Wacker Art Geometric Algebra Wappen der Familie Wacker
Flower Power
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

Edersee
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 Bivector 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 Basic

Quaternion algebra is today a subset of geometric algebra.

Josiah Willard Gibbs (18391903)

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.

Edersee
Bild: "Edersee"

Introduction

This page is under construction!

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 Vector Bivector Trivector Trivector ... n-vector
Vector Bivector Trivector

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.

Scalars

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

Vector

  Vector

A vector is an oriented length.

Bivector

A Bivector: Bivector can be visualised by an oriented area.

  Bivector = Bivector

A bivector is an oriented area element. It contains the area and the orientation of the area. It is not a vector that stands orthogonal to this area (like the cross product.)

Trivector

A trivector Trivector is an oriented volume.

  Trivector  = Triivector

n-vector

n-vector

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

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.

Multivector

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 R n.

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

Example Multivectorspace Multivectorspace G3

The Multivectorspace Multivectorspace G3 is build with vectors from R3

A multivector m element of g3 is build from a scalar, three vectors, three bivectors and one trivector.

Multivector 3D Base

The base vectors take the following form:

Multivector 3D Base

A multivector decomposed in its base elements looks as follows:

M = mo + m1e1 + m2e2 + m3e3 + m4(e1e2) + m5(e2e3) + m6(e3e1) + m7(e1e2e3)

The Multivectorspace G3 can be described abstract as the direct sum of the building subspaces:

Multivector 3D Base

Canonic Base

The canonic base of the space R^n are the base vectors unit vectors.

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

Outer Product Short

A vector V element of R has the following decomposition in base vectors ei.

Vector Components

Basic laws for the calculus with unit vectors

Inner product of a unit vector:

Inner product of unit vectors

Anticommutativity

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

Cannonic Product is Anicomutative

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):

Square of the Unitvectors

Complex Unit


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

Outer Product Short

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.

Pseudoscalar

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 2-Dimensions I(2)

Pseudoscalar Dimension 2

Pseudoscalar Diemension 2 Square

Unit Pseudoscalar in 3-Dimensions I(3)

Pseudoscalar

Square Trivector

Square Trivector

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:

Scalar Product

Inner Product

Outer Product

Basic relations of the outer product of two Vectors.

Commutative Law:

Outer Product Commutative Law

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

Outer Product Square

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

Outer Product

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

Outer Product Canonic

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

Relation to the Crossproduct

Example:

Relation to the Crossproduct

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

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:

Geometric Product

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

Commutating the two vectors in the geometric product gives:

Commutating Geometric Product

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

Inner Product - Geometric Product

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

Outer Product - Geometric Product

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.

uv depend on dot

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

uv depend on wedge

Application of the Geometric Product on Base Vectors

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

base vector product

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

base vector product othogonal

Canonic Form in two Dimensions

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

u vector canonic

v vector canonic

The geometric product of u and v can then be expressed as follows:

u vector canonic

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

The Geometric Product and Eulers Formular

Geometric Product and Euler Formular

Inverse of a Vector

Square of a vector:

V Square

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!):

Inverse of a Vector

Blades

Definition of a Blade

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

k-blade

Reversion of a Blade

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

k-blade-reverse

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

Inversion of a Blade

The invers of a blade is defined as follows

invers blade

Inverse of the Unit Pseudo Scalar

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

Inverse Unit Pseudo Scalar

Pseudo Scalar Inversion

In 3 Dimensions we have the following Example:

Pseudo Scalar Inversion 3D

Dual Vector Space - Hodge Star Operator

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

V Square

I is the unit pseudoscalar of the geometric algebra.

The Meet Operation

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

meet

Frames and reciprocal frames

Any set of lineare independent vectors form a basis.

Frame Vectors

The defining vectors do not have to be orthonormal vectors.

The volume element for this vectors is defined as follows:

Volume Element

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

Reciprocal Vector

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.

Kronecker Relation

Edersee
Bild: "Edersee - Weißerstein"

Clifford Algebra

The Inner Product

We start with the geometric product of two vectors:

Geometric Product

Exchanging the vectors u and v gives the relation:

Geometric Product Commutated

By adding the two equations we get the scalar product:

Scalar Product

Or in a more general form:

Scalar Product general Form.

The relation of geometric products of the ortogonormal basic vectors, can be written in the form of a clifford product as follows:

Clifford Product

If we square the sum of two vectors (using the geometric product) we get:

Sum of two vectors squared   (1)

Reordering the elements gives:

Clifford Product   (2)

If we square the difference of two vectors we get:

Difference of two vectors squared   (3)

Reordering the elements gives:

Clifford Product   (4)

Summing up equation (2) and (4) gives equation (5):

Clifford Product   (5)

We now have the following relations for the scalar product of two vectors:

Clifford Product   (2)

Clifford Product   (4)

Clifford Product   (5)

The Outer Product

The outer product can be defined in dependence from the geometric product as follows:

Outer Product

Using the following product relation:

Complex Length   (6)

Gives the following relation for the outer product:

Outer Product   (7)

Using the following product relation:

Complex Length   (8)

Gives a further relation for the outer product:

Outer Product   (9)

The diverens between the given two relations for the outer product results in the folowing expression:

Outer Product   (10)

We now have the following relations for the outer product of two vectors:

Outer Product

Outer Product

Outer Product

Geometric Product

When we sum up the results for the inner (5) and the outer product (10) we come back to the geometric product:

Geometric Product

Geometric Product

Geometric Product

Geometric Product

Edersee
Bild: "Edersee"

Geometric Calculus

Vector Derivative

Partial Derivation

Partial Derivative

Nabla Operator

Nabla Operator

Vector Derivation with the Geometric Product

Vector Derivative

Inner Derivative or Divergence of a Vector

Inner Derivative

Exterior Derivative of a Vector

Outer Derivative

The exterior derivative generalises the curl to arbitary dimensions.

Rain
Bild: "Rain"

Base Vector Tables

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

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
Edersee
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
Edersee
Bild: "Edersee"

Programing through Rainy Days

This software is under construction and not for commercial use!

Disclaimer

Double Rainbow
Bild: "Double Rainbow"

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

Complex Numbers

Relation between vectors from R^2 and complex numbers Complex Set

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

Complex Number

A vector a element of R^2 can be expressed as multiples of the unit vectors e1, e2

Vector R^2

Converting a vector in the plane to a complex number.

Vector to complex number

Vector to complex number

Converting a complex number to a vector in the plane.

Complex number to vector

The product of vectors from R^2

The result of the geometric product of two vectors from R^2 can be interpreted as a complex number:

u vector canonic

v vector canonic

u vector canonic

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

Quaternion Basic

From this fundamental relations we get:

i equations

j equations

k equations

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

Complex Conjugate of a Quaternion

Square of the Length

The Product of two Quaternions

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.

i quaternion

j quaternion

k quaternion

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

i, j, k, product

three bivectors left handed

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

three bivectors right handed

Confusion in the Application of Quarternions

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 quarternion. Hamilton introduced vectors into the main stream of mathematics in this form.

Octonions

1

o1

o2

o3

o4

o5

o6

o7

1

1

o1

o2

o3

o4

o5

o6

o7

o1

o1

-1

o3

-o2

o5

-o4

-o7

o6

o2

o2

-o3

-1

o1

o6

o7

-o4

-o5

o3

o3

o2

-o1

-1

o7

-o6

o5

-o4

o4

o4

-o5

-o6

-o7

-1

o1

o2

o3

o5

o5

o4

-o7

o6

-o1

-1

-o3

o2

o6

o6

o7

o4

-o5

-o2

o3

-1

-o1

o7

o7

-o6

o5

o4

-o3

-o2

o1

-1

Sedenions

1 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15
1 1 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15
s1 s1 -1 s3 -s2 s5 -s4 -s7 s6 s9 -s8 -s11 s10 -s13 s12 s15 -s14
s2 s2 -s3 -1 s1 s6 s7 -s4 -s5 s10 s11 -s8 -s9 -s14 -s15 s12 s13
s3 s3 s2 -s1 -1 s7 -s6 s5 -s4 s11 -s10 s9 -s8 -s15 s14 -s13 s12
s4 s4 -s5 -s6 -s7 -1 s1 s2 s3 s12 s13 s14 s15 -s8 -s9 -s10 -s11
s5 s5 s4 -s7 s6 -s1 -1 -s3 s2 s13 -s12 s15 -s14 s9 -s8 s11 -s10
s6 s6 s7 s4 -s5 -s2 s3 -1 -s1 s14 -s15 -s12 s13 s10 -s11 -s8 s9
s7 s7 -s6 s5 s4 -s3 -s2 s1 -1 s15 s14 -s13 -s12 s11 s10 -s9 -s8
s8 s8 -s9 -s10 -s11 -s12 -s13 -s14 -s15 -1 s1 s2 s3 s4 s5 s6 s7
s9 s9 s8 -s11 s10 -s13 s12 s15 -s14 -s1 -1 -s3 s2 -s5 s4 s7 -s6
s10 s10 s11 s8 -s9 -s14 -s15 s12 s13 -s2 s3 -1 -s1 -s6 -s7 s4 s5
s11 s11 -s10 s9 s8 -s15 s14 -s13 s12 -s3 -s2 s1 -1 -s7 s6 -s5 s4
s12 s12 s13 s14 s15 s8 -s9 -s10 -s11 -s4 s5 s6 s7 -1 -s1 -s2 -s3
s13 s13 -s12 s15 -s14 s9 s8 s11 -s10 -s5 -s4 s7 -s6 s1 -1 s3 -s2
s14 s14 -s15 -s12 s13 s10 -s11 s8 s9 -s6 -s7 -s4 s5 s2 -s3 -1 s1
s15 s15 s14 -s13 -s12 s11 s10 -s9 s8 -s7 s6 -s5 -s4 s3 s2 -s1 -1

Links

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Geometric Algebra in Physics

Computer Graphics with Geometric Algebra

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Bild: "Edersee"
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