Wacker Art Geometric Algebra Wappen der Familie Wacker
Flower Power
Bild: "Flower Power"


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 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 (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"


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. The scalar product is sometimes visualised as a dot at the origin.


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



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

  Bivector = Bivector

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:

Noncommutative Product

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:




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:

Outer Product Canonic

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.


A trivector Trivector is an oriented volume.

  Trivector  = Triivector



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.

Geometric Product

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.


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

Multivector Components Multivectorspace G3

Multivector Subspaces

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

Multivector 3D Base

Canonic Base System

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 ei is one:

Inner product of unit vectors

Outer product of a unit vector ei is zero.

Inner product of unit vectors

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

Inner product of unit vectors

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

Inner product of unit vectors


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

Cannonic Product is Anicomutative

The Geometric Product of Canonic Vectors

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

geometric product unit vectors parallel.

Geometric product of orthogonal unit vectors will be a bivector.

geometric product unit vectors orthogonal

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

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.

Complex 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:

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.


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)


Square Trivector

Unit Pseudoscalar in 4-Dimensions I(4)

Pseudoscalar 4

Pseudoscalar 4 Square

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.

Pseudoscalar Square

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 -
... ... ... ...


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:


= (u1e1 + u2e2) ∧ (v1e1 + v2e2)

= u1v1e1e1 + u2v1e2e1 + u1v2e1e2 + u2v2e2e2

= (u1v2 - v1u2)e1e2

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

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


Outer Product Canonic

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

Relation to the Crossproduct


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

Building the sum of the matrix elements will give:

Geometric Product Canonic

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.

The geometric product is noncommutative. 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

Associative Property of the Geometric Product

Associative property of the Geometric Product

Distributive Property of the Geometric Product

Distributive property of the Geometric Product

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 (a,b,c,d are scalare values):

u vector canonic

u vector canonic

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.

The Geometric Product and Eulers Formular

Geometric Product and Euler Formular

Inverse of a Vector

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

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.

Reverse k-Blade

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

Releation between a Blade and it's Reverse

Reversion of a 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

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.

Kronecker Delta Function


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.


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:

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.


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 - Weißer Stein
Bild: "Edersee - Weißer Stein"

Clifford Algebra

Using the Inner Product

Clifford Product

Clifford Product

Clifford Product

Using the Outer Product

Outer Product

Outer Product

Outer Product

Using the Geometric Product

Geometric Product

Geometric Product

Geometric Product

Geometric Product

Bild: "Edersee"

Geometric Calculus

Partial Derivation

Partial Derivative

Nabla Operator

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.

Overdot Notation

Overdot Notation


The gradient of a scalar function is defined as follows:

Vector Derivative

Vector Derivation with the Geometric Product

Vector Derivative

Inner Derivative or Divergence of a Vector

Inner Derivative

Inner Derivative

Exterior Derivative of a Vector

Outer Derivative

Outer Derivative

The exterior derivative generalises the curl to arbitary dimensions.

Integration Theory


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:

Fundamental Theorem of Geometric Calculus

The m-dimensional manifold M is in R<sup>n</sup> 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 Vector Components 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:

Total numbers of k-vectors

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.


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


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


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!


Double Rainbow
Bild: "Double Rainbow"

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 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

Quaternion Conjugated

Square of the Length

Length Square

The Product of two Quaternions



Quaternions Product

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

If we build the geometric product of two vectors u and v from R^2

Geometric Product Canonic

Will result in the following quarternion:

Geometric Product Canonic

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.


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.


Geometric Algebra

Geometric Algebra in Physics

Computer Graphics with Geometric Algebra

Hermann Grassmann

Online Videos

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Bild: "Edersee"

The next page is about Clifford Algebra.

6. September 2014 Version 1.0
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