GEOMETRIC ALGEBRA

"Geometry without algebra is dumb. Algebra without geometry is blind" (David Hestenes and Garret Sobczyk).

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Clifford's Algebra
Vector algebra, despite its success in many fields of application, is very limited:

• It is not very generic, since it only works with one type of geometric object (the vector).
  
  
• There are two product operations defined between vectors (inner product and outer product). There is no unified product operation.
  
  
• Operations such as inverse of a vector, power, etc. cannot be performed.

Clifford algebra, also called "geometric algebra", is a richer and more generic algebra than vector algebra:

• It handles objects of different ranges or dimensions and geometric magnitudes. A geometric magnitude, as in physics, is specified by a quantity (real number) and a unit. The latter is a unitary geometric object, which can be of dimension or range 1, 2, 3, etc.
  
  
• All possible objects in an n-dimensional space are generated from n linear unitary objects (of rank or dimension 1), called "generators", which are associated to each of the dimensions: e₁, e₂, . .. , e_n. The generators represent the degrees of freedom of the geometric space and are the equivalent to the basis of a vector space or primitive or elementary magnitudes, from which the rest of magnitudes (which are derived magnitudes) are obtained.
  
  From the generating objects of the space of n dimensions (nD), all the other basic objects are generated, of dimensions 2, 3, ... , n, also unitary. In addition to these objects, there is the unitary scalar (1), which is interpreted as a geometric object of dimension 0 (geometric point).
  
  
• All basic (unitary) objects have an orientation or sense, not only those of one dimension (oriented linear segment), but also those of two dimensions (oriented plane segment), three dimensions (oriented volume segment), and so on. These senses are equivalent to those of multivectors [see previous chapter].

Specification in MENTAL
Geometric product of two vectors

The geometric product of two vectors, symbolized by "*·", is the sum of the inner product and the outer product of the two vectors:

⟨( v·w = (v */vi w) + (v */ve w) )⟩

The expressions */vi and */ve represent the inner and outer products of two vectors, respectively.


Ejemplo

(v =: v(a1 a2) // rep. (a1*e1 + a2*e2)

(w =: v(b1 b2)) // rep. (b1*e1 + b2*e2)

(v */vi w) // ev. (a1*b1 + a2*b2) (inner product)

(v */ve w) // ev. (a1*b2 - a2*b1)*e1*e2 (outer product)

v·w // ev. ((a1*b1 + a2*b2) + (a1*b2 - a2*b1)*e12)

where e12 = e1·e2 = e1*e2


Geometric objects in 1D space

In this space there are 2¹ = 2 basic geometric objects:

• The unit scalar (1), object of rank or dimension 0.
  
  
• e1, object of rank or dimension 1, which is the space generator.

The multiplication table (of the geometric product) is:

·	1	e1
1	1	e1
e1	e1	1

All geometric objects in 1D space are of the form (a + b*e1), where a and b are scalars (real numbers).

The object e1 is also called a "pseudoscalar" because multiplying it by itself produces a scalar.


Geometric objects in 2D space

In this space there are 2² = 4 basic (unitary) geometric objects:

• The unitary scalar (1), object of dimension 0.
  
  
• e1 and e2, which are the linear segments (objects of rank 1) generators of the space.
  
  
• e12 is a flat segment (rank 2 object). It is the highest rank object and is also called a pseudoscalar, because multiplying it by itself produces a scalar).

The multiplication table for these basic objects is:

·	1	e1	e2	e12
1	1	e1	e2	e12
e1	e1	1	e12	e2
e2	e2	−e12	1	−e1
e12	e12	−e2	e1	−1

Indeed, for example:

e1·e1 // ev. (e1 */vi e1) + (e1 */ve e1) ev. 1 + 0 ev. 1

e2·e1 // ev. (e2 */vi e1) + (e2 */ve e1) ev. 0−e12 ev. −e12

e1·e12 // ev. ev. e1·e1·e2 ev. 1·e2 ev. e2

e2·e12 // ev. e2·e1·e2 ev. −e2·e2·e1 ev. −1·e1 ev. −e1

e12·e12 // ev. e1·e2·e1·e1·e2 ev. −e2·e1·e1·e2 ev. -e2·e2 ev. −1

where (e12 =: e1·e2)

e12 behaves as the imaginary unit i: (e12·e12 = −1)

All geometric objects in 2D space are of the form
a + b*e1 + c*e2 + d*e12,
where a, b, c and d are scalars (real numbers).


Geometric objects in 3D space

In this space there are 2³ = 8 basic geometric objects:

• The unit scalar (1), object of dimension 0.
  
  
• e1, e2 and e3, which are linear segments (rank 1 object) and space generators.
  
  
• e12, e13 and e23 are flat segments (rank 2 objects).
  
  
• e123 is a volume segment (rank 3 object). It is the highest rank object and is also called pseudoscalar.

The multiplication table is:


being
⟨( (eij =: ei·ej) ← i≠j )⟩

All geometric objects in 3D space are linear combinations of the 8 basic objects.


Recursive generation of the basic objects

The basic objects of the space of n dimensions are generated from the space of n−1 dimensions by adding to them the product of themselves by in:

Dimension	Basic Objects		Total No.
	From previous level	Added
0		1	1
1	1	e1	2
2	1 e1	e2 e12	4
3	1 e1 e2 e12	e3 e13 e23 e123	8

For example, from 1 and e1, multiplying them by e2, we get e2 and e12.

In general, the formula is:

⟨( objs(n) = ( { {objs(n−1)↓ [objs(n−1)↓]·en]} ← n>0 →' {1}) )⟩

Examples:

objs(0) // ev. {1}
objs(1) // ev. {1 e1}
objs(2) // ev. {1 e1 e2 e2 e12}
objs(3) // ev. {1 e1 e2 e12 e3 e13 e23 e123}

Properties

1. Asociativa:
   ⟨( x·(y·z) = (x·y)·z )⟩
   
   
2. Distributive with respect to the vector sum:
   ⟨( x·(y +/v z) ≡ (x·y +/v x·z) )⟩
   
   
3. Multiplication by a scalar:
   
   Commutative:
   ⟨( r*x ≡ x*r )⟩
   
   Associative:
   ⟨( r1*(r2*x) ≡ (r1*r2)*x )⟩
   
   Unit element:
   ⟨( 1*x = x )⟩
   
   
4. If v is a vector in the space nD,
   v = v1*e1 + ... + vn*en
   then
   v·v = v1*v1 + ... + vn*vn

v2 = v12 + ... + vn2
   
   For example, in 2D space,
   
   
   (a*e1 + b*e2)·(a*e1 + b*e2) ev. (a*a + a*b*e12 + b*a*e21 + b*b) ev. (a*a + a*b*e12 − a*b*e12 + b*b) ev. (a*a + b*b)

Inner and outer products from the geometric product

Since:

x·y = (x */vi y) + (x */ve y)

y·x = (y */vi x) + (y */ve x) = (x */vi y) - (x */ve y)

it follows that the inner and outer product can be calculated from the geometric product:

⟨( (x */vi y) = (x·y + y·x)÷2 )⟩

⟨( (x */ve y) = (x·y − y·x)÷2 )⟩

Example:

(x = e1 + 3*e2)
(y = 6*e2 − 7*e3)

x·y // ev. 6*e12 - 7*e13 + 18 - 21*e23

y·x // ev. −6*e12 + 7*e13 + 18 + 21*e23

(x */vi y) // ev. 18 (inner product)

(x */ve y) // ev. 6*e12 - 7*e13 - 21*e23 (external product)

Inverse objects

The inverse object of an object x is another object inv(x) such that (x·inv(x) = 1) or (inv(x)·x = 1).

The division between two geometric objects x and y is x·inv(y).

The inverse objects of the basic objects in 1D space are:

Objeto	Inverso
1	1
e1	e1

In 2D space they are:

Objeto	Inverso
1	1
e1	e1
e2	e2
e12	−e12

And in 3D space:

Objeto	Inverso
1	1
e1	e1
e2	e2
e3	e3
e12	−e12
e13	−e13
e23	−e23
e123	−e123

Examples:

1. (x = (2 + 3*e1)) // 1D object
(inv(x) = −2÷5 + (3÷5)*e1) // inverse of x
x·inv(x) // ev. 1
   
   
2. (x = (2 + 3*e1 + 4*e2)) // 2D object
(inv(x) = −2÷21 + (3÷M21)*e1 + (4÷21)*e2) // inverse of x
x·inv(x) // ev. 1

Dual objects

There is symmetry in the number of objects of each rank in nD space:

Range	Number of objects
0 (scalar)	comb(n, 0) = 1
1 (linear segments)	comb(n, 1) = n
2(flat segments)	comb(n, 2)
...	...
n−2	comb(n, n−2) = comb(n, 2)
n−1	comb(n, n−1) = comb(n, 1) = n
n (pseudoscale)	comb(n, n) = 1

where comb(n, k) are the combinations of n elements taken from k in k. For example, for n = 3: 1, 3, 3, 3, 1. And for n = 4: 1, 4, 6, 4, 4, 1.

That is, there are the same number of objects from the minimum and maximum range, from the range next to the minimum and before the maximum, etc.

The dual operation allows to relate objects of rank r to those of the complementary rank n−r and is defined as follows: the dual object of an object x is the object dual(x) such that x·dual(x) is the geometric object of maximum rank.

In the case of 3D space, (x·dual(x) = e123)

Multiplying both sides by inv(x), we have:
(dual(x) = inv(x)·e123)

The dual objects of the basic objects in 3D space are:

Objet	Dual
1	e123
e1	e23
e2	−e13
e3	e12
e12	e3
e13	−e2
e23	e1
e123	1

Properties:

1. ⟨( x·dual(x) = (e 1...n) )⟩ // for every object of nD
⟨( x·dual(x) = e123 )⟩ // for every 3D object
   
   
2. ⟨( dual(x) = inv(x)·(e 1...n) )⟩ // for every nD object
⟨( dual(x) = inv(x)·e123 )⟩ // for every 3D object
   
   
3. ⟨(dual(dual(x)) = x )⟩

Example:

(x = (2 + 3*e1 + 4*e2))

(inv(x) = &minis;221 + (3÷21)*e1 + (4÷21)*e2) // inverse of x

dual(x) = inv(x)·e12 // ev. -(2÷21)*e12 + (3÷21)*e2 - (4÷21)*e1)

x·dual(x) // ev. e12

Reverse of an object

The reverse of an object of order 0 or 1 is the same object. The reverse of an object of the form x·y is y·x.

Definition: ⟨( (x·y)∼ = y·x )⟩

To obtain the reverse expression of the basic objects, just specify the indices of the objects of rank greater than or equal to 2 in reverse order (which is equivalent to changing their sign).

Objet	Reverse
1	e123
e1	e23
e2	−e13
e3	e12
e12	e21 = −e12
e13	e31 = −e13
e23	e32 = −e23
e123	e321 = −e123

Example:

(x = (3 + 7*e2 − 6*e13 + e123))
x∼ // ev. (3 + 7*e2 + 6*e13 − e123)

By definition. the norm of an object x (generalization of the modulus of a vector) is the square root of the scalar part of x·(x∼). For example:

(x = 3*e1 + 4*e12)
(x∼ = 3*e1 − 4*e12)
x·(x∼) // ev. 25 + 24*e2 (standard is 5)

Idempotence and nullpotence

An idempotent geometric object is one that has the property: (x·x = x). Examples:

1. (x =: (1÷2)*(1+e1))
x·x // ev. x
   
   
2. (x =: (1÷)*(1+e13))
x·x // ev. x

A nulpotent geometric object is one that has the property: (x·x = 0). Example:

(x = e2·(1+e1))
x·x // ev. 0

Complex numbers as geometric objects

We have seen that the basic object e12 in 2D space behaves as the imaginary unit. Therefore, in 2D, geometric algebra reproduces the properties of complex numbers, but referring only to geometric objects.

• The object of degree 2 (z = (a + b*e12)) is equivalent to the complex number (z = (a + b*i)), where i is the imaginary unit (i*i = −1).
  
  
• The inverse of the geometric object z is the conjugate complex number:
  
  (a + b*e12)∼ = (a + b*e21) = (a - b*e12)
  
  
• When multiplying by e12 a linear object, the result is the same object rotated 90º (counterclockwise). If the multiplication is counterclockwise, the rotation is clockwise:
  
  
  (z = a*e1 + b*e2)

z·e12 // ev. (a*e1 + b*e2)·e12 ev. a*e2 − b*e1

e12·z // ev. e12·(a*e1 + b*e2) ev. −a*e2 + b*e1

Alternative Clifford algebras

The Clifford algebra presented so far is one of the possible ones that can be defined. The parameterized Clifford algebra, symbolized by Cl(p, q), on the space nD, where p+q = n represents the algebra in which p basic linear objects (basic vectors) ei fulfill e_i² = +1 and q basic vectors e_j meet e_j² = −1 .

The geometric algebra studied so far is Cl(n, 0), where n is the dimension of space.

• Cl(0, 0) is the algebra of the real numbers.
• Cl(0, 1) is the algebra of complex numbers.
• Cl(0, 3) is the algebra of Hamilton quaternions.

Advantages of geometric algebra

Geometric algebra has great advantages over vector algebra:

• It unifies and generalizes multiple concepts: vectors, tensors, matrices, complex numbers, quaternions, as well as different formal algebras: synthetic, projective, Pauli, Dirac and spinor.
  
  
• It is more powerful. Much of its power lies in its operational possibilities with geometric objects.
  
  
• It reduces complexity, since it allows to express physical and geometric magnitudes and relationships in a conceptual, compact and efficient way.
  
  
• Allows to represent any geometric object in space as a linear combination of the basic objects.
  
  
• It does not necessarily require a representation system or the use of coordinates.
  
  
• Allows to operate with objects of different dimensions.
  
  
• Encourages the discovery of new mathematical structures.

In addition, geometric algebra is linked to consciousness by linking the two dual operations of scalar product and vector product through the sum of both operations.

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Addenda
A brief history of geometric algebras

The concept of geometric algebra dates back to 1797 when Caspar Wessel interpreted the imaginary unit as a unit element of another dimension, representing it perpendicular to the real line. From this idea arose the complex numbers, with two dimensions (the real and the imaginary).

In 1843, William Rowen Hamilton presented the algebra of quaternions, a generalization of complex numbers for 4 dimensions. This algebra contains 4 elements (1, i, j, k) with the properties: i² = j² = k² = ijk = −1. Quaternions were the germ of the vector concept and have been useful for representing rotations in 3D space.

In 1844, Hermann Grassmann published "Theory of Linear Extension", a theory of extensive geometric quantities.

In 1878, Clifford unified and generalized Hamilton's and Grassmann's algebras, presenting his own geometric algebra. Because it is the most generic geometric algebra, Clifford's algebra is simply called "geometric algebra".

In the 1880s, Gibbs presented his vector algebra, which because of its clarity and simplicity was a great success, leaving Clifford's algebra in oblivion.

In 1920, Clifford's algebra resurfaced as the underlying algebra of quantum spin.

In the 1960s David Hestenes promoted and popularized Clifford algebra as a unified language for mathematics, physics and engineering. It has now gained great importance as an alternative to vector, matrix and tensor algebra. It is being used in numerous areas: cosmology, quantum and electromagnetic physics, CAD (computer-aided design), computer vision, robotics, neural networks, etc.

Clifford algebra is especially useful in theoretical physics, since many of its formulations have a geometric interpretation, providing clearer and more understandable conceptual models and generalizing vector, tensor and matrix expressions. In addition, the complex numbers that appear in the formulations also have a natural geometric interpretation. Some examples are:

• Electromagnetism. Maxwell's equations, which in vector notation are four equations, are reduced to a single Clifford algebra. In this equation appears a complete electromagnetic vector, that is, with its components (electric and magnetic) integrated.
  
  
• Theory of relativity. The space-time geometry of relativity theory can be described with 3D geometry more simply than with 4D geometry. This is achieved by defining the geometric product with e₁*e₁ = −1 instead of e ₁*e₁ = 1, since e₁ behaves as a temporal dimension instead of a spatial one.
  
  
• Quantum physics. Clifford algebras are very useful for the definition of spinors. Spinors made their appearance with the famous Dirac equation in 1928. A spinor is an imaginary object that becomes its negative when it undergoes a complete rotation (2π). A rotation of 4π returns the object to its original state.

Bibliography

• Clifford Algebra Society: www.clifford.org.
  
  
• Clifford, William Kingdon (autor); Stephen, Leslie (ed.); Pollock, Frederick (ed.). Lectures and Essays. Cambridge University Press, 2011.
  
  
• Doren, Chris; Lasenby, Anthony. Geometric Álgebra for Physicists. Cambridge University Press, 2003.
  
  
• Dorst, Leo; Fontijne, Daniel; Mann, Stephen. Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann, 2007.
  
  
• Hartshorne, Robin. Algebraic Geometry. Springer, 1997.
  
  
• Hestenes, David. A Unified Language for Mathematics and Physics. Internet.
  
  
• Hestenes, David. Space-Time Algebra. Gordon & Breach, New York, 1967.
  
  
• Hestenes, David y Sobczyk, Garret. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Springer, 1987. ). (Es muy teórico. Cubre el álgebra y cálculo de multivectores de cualquier dimensión).
  
  
• Hestenes, David. New Foundations for Classical Mechanics. Springer, 1999. (Proporciona buenos ejemplos sobre la aplicación del álgebra geométrica a la mecánica de los cuerpos sólidos).
  
  
• Jancewicz, Bernard. Multivectors and Clifford Algebra in Electrodynamics. World Scientific Pub. Co. Inc., 1989. (Incluye un capítulo introductorio dedicado a los multivectores y al álgebra Clifford en el espacio 3D).
  
  
• Lasenby, Joan; Lasenby, Anthony N. Doran; Chris, J.L. A unified mathematical language for phisics and engineering in the 21st century. Phil. Trans. Royal Society, London. A, 358, pp. 21-39, 2005. Disponible online. (Es una introducción al álgebra geométrica.)
  
  
• Snygg, John. Clifford Algebra: A Computational Tool for Physicists. Oxford University Press, 1997.