be used as imaginary units in Complex Number theory. Not the other way around. Imaginary units always communicate in geometric algebra products, because they are handled as scalar. But pseudoscalars do not!

Further I will often use different short forms for the outer products:

### The Unit Pseudoscalar I(n)

The product of the n orthogonal unit base vectors of a n-dimensional vector space is called the unit pseudoscalar I.

It is the base element of the highest grade element of a multivector. The dimension of the vector space maybe indicated as indices I(n).

### Unit Pseudoscalar in 4-Dimensions I(4)

Attention: In 4 dimensions, the square of the unit pseudoscalar is 1!

### The Square of the Unit Pseudoscalar I(n) in Higher Dimensions

The square of the pseudoscalar can be +1 or -1. Depending on the dimension n of the vectorspace.

The result of this formular is a ++--++--++... pattern depending on the dimension n of the bulding vector space.

An overview on the first values gives the following table:

 n I (I(n))2 Sign 0 1 +1 + 1 e1 +1 + 2 e1e2 -1 - 3 e1e2e3 -1 - 4 e1e2e3e4 +1 + 5 e1e2e3e4e5 +1 + 6 e1e2e3e4e5e6 -1 - 7 e1e2e3e4e5e6e7 -1 - ... ... ... ...

## Products

Geometric Algebra also introduces several new products. Like the outer product and the geometric product.

#### Scalar Product, Inner Product

The inner product or scalare product can be calculated as follows:

#### Outer Product

Basic relations of the outer product of two Vectors.

Commutative Law:

The outer product of two parallele vectors is zero. (The area becomes zero.)

The relation between the outer product of two vectors and the angle between two vectors u and v is given in the following formular:

If we have only two dimensions or if the two vectors are in the same plane as e1 and e2 we have:

 u∧v = (u1e1 + u2e2) ∧ (v1e1 + v2e2) = u1v1e1∧e1 + u2v1e2∧e1 + u1v2e1∧e2 + u2v2e2∧e2 = (u1v2 - v1u2)e1∧e2

uv = (u1e1 + u2e2) ∧ (v1e1 + v2e2) = (u1v2 - u2v1)(e1e2)

We get the following relation between the outer product and the determinant:

In three dimensions the following relation between a bivector and the cross product exists:

Example:

#### Outer Product in Higher Dimensons

If a vector space has n dimensions and an orthogonal basis of unit vectors:

If we have a set of n vectors in this n dimensional vector space:

These vectors can be represented by the orthogonal base of unit vectors as follows::

The outer product of these n vectors can be represented as follows:

Remember, that the determinante is only none zero when all vectors are linear independet.

#### Tensor Product of two Vectors

Building the sum of the matrix elements will give:

## The Geometric Product

The geometric product is a fundamental operation of geometric algebra. In this section the basic operations are given.

The geometric product of two vectors is the sum of the scalar product and the outer product:

So the result of the geometric product is a multi vector with two parts: a scalar and a bivector.

The geometric product is noncommutative. Commutating the two vectors in the geometric product gives:

Expressing the inner product with the geometric product gives the following formular:

Expressing the outer product with the geometric product gives the following formular:

The inner and outer products are the symmetric and antisymmetric parts of the geometric product. Rearranging the two equations above gives the following two formulas:

Dependens of the geometric product of two vectors on the dot product of these vectors.

Dependens of the geometric product of two vectors on the wedge product of these vectors.

### Application of the Geometric Product on Base Vectors

Vector product of base vectors that are equal has the result 1.

Vector product of base vectors that are orthogonal produces a bivector. (i is not equal to j)

### Canonic Form in two Dimensions

Having u and v expressed by their canonic base vectors in two dimensions gives:

The geometric product of u and v can then be expressed as follows (a,b,c,d are scalare values):

The result of the geometric product of two vectors can be interpreted as a complex number.

### Canonic Form in three Dimensions

In three dimensions the geometric product of two vectors can be interpreted as a quaternion.

### Inverse of a Vector

Square of a vector results in a scalar value, being the square length of the vector:

For the geometric product the inverse of a vector that is not zero is defined (which is not the case for the dot and the cross product!):

It is possible to present the basic elements, one basic scalar 1 = I(0), two basic vectors e1, e2 and a basic bivector or pseudo scalar e1e2 = I(2) of the geometric algebra 𝔾2 as matrices:

### Definition of a Blade

The outer product of k linear independent vectors is called a k-blade.

If one or several vectors are not linear independent the blade will be Bk = 0.

Another way to define a blade is to use the geometric product of anticommunicating vectors.
Vectors are anticommunicating if they fulfill the following equation for all the vectors used for the construction of the blade:

Then a blade Bk can be defined by the following geometric product:

### Reversion of a Blade

The reversion of a k-blade is defined by the reversion of the order of the defining vectors.

The ˜ tilde symbol is used to mark a reversed k-blade.

### Inversion of a Blade

The invers of a blade is defined as follows

### Inverse of the Unit Pseudo Scalar

The inverse of the unit pseudo scalar is equal to the reverse of the unit pseudo scalar.

In 3 Dimensions we have the following Example:

## Kronecker Delta

The Kronecker Delta function can be expressed with the help of the inner product of orthogonal unit vectors.

• If the unit vectors are parallel the inner product is one. (The two unit vectors are equal)
• If the unit vectors are orthogonal the inner product is zero.

## Levi-Civita-Symbol

The Levi-Civita-Symbol also known as the total antisymetric tensor or ε tensor, is defined in general form as follows:

A specific Levi-Civita-Symbol has to be defined for a specific value n. And has then n indices, where each index values can lay in a range between 1 and n.

#### Example:

In 2-Dimensions the possible values can be presented in matrix form:

In 3-Dimensions the result can be thought of as staging of three matrices.

### Levi-Civita-Symbol expressed by Linear Independent Orthogonal Unit Vectors

The n-dimensional Levi-Civita-Symbol can be expresse by n linear independent orthogonal unit vectors and the unit pseudo scalar I(n):

Because of the antisymetric property of the outerproduct the expression in the nominator becomes zero if any of the indices are the same: eiei=0.
A permutation of two vectors means the exchange of a pair of vectors in a wedge product. Exchanging two vectors in a wedge product results in a change of the sign: eiej=-(ejei). If an even number of permutations (pair wise exchange of vectors) is required to bring the nominator in the order of e1e2∧...∧en the sign will be positiv. If an odd number of permutations is required the sign will be negative.

## Dual Vector Space - Hodge Star Operator

The dual Multivector A* of a given Multivector A is defined as follows:

I is the unit pseudoscalar of the geometric algebra.

### The Meet Operation

The meet operation provides a way to calculate intersections of geometric elements.

## Frames and reciprocal frames

Any set of lineare independent vectors form a basis.

The defining vectors do not have to be orthonormal vectors.

The volume element for this vectors is defined as follows:

The reciprocal vector ei is defined by the following formula. The check ei is left out on the right hand side of the formular.

If i = j the Kronecker delta is 1 in all other cases it is 0. That means that ei is orthogonal to all other vectors ej with the exception of ei.

Bild: "Edersee - Weißer Stein"

Bild: "Edersee"

## Geometric Calculus

### Nabla Operator

Einstein sum convention: Elements with repeated upper and lower indices are implicitly summed over. This convetion allows the avoidence of summation sysmbols in many cases.

#### Overdot Notation

1. In the absence of Brackets, the nabla operator acts on the element on the right.
2. When the objects are in brackets the nabla operator acts on the all elements.
3. When the nabla oberator acts on elements which are not adjacent the overdot notation is used to identify the element the operator is applied to.

The gradient of a scalar function is defined as follows:

### Exterior Derivative of a Vector

The exterior derivative generalises the curl to arbitary dimensions.

### Fundamental Theorem of Geometric Calculus

Let M be an oriented and bounded m-dimensional manifold with boundary ∂M (The ∂ symbol is used for the boundary!).

Let F be a continous multivector field on M∪∂M with a continous vector derivative ∂F on M (The ∂ symbol is used for the vector derivative!).

Then we have:

The m-dimensional manifold M is in with m ≤ n.

Bild: "Rain"

## Base Vector Tables

Canonical base elements for the construction of multivectors in different dimensions.

An n-dimensional vector space has base elements (k-vectors) of grade k.

The total number of independent k-vectors supported by the vector space of dimension n is given by the following formular:

The following tables contain the basic elements for the construction of multivectors in a given dimension of the vector space.

### 2 Dimensions

 Grade Elements 0 1 1 1 2 e1 e2 2 1 e1∧e2

### 3 Dimensions

 Grade Elements 0 1 1 1 3 e1 e2 e3 2 3 e1∧e2 e1∧e3 e2∧e3 3 1 e1∧e2∧e3

### 4 Dimensions

 Grade Elements 0 1 1 1 4 e1 e2 e3 e4 2 6 e1∧e2 e1∧e3 e2∧e3 e1∧e4 e2∧e4 e3∧e4 3 4 e1∧e2∧e3 e1∧e2∧e4 e1∧e3∧e4 e2∧e3∧e4 4 1 e1∧e2∧e3∧e4

### 5 Dimensions

 Grade Elements 0 1 1 1 5 e1 e2 e3 e4 e5 2 10 e1∧e2 e1∧e3 e2∧e3 e1∧e4 e2∧e4 e3∧e4 e1∧e5 e2∧e5 e3∧e5 e4∧e5 3 10 e1∧e2∧e3 e1∧e2∧e4 e1∧e3∧e4 e2∧e3∧e4 e1∧e2∧e5 e1∧e3∧e5 e2∧e3∧e5 e1∧e4∧e5 e2∧e4∧e5 e3∧e4∧e5 4 5 e1∧e2∧e3∧e4 e1∧e2∧e3∧e5 e1∧e2∧e4∧e5 e1∧e3∧e4∧e5 e2∧e3∧e4∧e5 5 1 e1∧e2∧e3∧e4∧e5
Bild: "Edersee"

## Multiplication Tables

Multiplication tables of multivectors in different dimensions.

### 2-Dimensions

 1 e1 e2 e1∧e2 1 1 e1 e2 e1∧e2 e1 e1 1 e1∧e2 e2 e2 e2 -e1∧e2 1 -e1 e1∧e2 e1∧e2 -e2 e1 -1

### 3-Dimensions

 1 e1 e2 e1∧e2 e3 e1∧e3 e2∧e3 e1∧e2∧e3 1 1 e1 e2 e1∧e2 e3 e1∧e3 e2∧e3 e1∧e2∧e3 e1 e1 1 e1∧e2 e2 e1∧e3 e3 e1∧e2∧e3 e2∧e3 e2 e2 -e1∧e2 1 -e1 e2∧e3 -e1∧e2∧e3 e3 -e1∧e3 e1∧e2 e1∧e2 -e2 e1 -1 e1∧e2∧e3 -e2∧e3 e1∧e3 -e3 e3 e3 -e1∧e3 -e2∧e3 e1∧e2∧e3 1 -e1 -e2 e1∧e2 e1∧e3 e1∧e3 -e3 -e1∧e2∧e3 e2∧e3 e1 -1 -e1∧e2 e2 e2∧e3 e2∧e3 e1∧e2∧e3 -e3 -e1∧e3 e2 e1∧e2 -1 -e1 e1∧e2∧e3 e1∧e2∧e3 e2∧e3 -e1∧e3 -e3 e1∧e2 e2 -e1 -1

### 4-Dimensions

 1 e1 e2 e12 e3 e13 e23 e123 e4 e14 e24 e124 e34 e134 e234 e1234 e1 1 e12 e2 e13 e3 e123 e23 e14 e4 e124 e24 e134 e34 e1234 e234 e2 -e12 1 -e1 e23 -e123 e3 -e13 e24 -e124 e4 -e14 e234 -e1234 e34 -e134 e12 -e2 e1 -1 e123 -e23 e13 -e3 e124 -e24 e14 -e4 e1234 -e234 e134 -e34 e3 -e13 -e23 e123 1 -e1 -e2 e12 e34 -e134 -e234 e1234 e4 -e14 -e24 e124 e13 -e3 -e123 e23 e1 -1 -e12 e2 e134 -e34 -e1234 e234 e14 -e4 -e124 e24 e23 e123 -e3 -e13 e2 e12 -1 -e1 e234 e1234 -e34 -e134 e24 e124 -e4 -e14 e123 e23 -e13 -e3 e12 e2 -e1 -1 e1234 e234 -e134 -e34 e124 e24 -e14 -e4 e4 -e14 -e24 e124 -e34 e134 e234 -e1234 1 -e1 -e2 e12 -e3 e13 e23 -e123 e14 -e4 -e124 e24 -e134 e34 e1234 -e234 e1 -1 -e12 e2 -e13 e3 e123 -e23 e24 e124 -e4 -e14 -e234 -e1234 e34 e134 e2 e12 -1 -e1 -e23 -e123 e3 e13 e124 e24 -e14 -e4 -e1234 -e234 e134 e34 e12 e2 -e1 -1 -e123 -e23 e13 e3 e34 e134 e234 e1234 -e4 -e14 -e24 -e124 e3 e13 e23 e123 -1 -e1 -e2 -e12 e134 e34 e1234 e234 -e14 -e4 -e124 -e24 e13 e3 e123 e23 -e1 -1 -e12 -e2 e234 -e1234 e34 -e134 -e24 e124 -e4 e14 e23 -e123 e3 -e13 -e2 e12 -1 e1 e1234 -e234 e134 -e34 -e124 e24 -e14 e4 e123 -e23 e13 -e3 -e12 e2 -e1 1
Bild: "Edersee"

## Programing through Rainy Days

This software is under construction and not for commercial use!

Disclaimer

Bild: "Double Rainbow"

## Complex Numbers

### Relation between vectors from and complex numbers

A complex number can be expressed by using the pseudoscalar of the plane, the bivector e1e2, as imaginary unit i = I(2) = e1e2.

A vector a element of can be expressed as multiples of the unit vectors e1, e2

Converting a vector in the plane to a complex number.

Converting a complex number to a vector in the plane.

### The product of vectors from

The result of the geometric product of two vectors from can be interpreted as a complex number:

## Quaternions

Quaternions are elements of the Form:

q0, q1, q2, q3 are real numbers. i, j, k are three different square roots of -1 and are the new elements used for the construction of quaternions. They have the following algebraic properties:

### Algebraic Properties

From this fundamental relations we get:

From this relations a multiplication table can be constructed.

### Multiplication Table

 1 i j k 1 1 i j k i i -1 k -j j j -k -1 i k k j -i -1

### The Relation between Quaternions and Geometic Algebra

The basic elements i, j, k of the quarterions can be indentified with the following bivectors in geometric algebra.

The elements i, j, k form a left handed System.

On the other hand if I build the product of bivectors for a right handed system I get:

If we build the geometric product of two vectors u and v from

Will result in the following quarternion:

### Confusion in the Application of Quaternions

Hamilton tried to apply i, j, k as Vectors in a right handed System, which lead to much confusion in such applications. This confusion can be avoided if the the i, j, k are interpretated as bivectors in a left handed system.

In older books sometimes i, j. k can be found to name the three spatial unit vectors in a right handed system. Which is correct, if one does not interpret them as elements of a quaternion. Hamilton introduced vectors into the main stream of mathematics in this form. Gibbs developed the vector analysis to overcome the difficulties introduced by the misinterpretation of quaternions. On the other hand quaternions are an powerfull tool to perform rotations.

### Biquaternions

Quaternions and Biquaternions are elements of the Form:

For quaternions   q0, q1, q2, q3 are real numbers.

For biquaternions q0, q1, q2, q3 are complex numbers.

## Hypercomplex Numbers

ak's are real numbers, ik are different complex units with the property (ik)2 = -1 for all k.

 n Complex Units Name Structure 0 0 Real Numbers a0 1 1 Complex Numbers a0 + a1 i1 2 3 Quaternions a0 + a1 i1 + a2 i2 + a3 i3 3 7 Octonions a0 + a1 i1 + a2 i2 + a3 i3 + a4 i4 + a5 i5 + a6 i6+ a7 i7 4 15 Sedenions a0 + a1 i1 + a2 i2 + a3 i3 + a4 i4 + a5 i5 + a6 i6+ a7 i7 + a8 i8 + a9 i9 + a10 i10 + a11 i11 + a12 i12 + a13 i13 + a14 i14 + a15 i15 n 2n-1 a0 + ... + an2-1 in2-1

Hypercomplex Numbers can be generated by the Cayley-Dickson Construction.

### More on Hypercomplex Numbers

More on hypercomplex numbers can be found on my hypercomplex numbers page.

### Others

Bild: "Edersee"

The next page is about Clifford Algebra.

6. September 2014 Version 1.0
Copyright: Hermann Wacker Uhlandstraße 10 D-85386 Eching bei Freising Germany Haftungsausschluß