Geometric Algebra

This page is a sub-page of our page on Algebra.


The sub-pages of this page are:


Related KMR-pages:

New Foundations for Classical Mechanics by David Hestenes


Other related sources of information:

Geometric Algebra: New Foundations, New Insights,
advanced course at SIGGRAPH 2000.
A Swift Introduction to Geometric Algebra, by sudgylacmoe on YouTube.
Addendum to A Swift Introduction to Geometric Algebra, by sudgylacmoe on YouTube.
A Swift Introduction to SpaceTime Algebra, by sudgylacmoe on YouTube.


/////// Formulas from Ambjörn’s presentation at SIGGRAPH 2000 and 2001.

\, \mathrm{V}^n \, , \, e_1 \, , e_2 \, , \dots , e_n \, \, \mathrm{G} = \mathrm{G}_n = \mathrm{G}(\mathrm{V}^n) \, \, M = \sum\limits_{k = 0}^{n} {\langle \, M \, \rangle}_k \, \, {\langle \, M \, \rangle}_k = B_1 + B_2 + \cdots \, \, A_k = a_1 \wedge a_2 \wedge \cdots \wedge a_k \, \, A_k \neq 0 \, \iff \, \{ a_1, a_2, \dots , a_k \} \, \, P \in \mathrm{G}_n \, \, P = p_1 \wedge p_2 \wedge \cdots \wedge p_n \, \, I = e_1 \wedge e_2 \wedge \cdots \wedge e_n \, \, [P] = PI^{-1} \, \, \text{dual}(X) = XI^{-1} \, \, \text{dual}(X) = X^* \, \, B = b_1 \wedge b_2 \wedge \cdots \wedge b_m \, \, \overline{B} \subseteq \mathrm{V}^n \, \, \overline{B} = \text{Linspan}\{ b_1, b_2, \dots , b_m \} = \, \, = \text{Linspan}\{ b \in \mathrm{G}_n : b \wedge B = 0 \} \, \, \{ e_1, \dots , e_m \} \, \, b_i = \sum\limits_{k = 0}^{m} b_{ik}e_k \; \text{for} \; i = 1, \, \dots \, , m \; , \, \, B =( \det{b_{ik}}) \, e_1 \wedge e_2 \wedge \cdots \wedge e_m = \, \, = (\det{b_{ik}}) \, e_1 e_2 \cdots e_m = \,


The sub-pages of this page are:

• Geometric Numbers in the Line and in the Plane
Geometric Numbers in Euclidean 3D-space
Geometriska tal i den Euklidiska 3D-rymden
Blades in Geometric Algebra
The Evolution of Geometric Arithmetic
The Vector Algebra War
Historical Remarks
History and Properties of Geometric Algebra


Related KMR pages:

Clifford Algebra
Combinatorial Clifford Algebra
David Hestenes.
David Hestenes on Geometric Algebra.
David Hestenes on Geometric Calculus.
David Hestenes on Modeling.


Metric Geometry
Affine Geometry
Projective Geometry
Differential Geometry


Papers and conference presentations by KTH:

• Ambjörn Naeve and Lars Svensson, (2001), Geo-MAP unification, in Geometric Computing with Clifford Algebras – Theoretical Foundations and Applications in Computer Vision and Robotics, Sommer, G. (ed.), pp. 105-126, Springer Verlag, ISBN 3-540-41198-4.
• Ambjörn Naeve and Lars Svensson, (1999), Projective Geometric Computing, The 5:th International Conference on Clifford Algebras and their Applications in Mathematical Physics, Ixtapa-Zihuatanejo, Mexico, June 27-July 4, 1999.
• Ambjörn Naeve and Lars Svensson, (1999), Discrete Integration and the Fundamental Theorem, The 5:th International Conference on Clifford Algebras and their Applications in Mathematical Physics, Ixtapa-Zihuatanejo, Mexico, June 27-July 4, 1999.
• Lars Svensson and Ambjörn Naeve, (2002), Combinatorial Aspects of Clifford Algebra, presented at the International Workshop on Applications of Geometric Algebra, Cambridge, 5-6 Sept. 2002.
• Douglas Lundholm and Lars Svensson, (2009), Clifford algebra, geometric algebra, and applications.

Papers by others:

Applications of Grassmann’s Extensive Algebra, by William Kingdon Clifford, American Journal of Mathematics, Vol. 1, No. 4 (1878), pp. 350-358.
Articulating Space : Geometric Algebra for Parametric Design Symmetry, Kinematics and Curvature, Dissertation thesis by Pablo Colapinto, March 2016.
Geometric and Clifford Algebras, by Aziz Budiman Bin Kamlan, November 2013.
• VERSOR: Spatial Computing with Conformal Geometric Algebra, M.Sc. thesis of Pablo Colapinto, March 2011.
• Geometric Algebra and its Application to Computer Graphics, by D. Hildenbrand, D. Fontijne, C.Perwass, and L. Dorst, 2004.
• Geometric Algebra: A powerful tool for solving geometric problems in visual computing, by Leandro A. F. Fernandes and Manuel M. Oliveira, 2007.
• Analysis of Point Clouds using Conformal Geometric Algebra, by Dietmar Hildenbrand and Eckhard Hitzer, 2008.
• Composing Surfaces with Conformal Rotors, by Pablo Colapinto, 2017.
• Boosted Surfaces using Point Pair Generators as Curvature Operators, by Pablo Colapinto, 2013.
• Geometric Algebra in Linear Algebra and Geometry, by by José María Pozo and Garret Sobczyk, 2001.
• Multiplication of vectors and structure of 3D Euclidean space, by Miroslav Josipović, 2017.
• Associative Composition Algebra: Quaternions, by Wikibooks, 2019.
• Some remarks on Cl3 and Lorentz transformations, by Miroslav Josipović, June, 2015.



Hermann Grassmann: A New Branch Of Mathematics – The Ausdehnungslehre of 1844 and Other Works, translated by Lloyd C. Kannenberg (1995), ISBN 0-8126-9276-4.
David Hestenes (1993, (1986)), New Foundations for Classical Mechanics
Pertti Lounesto: Clifford Algebras and Spinors
Kluwer Academic Publishers, ISBN 90-277-2090-8.
Leo Dorst, Daniel Fontijne and Stephen Mann (2009), Geometric Algebra for Computer Science – An Object-Oriented Approach To Geometry, Morgan Kaufmann Publishers,

Felix Klein (1926), Vorlesungen über Nichteuklidische Geometrie,
Verlag von Julius Springer in Berlin, (1928).
Jürgen Richter-Gebert (2011), Perspectives on Projective Geometry – A Guided Tour Through Real and Complex Geometry, Springer, ISBN 978-3-642-17285-4.
D.M.Y. Sommerville (1914), The Elements of Non-Euclidean Geometry, Dover (1958, 2005).
Henry Parker Manning (1901), Introductory Non-Euclidean Geometry, Dover (1963, 2005).
H.S.M. Coxeter (1947 (1942)), Non-Euclidean Geometry.
W. T. Fishback (1969), Projective and Euclidean Geometry, John Wiley & Sons, Inc.,
ISBN 13: 978-047126-053-0.
John Stillwell (2016), Elements of Mathematics – From Euclid to Gödel,
Princeton University Press, ISBN 978-0-691-17854-7.
• John Stillwell (1999, (1989)), Mathematics and Its History, Springer, ISBN 0-387-96981-0.
• John Stillwell (1980), Classical Topology and Combinatorial Group Theory,
Springer Verlag, ISBN 0-387-90516-2.
Jeremy Gray (2007), Worlds Out of Nothing – A Course in the History of Geometry in the 19th Century, Springer, ISBN 1-84628-632-8.
A new branch of mathematics/Algebra books.


Other relevant sources of information:

Geometric Algebra.
The right-hand screw rule.
Geometric Calculus.
All Hail Geometric Algebra.


The ancient Greeks had the beginning of a geometric algebra,
namely the exterior product \, \wedge \, . In modern notation their algebra worked like this:

\, l_{ength} \wedge l_{ength} \, \equiv \, a_{rea} \, \, l_{ength} \wedge l_{ength} \wedge l_{ength} \equiv \, v_{olume} \, \, l_{ength} \wedge l_{ength} \wedge l_{ength} \wedge l_{ength} \equiv \, n_{othing} \,

since there were only three dimensions at that time.

The person that changed this fact was a German-speaking highschool teacher in Stettin by the name of Hermann Günther Graßmann. In 1844 he published his masterpiece Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik [The Theory of Linear Extension, a New Branch of Mathematics]. This book is normally referred to as Die Ausdehnungslehre, which can be translated as “The extension theory” or “The theory of extensive magnitudes.”

In this remarkable book – which nobody read since it was written in an obscure philosophical style that no mathematicians of the time were familiar with – Graßmann introduced the concept of abstract vector spaces with n-dimensional algebra and geometry. By doing so he singlehandedly created the mathematical subject that we today call linear algebra.

During the 1870:ies William Kingdon Clifford developed an algebra – later called Cliffordalgebra – which built on Graßmann’s exterior algebra.

/////// Quoting Wikipedia on William Kingdon Clifford:

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

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

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. The resulting geometric algebra, as Clifford called it, eventually realized the long sought goal[13] of creating an algebra that mirrors the movements and projections of objects in 3-dimensional space.[14]

Moreover, Clifford’s algebraic schema extends to higher dimensions. 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 classes – as real algebras – have been identified in other mathematical systems beyond simply the quaternions.

The realms of real analysis and complex analysis have been expanded through the algebra H of quaternions, thanks to its notion of a three-dimensional sphere embedded in a four-dimensional space.

Quaternion versors, which inhabit this 3-sphere, provide a representation of the rotation group SO(3).

Clifford noted that Hamilton’s biquaternions were a tensor product H ⊗ C of known algebras, and proposed instead two other tensor products of H:

Clifford argued that the “scalars” taken from the complex numbers C might instead be taken from split-complex numbers D or from the dual numbers N. In terms of tensor products, H ⊗ D produces split-biquaternions, while H ⊗ N forms dual quaternions.

The algebra of dual quaternions is used to express screw displacement, a common mapping in kinematics.

/////// End of quote from Wikipedia (on William Kingdon Clifford)


Some notation and background from combinatorial clifford algebra

Since we are here only interested in combinatorial and algebraic aspects of Clifford algebra, we will allow our scalars to lie in an arbitrary commutative ring \mathcal {R} with unit element. We will also take a slightly different point of view regarding the Clifford algebra and its interpretations.

Let X be a finite set which is totally ordered, i.e., X = \{x_1, \ldots, x_n\} where x_1 < x_2 < \ldots < x_n. We will identify the k base-blades x_{n_1} x_{n_2} \ldots x_{n_k}, where n_1 < n_2 < \ldots < n_k \leq n, with the \, k -subsets \{x_{n_1}, \ldots, x_{n_k}\} \subseteq X and we will denote the pseudoscalar x_1 x_2 \ldots x_n by X (by an unproblematic change of context). Moreover, the ring unit 1 is identified with the empty set \emptyset.

We will view the Clifford algebra C_l(X) as the free R module generated by the power set \wp(X) of all subsets of X, i.e.,

C_l(X) = {\oplus \atop {x \in \wp(X) } } \mathcal {R}.

Note that if X \rightarrow Y is a bijection, then C_l(X) is isomorphic to C_l(Y).

We will always assume that x^2 = 1, \forall x \in X. The set of k -vectors is denoted by {C_l}^k(X). We observe that every bilinear map C_l(X) \times C_l(X) \longrightarrow C_l(X) is uniquely determined by its values on \wp(X) \times \wp(X). Moreover, if P is a proposition, we will use (P) to denote 1 or 0 depending on whether P is true or false.

Let A,B \in \wp(X). The following notation will be used:

Geometric product: AB = \epsilon A \Delta B, where \epsilon = \pm 1 and \Delta is symmetric difference.

Outer product: A \wedge B = (A \cap B = \emptyset) A B.

Inner product: A \cdot B = ((A \subseteq B) \ \text {or} \ (A \supseteq B)) A B.

Left inner product: A \; \llcorner \; B = (A \subseteq B) A B.

Scalar product: A \ast B = (A = B) A B.

Reverse: A^\dagger = (-1)^\epsilon A, where \epsilon ={ {\mid A \mid } \choose { 2 } }.

Complement: \tilde {A} = AX^{-1}.

All of these definitions are extended to C_l(X) by linearity.