{"id":27894,"date":"2020-12-01T16:35:19","date_gmt":"2020-12-01T15:35:19","guid":{"rendered":"http:\/\/kmr.dialectica.se\/wp\/?page_id=27894"},"modified":"2022-10-18T11:37:05","modified_gmt":"2022-10-18T09:37:05","slug":"the-history-of-geometric-algebra","status":"publish","type":"page","link":"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/algebra\/geomtric-algebra\/the-history-of-geometric-algebra\/","title":{"rendered":"History and Properties of Geometric Algebra"},"content":{"rendered":"

This page is a sub-page of our page on Geometric Algebra<\/a><\/p>\n

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Sub-pages of this page<\/strong>:<\/p>\n

\u2022 Origins of Geometric Algebra<\/a><\/p>\n

\/\/\/\/\/\/\/<\/p>\n

Related KMR-pages<\/strong>:<\/p>\n

\u2022 Origins of Geometric Algebra<\/a><\/p>\n

\u2022 Clifford Algebra<\/a>
\n\u2022 Geometric Algebra<\/a>
\n\u2022 Complex Numbers<\/a>
\n\u2022 Quaternions<\/a><\/p>\n

\/\/\/\/\/\/\/<\/p>\n

Books<\/strong>:<\/p>\n

\u2022 Ren\u00e9 Descartes (1637), La G\u00e9ometrie<\/a>
\n\u2022 Hermann G\u00fcnther Gra\u00dfmann<\/a> (1844), Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik<\/a>
\n\u2022 A New Branch Of Mathematics – The Ausdehnungslehre of 1844 and Other Works<\/a>, translated by Lloyd C. Kannenberg (1995)
\n\u2022 William Kingdon Clifford (1878), Applications of Grassmann’s Extensive Algebra<\/a>, American Journal of Mathematics Vol. 1, No. 4 (1878), pp. 350-358 (9 pages), Published by: The Johns Hopkins University Press
\n\u2022 David Hestenes<\/a> (1993, (1986)), New Foundations for Classical Mechanics<\/a><\/p>\n

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Other relevant sources of information<\/strong>:<\/p>\n

\u2022 Associative Composition Algebra<\/a> (at Wikibooks)
\n\u2022 Quaternion<\/a> (at Wikipedia)
\n\u2022 Rotor<\/a> (at Wikipedia)
\n\u2022 Versor<\/a> (at Wikipedia)
\n\u2022 Hyperbolic versor<\/a> (at Wikipedia)
\n\u2022 Rotation formalisms in three dimensions<\/a> (at Wikipedia)
\n\u2022 Elias Riedel G\u00e5rding (2017), Geometric algebra, conformal geometry and the common curves problem<\/a>, Bachelors Thesis at the Royal Institute of Technology, Stockholm, Sweden<\/p>\n

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The interactive simulations on this page can be navigated with the Free Viewer<\/a>
\nof the Graphing Calculator<\/a>.<\/p>\n

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Anchors into the text below<\/strong>: ????????<\/p>\n

\u201eThe World Health Organization has announced a world-wide epidemic of the Coordinate Virus in mathematics and physics courses at all grade levels. Students infected with the virus exhibit compulsive vector avoidance behavior, unable to conceive of a vector except as a list of numbers, and seizing every opportunity to replace vectors by coordinates. At least two thirds of physics graduate students are severely infected by the virus, and half of those may be permanently damaged so they will never recover. The most promising treatment is a strong dose of Geometric Algebra\u201c. (Hestenes)<\/p>\n

From the paper
\nMultiplication_of_vectors_and_structure_of_3D-Euclidean_space.pdf<\/a>
\nby Miroslav Josipovic\u0301, Zagreb, 2017.<\/p>\n

A brief history of geometric algebra<\/strong><\/p>\n

The ancient greeks were in possession of the beginnings of a form of geometric algebra<\/a>, namely the embryo of an exterior algebra<\/a>. In terms of multiplication of units, it worked like this:<\/p>\n

\\, l_{ength} \\times l_{ength} \\equiv \\, a_{rea} <\/span>,<\/p>\n

\\, l_{ength} \\times l_{ength} \\times l_{ength} \\, \\equiv \\, v_{olume} <\/span>,<\/p>\n

\\, l_{ength} \\times l_{ength} \\times l_{ength} \\times l_{enth} \\equiv n_{othing} <\/span>,<\/p>\n

since there were no more than three dimensions at that time. The idea of the existence of higher dimensions was hindered by a very powerful and concrete “reality block” that was impossible to overcome with the extremely clumsy and difficult-to-handle algebraic notation that the greeks had access to.<\/p>\n

A fundamentally important contribution to removing this mental roadblock was provided by Ren\u00e9 Descartes<\/a> in his book La G\u00e9ometrie<\/a> from 1637. As brilliantly described<\/a> by David Hestenes<\/a> in his book New Foundations for Classical Mechanics<\/a>, Descartes introduced a cleverly chosen binary operation<\/a> so that \\, l_{ength} \\times l_{ength} \\, <\/span> became equal to<\/em> \\, l_{ength} <\/span>.<\/p>\n

The person that threw open “the algebraic flood gates” to higher dimensions was a German high-school teacher in Stettin<\/a> by the name of Hermann G\u00fcnther Gra\u00dfmann<\/a>. In 1844 he published his masterpiece Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik<\/em> [A New Branch Of Mathematics – The Ausdehnungslehre of 1844 and Other Works<\/a>, translated by Lloyd C. Kannenberg (1995)]. In English, this book is most often denoted by “Extension Theory” or “Theory of Extensions.”<\/p>\n

In his Extension Theory – which nobody read at the time because it was written in an obscure philosophical style that no mathematicians were accustomed to – Gra\u00dfmann introduced the idea of abstract vector spaces<\/a> with n-dimensional algebra and geometry<\/a>. In so doing, he created the domains of mathematics that we today refer to as linear algebra<\/a> and exterior algebra<\/a> (also known as multilinear algebra<\/a>).<\/p>\n

In 1878, William Kingdon Clifford<\/a>, who tragically died the same year at the age of only 34, published the basics of an algebra, which today is called clifford algebra<\/a> (but which he himself referred to as geometric algebra<\/a>), and which builds upon the exterior algebra of Gra\u00dfmann.<\/p>\n

\/\/\/\/\/\/\/ Quoting Wikipedia<\/strong> on William Kingdon Clifford:<\/p>\n

In 1878 Clifford published a seminal work, Applications of Grassmann’s extensive algebra<\/a>, building on Grassmann’s algebraic work. He had succeeded in unifying the quaternions<\/a>, developed by William Rowan Hamilton<\/a>, with Grassmann’s outer product (also known as the exterior product<\/a>). Clifford understood the geometric nature of Grassmann’s creation, and that the quaternions fit cleanly into the algebra Grassmann<\/a> had developed. The versors<\/a> in quaternions facilitate representation of rotation<\/a>.<\/p>\n

Clifford laid the foundation for a geometric product<\/a>, composed of the sum of the inner product<\/a> and Grassmann’s outer product<\/a>. The geometric product was eventually formalized by the Hungarian mathematician Marcel Riesz<\/a>. The inner product<\/a> equips geometric algebra with a metric<\/a>, fully incorporating distance<\/a> and angle<\/a> relationships for lines, planes, and volumes, while the outer product<\/a> gives those planes and volumes vector<\/a>-like properties, including a directional sensitivity<\/a>.<\/p>\n

Combining the two brought the operation of division into play. This greatly expanded our qualitative understanding of how objects interact in space. Crucially, it also provided the means for quantitatively calculating the spatial consequences of those interactions.<\/em> The resulting geometric algebra<\/a>, as Clifford called it, eventually realized the long sought goal[13] of creating an algebra that faithfully represents the movements and projections of objects in \\, 3 <\/span>-dimensional space.[14]<\/p>\n

Moreover, Clifford’s algebraic schema extends to higher dimensions<\/strong>. The algebraic operations have the same symbolic form as they do in 2 or 3-dimensions. The importance of general Clifford algebras has grown over time, while their isomorphism<\/a> classes – as real algebras – have been identified in other mathematical systems beyond simply the quaternions<\/a>.<\/p>\n

\/\/\/\/\/\/\/ End of quote from Wikipedia<\/strong> (on William Kingdon Clifford)<\/p>\n

The definition of the geometric product<\/strong><\/p>\n

The geometric product<\/strong> of two vectors \\, a \\, <\/span> and \\, b \\, <\/span> is defined as the direct sum<\/a>
\nof their inner product<\/a> and their exterior product<\/a>:<\/p>\n

\\, a \\, b \\stackrel {\\mathrm{def}}{=} a \\cdot b + a \\wedge b <\/span>.<\/p>\n

By making use of the distributive property of the geometric product (see below), this definition can be extended to products of multivectors.<\/p>\n

Terminology<\/strong>: Sums of products or products of sums (i.e., polynomials<\/a>) of vectors from the same vector space are called multivectors<\/a>.<\/p>\n

Some properties of the geometric product<\/strong><\/p>\n

The geometric product of two vectors \\, a \\, <\/span> and \\, b \\, <\/span> “encodes” both their “degree of parallelity” and their “degree of orthogonality”:<\/p>\n

Lemma 1<\/strong>: \\, a \\, b = a \\cdot b \\, <\/span> if and only if the vectors are parallel.<\/p>\n

Lemma 2<\/strong>: \\, a \\, b = a \\wedge b \\, <\/span> if and only if the vectors are perpendicular.<\/p>\n

Lemma 3<\/strong>: The inner product is symmetric: \\, a \\cdot b = b \\cdot a <\/span>,
\nand the exterior product is antisymmetric: \\, a \\wedge b = - b \\wedge a \\, <\/span>.<\/p>\n

Therefore we have:<\/p>\n

\\, a \\cdot b = \\frac {1}{2} (a b + b a) <\/span>, and<\/p>\n

\\, a \\wedge b = \\frac {1}{2} (a b - b a) <\/span>.<\/p>\n

Hence the geometric product of two vectors is naturally separated
\ninto a symmetric and an antisymmetric part:<\/p>\n

\\, a \\, b = \\frac {1}{2} (a b + b a) + \\frac {1}{2} (a b - b a) <\/span>.<\/p>\n

\/\/\/\/\/\/\/<\/p>\n

The geometric product is associative<\/a>, i.e., it satisfies \\, (a \\, b) \\, c \\, = \\, a \\, (b \\, c) <\/span>.<\/p>\n

The geometric product is distributive<\/a> over multivector addition (both from the left and from the right).<\/p>\n

Let \\, b = b_{ \\, \\shortparallel \\, a} + b_{ \\perp a} \\, <\/span> denote the splitting of the vector \\, b \\, <\/span> into two components that are parallel respectively perpendicular to the vector \\, a <\/span>.<\/p>\n

The distributivity from the left gives:<\/p>\n

\\, a \\, b \\, = <\/span>
\n = \\, a \\, ( b_{ \\, \\shortparallel \\, a} + b_{ \\perp a} ) \\, = <\/span>
\n = \\, a \\, b_{\\, \\shortparallel \\, a} \\, + \\, a \\, b_{ \\perp a} \\, = <\/span>
\n = \\, a \\cdot b_{\\, \\shortparallel \\, a} + a \\wedge b_{\\, \\shortparallel \\, a} + a \\cdot b_{ \\perp a} + a \\wedge b_{ \\perp a} \\, = <\/span>
\n = \\, a \\cdot b_{\\, \\shortparallel \\, a} + a \\wedge b_{ \\perp a} \\, = <\/span>
\n = \\, a \\cdot b + a \\wedge b <\/span>,<\/p>\n

as it should be.<\/p>\n

With \\, a = a_{ \\, \\shortparallel \\, b} + a_{ \\perp b} <\/span>, the distributivity from the right gives:<\/p>\n

\\, a \\, b \\, = <\/span>
\n = \\, (a_{ \\, \\shortparallel \\, b} + a_{ \\perp b}) \\, b \\, = <\/span>
\n = \\, a_{ \\, \\shortparallel \\, b} \\, b + a_{ \\perp b} \\, b \\, = <\/span>
\n = \\, a_{ \\, \\shortparallel \\, b} \\cdot b + a_{ \\, \\shortparallel \\, b} \\wedge b + a_{ \\perp b} \\cdot b + a_{ \\perp b} \\wedge b \\, = \\, <\/span>
\n = \\, a_{ \\, \\shortparallel \\, b} \\cdot b + a_{ \\perp b} \\wedge b \\, = <\/span>
\n = \\, a \\cdot b + a \\wedge b <\/span>,<\/p>\n

as it should be.<\/p>\n

\/\/\/\/\/\/\/<\/p>\n

SOME EXAMPLES OF WHAT A DIRECTED MAGNITUDE CAN ACHIEVE<\/strong><\/p>\n

\/\/\/\/\/\/\/<\/p>\n


\nDirected line segments<\/a><\/p>\n

A directed line segment can point out an end point relative to a given start point<\/strong><\/p>\n

If the start point is regarded as the “zero-point” one gets a Euclidean vector which is a so-called direction vector<\/a>. If the start point is included one gets an affine<\/a> vector, also called a position vector<\/a>.<\/p>\n

\/\/\/\/\/\/\/<\/p>\n

A directed planar area<\/a> can solve the equation \\, x^2 \\, = \\, -1 \\, <\/span> without the use of “imaginary” quantities<\/strong><\/p>\n

NOTE<\/strong>: The military solves the equation \\, x^2 \\, = \\, -1 \\, <\/span> without the use of imaginary quantities. Here is how they do it:<\/p>\n

\\, { \\text {left}}_{ \\text {turn}} \\; { \\text {left}}_{ \\text {turn}} \\; {\\text {start}}_{ \\text {direction}} \\, = \\, { \\text {left}}_{ \\text {turn}}^2 \\; {\\text {start}}_{ \\text {direction}} \\, = \\, { \\text{half} }_{ \\text {turn}} \\; {\\text {start}}_{ \\text {direction}} \\, = \\, - \\, {\\text {start}}_{ \\text {direction}} <\/span>,<\/p>\n

\\, { \\text {right}}_{ \\text {turn}} \\; { \\text {right}}_{ \\text {turn}} \\; {\\text {start}}_{ \\text {direction}} \\, = \\, { \\text {right}}_{ \\text {turn}}^2 \\; {\\text {start}}_{ \\text {direction}} \\, = \\, { \\text{half} }_{ \\text {turn}} \\; {\\text {start}}_{ \\text {direction}} \\, = \\, - \\, {\\text {start}}_{ \\text {direction}} <\/span>.<\/p>\n

Hence the operations \\, { \\text {left}}_{ \\text {turn}} \\, <\/span> and \\, { \\text {right}}_{ \\text {turn}} \\, <\/span> perform the respective functions of the imaginary entities \\, i \\, <\/span> and \\, -i \\, <\/span> in the algebra of complex numbers<\/a>.<\/p>\n

In fact, we can easily introduce an infinite number of “imaginary units” in clifford algebra:<\/p>\n

Let \\, \\mathcal {R} \\, <\/span> be a commutative ring<\/a> with unit<\/a> and let E <\/span> be a finite, totally ordered <\/a>set, i.e, E = \\{e_1, \\ldots, e_n\\} <\/span> where e_1 < e_2 < \\ldots < e_n. <\/span><\/p>\n

We define what is called \\, k <\/span>–base blades<\/a> \\, e_{n_1} e_{n_2} \\ldots e_{n_k}, \\, <\/span> where \\, n_1 < n_2 < \\ldots < n_k \\leq n, \\, <\/span> which we identify with the k <\/span>-subsets \\, \\{e_{n_1}, \\ldots, e_{n_k}\\} \\subseteq E.<\/span> Moreover, we define the unit pseudoscalar<\/a> \\, e_1 e_2 \\ldots e_n \\, <\/span> which we identify with the set \\, E \\, <\/span> by an unproblematic change of context. Finally, we identify the ring unit \\, 1 \\, <\/span> with the empty set \\, \\emptyset <\/span>.<\/p>\n

Then we have \\, (e_i e_j)^2 = -1 \\, <\/span> as soon as \\, i \u2260 j<\/span>. In fact, as is easily seen, the geometric product of any pair of perpendicular vectors of unit length is an imaginary unit, since it squares to \\, -1 <\/span>.<\/p>\n

We regard the Clifford Algebra \\, C_l(E) \\, <\/span> as the free \\mathcal {R} <\/span>-module<\/a> genererated by the powerset<\/a> \\, \\wp(E) \\, <\/span> of all subsets of \\, E <\/span>, i.e.,<\/p>\n

\\, C_l(E) \\, = \\, {\\oplus \\atop {e \\, \\in \\, \\wp(E) } } \\mathcal {R} <\/span>.<\/p>\n

For the multiplication in \\, C_l(E) \\, <\/span> we introduce, for the base elements, the following rules:<\/p>\n

\\, e_i^2 \\, = \\, 1 \\, <\/span> for \\, i = 1, ..., n <\/span>, and \\; e_j \\, e_i \\, = \\, - e_i \\, e_j \\, <\/span> if \\, i \\neq j <\/span>.<\/p>\n

These “basic” rules are then extended distributively to all multivectors of \\, C_l(E) <\/span>.<\/p>\n

A proof of the fact that these extended rules create a well-defined clifford algebra can be found in the appendix to the book-chapter:<\/p>\n

\u2022 Naeve, A., Svensson, L. (2001): Geo-MAP unification<\/em><\/a>, in Geometric Computing with Clifford Algebras – Theoretical Foundations and Applications in Computer Vision and Robotics<\/em>, Sommer, G. (ed.), pp. 105-126, Springer Verlag, ISBN 3-540-41198-4.<\/p>\n

If \\, i \\neq j \\, <\/span> we therefore obtain:<\/p>\n

\\,( e_i \\, e_j)^2 \\, = \\, e_i \\, e_j \\, e_i \\, e_j \\, = \\, - e_j \\, e_i \\, e_i \\, e_j \\, = \\, - e_j \\, e_j \\, = \\, -1 <\/span>.<\/p>\n

Hence, \\, e_i \\, e_j \\, <\/span> solves the equation \\, x^2 \\, = \\, -1 <\/span>. Each such bivector<\/a> therefore functions as an imaginary unit.<\/p>\n

\/\/\/\/\/\/\/<\/p>\n

The complex numbers represented as the even subalgebra<\/a><\/em>
\nof the clifford Algebra<\/a> \\, C_l(e_1, e_2) \\, <\/span> over the real numbers \\, \\mathbb{R} <\/span><\/strong>:<\/p>\n

\\, e_1, e_2 \\, <\/span>,<\/p>\n

\\, e_1^2 = e_2^2 = 1 \\, <\/span>,<\/p>\n

\\, e_2 e_1 = - e_1 e_2 \\, <\/span>.<\/p>\n

Hence we have: \\, (e_1 e_2)^2 = e_1 e_2 e_1 e_2 = - e_2 e_1 e_1 e_2 = - e_2 e_2 = -1 <\/span>.<\/p>\n

\/\/\/\/\/\/\/<\/p>\n

Multiplying two 1-vectors in the Clifford algebra \\, C_l(e_1, e_2) \\, <\/span> <\/strong>:<\/p>\n \\, (\\alpha_1 e_1 + \\alpha_2 e_2) (\\alpha'_1 e_1 + \\alpha'_2 e_2) = <\/span>\n \\, = \\alpha_1 e_1 \\alpha'_1 e_1 + \\alpha_1 e_1 \\alpha'_2 e_2 + \\alpha_2 e_2 \\alpha'_1 e_1 + \\alpha_2 e_2 \\alpha'_2 e_2 \\, = <\/span>\n = \\, \\alpha_1 \\alpha'_1 e_1 e_1 + \\alpha_2 \\alpha'_2 e_2 e_2 + \\alpha_1 \\alpha'_2 e_1 e_2 + \\alpha_2 \\alpha'_1 e_2 e_1 \\, = <\/span>\n = \\, \\alpha_1 \\alpha'_1 + \\alpha_2 \\alpha'_2 + \\alpha_1 \\alpha'_2 e_1 e_2 - \\alpha_2 \\alpha'_1 e_1 e_2 \\, = <\/span>\n

= \\, \\alpha_1 \\alpha'_1 + \\alpha_2 \\alpha'_2 + (\\alpha_1 \\alpha'_2 - \\alpha_2 \\alpha'_1) e_1 e_2 <\/span>.<\/p>\n

Multiplication in the even part of the Clifford algebra \\, C_l(e_1, e_2) \\, <\/span><\/strong>:<\/p>\n \\, (x + y e_1 e_2) (x' + y' e_1 e_2) \\, = <\/span>\n = \\, x x' + x y' e_1 e_2 + y e_1 e_2 x' + y e_1 e_2 y' e_1 e_2 \\, = <\/span>\n = \\, x x' + x y' e_1 e_2 + y x' e_1 e_2 + y y' e_1 e_2 e_1 e_2 \\, = <\/span>\n = \\, x x' + x y' e_1 e_2 + y x' e_1 e_2 - y y' e_2 e_1 e_1 e_2 \\, = <\/span>\n

= \\, x x' - y y' + (x y' + y x') e_1 e_2 <\/span>.<\/p>\n

Multiplication in the algebra of complex numbers<\/strong>:<\/p>\n \\, (x + iy ) (x' + iy') \\, = <\/span>\n = \\, x x' + i x y' + i y x' + i^2 y y' \\, = <\/span>\n

= \\, (x x' - y y') + i (x y' + y x') <\/span>.<\/p>\n

\/\/\/\/\/\/\/<\/p>\n

A directed volume can liberate the Jacobi determinant<\/a> from its surrounding modulus “straight jacket” when substituting variables in multiple integrals<\/a><\/strong>:<\/p>\n

The dirty little secret<\/a> of the multiple Riemann integral<\/a>.<\/p>\n

\/\/\/\/\/\/\/<\/p>\n

The outer product<\/a> can liberate the cross product<\/a> from its 3-dimensional “straight jacket”<\/strong><\/p>\n

\\, a \\times b \\, \\stackrel {\\mathrm{def}}{=} \\, (a \\wedge b) \\, I^{-1} <\/span>,<\/p>\n

where \\, I = \\, e_1 e_2 \\cdots e_n \\, <\/span> is the unit pseudoscalar<\/a> of \\, C_l(e_1, e_2, \\cdots, e_n) \\, <\/span>.<\/p>\n

\/\/\/\/\/\/\/<\/p>\n

Geometric operations can be carried out in a coordinate-free manner and have the same form in any dimension<\/strong><\/p>\n

Example<\/strong>:<\/p>\n

The algebraic structure of a reflection in a hyperplane<\/a><\/strong>:<\/p>\n

Let \\, \\pi \\, <\/span> be a hyperplane in \\, {\\mathbb{R}}^3 \\, <\/span> with the unit normal \\, n <\/span>.<\/p>\n

NOTE<\/strong>: In the plane, a hyperplane is a 1D-line, and in 3D-space a hyperplane is a 2D-plane.<\/p>\n

Theorem<\/strong>: A reflection \\, S_{\\pi} \\, <\/span> of the vector \\, x \\, <\/span> in the hyperplane \\, \\pi \\, <\/span> with unit normal \\, n \\, <\/span>
\ncan be expressed as \\, {\\mathbb{R}}^3 \\ni x \\mapsto - \\, n \\, x \\, n \\, = \\, \\bar{x} = S_{\\pi}(x) \\in {\\mathbb{R}}^3 <\/span>.<\/p>\n

\"Reflection<\/a>
\nProof<\/strong>: \\, \\bar{x} \\, = \\, - \\, n \\, x \\, n \\, = \\, - \\, n \\, (x_{\\, \\shortparallel \\, n} + x_{\\perp n}) \\, n \\, = \\, - \\, n \\, x_{\\, \\shortparallel \\, n} \\, n - \\, n \\, x_{\\perp n} \\, n \\, = - \\, n \\, n \\, x_{\\, \\shortparallel \\, n} + n \\, n \\, x_{\\perp n} \\, = \\, - \\, x_{\\, \\shortparallel \\, n} + x_{\\perp n} \\, = \\, S_{\\pi} (x) \\, <\/span> .\u220e<\/p>\n