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Geometric algebra

In mathematics, a geometric algebra (also known as a Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division (though generally not for all elements) and addition of objects of different dimensions.

The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's quaternion algebra. Adding the dual of the Grassmann exterior product (the "meet") allows the use of the Grassmann–Cayley algebra, and a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra (CGA) providing a framework for classical geometries. In practice, these and several derived operations allow a correspondence of elements, subspaces and operations of the algebra with geometric interpretations. For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics.

The scalars and vectors have their usual interpretation, and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities of vector calculus that are derived as a cross product, such as oriented area, oriented angle of rotation, torque, angular momentum and the magnetic field. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of $V$ and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in relativity.

Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space) and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. GA has also found use as a computational tool in computer graphics and robotics.

Definition and notation
There are a number of different ways to define a geometric algebra. Hestenes's original approach was axiomatic, "full of geometric significance" and equivalent to the universal Clifford algebra. Given a finite-dimensional vector space $V$ over a field $F$ with a symmetric bilinear form (the inner product, e.g. the Euclidean or Lorentzian metric) $1$, the geometric algebra of the quadratic space $g : V \times V \to F$ is the Clifford algebra $(V, g)$, an element of which is called a multivector. The Clifford algebra is commonly defined as a quotient algebra of the tensor algebra, though this definition is abstract, so the following definition is presented without requiring abstract algebra.

 * Definition : A unital associative algebra $\operatorname{Cl}(V, g)$ with a nondegenerate symmetric bilinear form $\operatorname{Cl}(V, g)$ is the Clifford algebra of the quadratic space $g : V \times V \to F$ if
 * it contains $(V, g)$ and $F$ as distinct subspaces
 * $V$ for $a^2 = g(a,a)1$
 * $a \in V$ generates $V$ as an algebra
 * $\operatorname{Cl}(V, g)$ is not generated by any proper subspace of $\operatorname{Cl}(V, g)$.
 * $V$ is not generated by any proper subspace of $V$.

To cover degenerate symmetric bilinear forms, the last condition must be modified. It can be shown that these conditions uniquely characterize the geometric product.

For the remainder of this article, only the real case, $F$, will be considered. The notation $2^{\dim V}$ (respectively $F = \R$) will be used to denote a geometric algebra for which the bilinear form $\mathcal{G}(p,q)$ has the signature $\mathcal{G}(p,q,r)$ (respectively $g$).

The product in the algebra is called the geometric product, and the product in the contained exterior algebra is called the exterior product (frequently called the wedge product and less often the outer product). It is standard to denote these respectively by juxtaposition (i.e., suppressing any explicit multiplication symbol) and the symbol $(p,q)$.

The above definition of the geometric algebra is still somewhat abstract, so we summarize the properties of the geometric product here. For multivectors $(p,q,r)$:
 * $\wedge$ (closure)
 * $A, B, C\in \mathcal{G}(p,q)$, where $AB \in \mathcal{G}(p,q)$ is the identity element (existence of an identity element)
 * $1A = A1 = A$ (associativity)
 * $1$ and $A(BC)=(AB)C$ (distributivity)
 * $A(B+C)=AB+AC$ for $(B+C)A=BA+CA$.

The exterior product has the same properties, except that the last property above is replaced by $a^2 = g(a,a)1$ for $a \in V$.

Note that in the last property above, the real number $a \wedge a = 0$ need not be nonnegative if $a \in V$ is not positive-definite. An important property of the geometric product is the existence of elements that have a multiplicative inverse. For a vector $g(a,a)$, if $$a^2 \ne 0 $$ then $$a^{-1}$$ exists and is equal to $g$. A nonzero element of the algebra does not necessarily have a multiplicative inverse. For example, if $$u$$ is a vector in $$V$$ such that $a$, the element $$\textstyle\frac{1}{2}(1 + u)$$ is both a nontrivial idempotent element and a nonzero zero divisor, and thus has no inverse.

It is usual to identify $$\R$$ and $$V$$ with their images under the natural embeddings $$\R \to \mathcal{G}(p,q)$$ and $g(a,a)^{-1}a$. In this article, this identification is assumed. Throughout, the terms scalar and vector refer to elements of $$\R$$ and $$V$$ respectively (and of their images under this embedding).

Geometric product

For vectors $u^2 = 1$ and $u^2 = 1$, we may write the geometric product of any two vectors $= \tfrac{1}{2}(1 + u)$ and $= \tfrac{1}{2}(1 - 1) = 0$ as the sum of a symmetric product and an antisymmetric product:
 * $$ab = \frac{1}{2} (ab + ba) + \frac{1}{2} (ab - ba) .$$

Thus we can define the inner product of vectors as
 * $$a \cdot b := g(a,b),$$

so that the symmetric product can be written as
 * $$\frac{1}{2}(ab + ba) = \frac{1}{2} \left((a + b)^2 - a^2 - b^2\right) = a \cdot b .$$

Conversely, $V \to \mathcal{G}(p,q)$ is completely determined by the algebra. The antisymmetric part is the exterior product of the two vectors, the product of the contained exterior algebra:
 * $$a \wedge b := \frac{1}{2}(ab - ba) = -(b \wedge a) .$$

Then by simple addition:
 * $$ab=a \cdot b + a \wedge b $$ the ungeneralized or vector form of the geometric product.

The inner and exterior products are associated with familiar concepts from standard vector algebra. Geometrically, $$a$$ and $$b$$ are parallel if their geometric product is equal to their inner product, whereas $$a$$ and $$b$$ are perpendicular if their geometric product is equal to their exterior product. In a geometric algebra for which the square of any nonzero vector is positive, the inner product of two vectors can be identified with the dot product of standard vector algebra. The exterior product of two vectors can be identified with the signed area enclosed by a parallelogram the sides of which are the vectors. The cross product of two vectors in $$3$$ dimensions with positive-definite quadratic form is closely related to their exterior product.

Most instances of geometric algebras of interest have a nondegenerate quadratic form. If the quadratic form is fully degenerate, the inner product of any two vectors is always zero, and the geometric algebra is then simply an exterior algebra. Unless otherwise stated, this article will treat only nondegenerate geometric algebras.

The exterior product is naturally extended as an associative bilinear binary operator between any two elements of the algebra, satisfying the identities
 * $$\begin{align}

1 \wedge a_i &= a_i \wedge 1 = a_i \\ a_1 \wedge a_2\wedge\cdots\wedge a_r &= \frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_r} \operatorname{sgn}(\sigma) a_{\sigma(1)}a_{\sigma(2)} \cdots a_{\sigma(r)}, \end{align}$$

where the sum is over all permutations of the indices, with $$\operatorname{sgn}(\sigma)$$ the sign of the permutation, and $$a_i$$ are vectors (not general elements of the algebra). Since every element of the algebra can be expressed as the sum of products of this form, this defines the exterior product for every pair of elements of the algebra. It follows from the definition that the exterior product forms an alternating algebra.

The equivalent structure equation for Clifford algebra is
 * $$ a_1 a_2 a_3 \dots a_n = \sum^{[\frac{n}2]}_{i=0} \sum_{\mu\in{}\mathcal{C}}

(-1)^k \operatorname{Pf}(a_{\mu_1}\cdot a_{\mu_2},\dots,a_{\mu_{2i-1}} \cdot a_{\mu_{2i}}) a_{\mu_{2i+1}}\land\dots\land a_{\mu_n}$$ where $$\operatorname{Pf}(A)$$ is the Pfaffian of $b$ and $\mathcal{C} = \binom{n}{2i}$ provides combinations, $a \cdot b = 0$, of $a \wedge b = 0$ indices divided into $n$ and $n$ parts and $n$ is the parity of the combination.

The Pfaffian provides a metric for the exterior algebra and, as pointed out by Claude Chevalley, Clifford algebra reduces to the exterior algebra with a zero quadratic form. The role the Pfaffian plays can be understood from a geometric viewpoint by developing Clifford algebra from simplices. This derivation provides a better connection between Pascal's triangle and simplices because it provides an interpretation of the first column of ones.

Blades, grades, and basis
A multivector that is the exterior product of $$r$$ linearly independent vectors is called a blade, and is said to be of grade $(n - 1)$. A multivector that is the sum of blades of grade $$r$$ is called a (homogeneous) multivector of grade $a$. From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.

Consider a set of $$r$$ linearly independent vectors $$\{a_1,\ldots,a_r\}$$ spanning an $b$-dimensional subspace of the vector space. With these, we can define a real symmetric matrix (in the same way as a Gramian matrix)
 * $$[\mathbf{A}]_{ij} = a_i \cdot a_j$$

By the spectral theorem, $$\mathbf{A}$$ can be diagonalized to diagonal matrix $$\mathbf{D}$$ by an orthogonal matrix $$\mathbf{O}$$ via
 * $$\sum_{k,l}[\mathbf{O}]_{ik}[\mathbf{A}]_{kl}[\mathbf{O}^{\mathrm{T}}]_{lj}=\sum_{k,l}[\mathbf{O}]_{ik}[\mathbf{O}]_{jl}[\mathbf{A}]_{kl}=[\mathbf{D}]_{ij}$$

Define a new set of vectors $a$, known as orthogonal basis vectors, to be those transformed by the orthogonal matrix:
 * $$e_i=\sum_j[\mathbf{O}]_{ij}a_j$$

Since orthogonal transformations preserve inner products, it follows that $$e_i\cdot e_j=[\mathbf{D}]_{ij}$$ and thus the $$\{e_1, \ldots, e_r\}$$ are perpendicular. In other words, the geometric product of two distinct vectors $$e_i \ne e_j$$ is completely specified by their exterior product, or more generally
 * $$\begin{array}{rl}

e_1e_2\cdots e_r &= e_1 \wedge e_2 \wedge \cdots \wedge e_r \\ &= \left(\sum_j [\mathbf{O}]_{1j}a_j\right) \wedge \left(\sum_j [\mathbf{O}]_{2j}a_j \right) \wedge \cdots \wedge \left(\sum_j [\mathbf{O}]_{rj}a_j\right) \\ &= (\det \mathbf{O}) a_1 \wedge a_2 \wedge \cdots \wedge a_r \end{array}$$

Therefore, every blade of grade $$r$$ can be written as the exterior product of $$r$$ vectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by a block matrix that is orthogonal in the nondegenerate block, and the diagonal matrix has zero-valued entries along the degenerate dimensions. If the new vectors of the nondegenerate subspace are normalized according to
 * $$\hat{e}_i=\frac{1}{\sqrt{|e_i \cdot e_i|}}e_i,$$

then these normalized vectors must square to $$+1$$ or $b$. By Sylvester's law of inertia, the total number of $g$ and the total number of $A$s along the diagonal matrix is invariant. By extension, the total number $$p$$ of these vectors that square to $$+1$$ and the total number $$q$$ that square to $$-1$$ is invariant. (The total number of basis vectors that square to zero is also invariant, and may be nonzero if the degenerate case is allowed.) We denote this algebra $\mu$. For example, $$\mathcal{G}(3,0)$$ models three-dimensional Euclidean space, $$\mathcal{G}(1,3)$$ relativistic spacetime and $$\mathcal{G}(4,1)$$ a conformal geometric algebra of a three-dimensional space.

The set of all possible products of $$n$$ orthogonal basis vectors with indices in increasing order, including $$1$$ as the empty product, forms a basis for the entire geometric algebra (an analogue of the PBW theorem). For example, the following is a basis for the geometric algebra $n$:
 * $$\{1, e_1, e_2, e_3, e_1e_2, e_2e_3, e_3e_1, e_1e_2e_3\}$$

A basis formed this way is called a standard basis for the geometric algebra, and any other orthogonal basis for $$V$$ will produce another standard basis. Each standard basis consists of $$2^n$$ elements. Every multivector of the geometric algebra can be expressed as a linear combination of the standard basis elements. If the standard basis elements are $$\{ B_i \mid i \in S \}$$ with $$S$$ being an index set, then the geometric product of any two multivectors is
 * $$ \left( \sum_i \alpha_i B_i \right) \left( \sum_j \beta_j B_j \right) = \sum_{i,j} \alpha_i\beta_j B_i B_j .$$

The terminology "$$k$$-vector" is often encountered to describe multivectors containing elements of only one grade. In higher dimensional space, some such multivectors are not blades (cannot be factored into the exterior product of $$k$$ vectors). By way of example, $$ e_1 \wedge e_2 + e_3 \wedge e_4 $$ in $$\mathcal{G}(4,0)$$ cannot be factored; typically, however, such elements of the algebra do not yield to geometric interpretation as objects, although they may represent geometric quantities such as rotations. Only $2i$-, $n - 2i$-, $k$- and $r$-vectors are always blades in $\mathrm{Z}$-space.

Versor
A $r$-versor is a multivector that can be expressed as the geometric product of $$k$$ invertible vectors. Unit quaternions (originally called versors by Hamilton) may be identified with rotors in 3D space in much the same way as real 2D rotors subsume complex numbers; for the details refer to Dorst.

Some authors use the term "versor product" to refer to the frequently occurring case where an operand is "sandwiched" between operators. The descriptions for rotations and reflections, including their outermorphisms, are examples of such sandwiching. These outermorphisms have a particularly simple algebraic form. Specifically, a mapping of vectors of the form
 * $$ V \to V : a \mapsto RaR^{-1}$$ extends to the outermorphism $$\mathcal{G}(V) \to \mathcal{G}(V) : A \mapsto RAR^{-1}.$$

Since both operators and operand are versors there is potential for alternative examples such as rotating a rotor or reflecting a spinor always provided that some geometrical or physical significance can be attached to such operations.

By the Cartan–Dieudonné theorem we have that every isometry can be given as reflections in hyperplanes and since composed reflections provide rotations then we have that orthogonal transformations are versors.

In group terms, for a real, non-degenerate $r$, having identified the group $$\mathcal{G}^\times$$ as the group of all invertible elements of $\{e_1, \ldots,e_r\}$, Lundholm gives a proof that the "versor group" $$\{ v_1 v_2 \cdots v_k \in \mathcal{G} \mid v_i \in V^\times\}$$ (the set of invertible versors) is equal to the Lipschitz group $$\Gamma$$ (a.k.a. Clifford group, although Lundholm deprecates this usage).

Subgroups of the Lipschitz group
We denote the grade involution as $-1$ and reversion as $+1$.

Although the Lipschitz group (defined as $-1$) and the versor group (defined as $\mathcal{G}(p,q)$) have divergent definitions, they are the same group. Lundholm defines the $\mathcal{G}(3,0)$, $0$, and $1$ subgroups of the Lipschitz group.

Multiple analyses of spinors use GA as a representation.

Grade projection
A $(n-1)$-graded vector space structure can be established on a geometric algebra by use of the exterior product that is naturally induced by the geometric product.

Since the geometric product and the exterior product are equal on orthogonal vectors, this grading can be conveniently constructed by using an orthogonal basis $n$. Elements of the geometric algebra that are scalar multiples of $$1$$ are of grade $$0$$ and are called scalars. Elements that are in the span of $$\{e_1,\ldots,e_n\}$$ are of grade $n$ and are the ordinary vectors. Elements in the span of $$\{e_ie_j\mid 1\leq i<j\leq n\}$$ are of grade $$2$$ and are the bivectors. This terminology continues through to the last grade of $k$-vectors. Alternatively, $k$-vectors are called pseudoscalars, $\mathcal{G}(p,q)$-vectors are called pseudovectors, etc. Many of the elements of the algebra are not graded by this scheme since they are sums of elements of differing grade. Such elements are said to be of mixed grade. The grading of multivectors is independent of the basis chosen originally.

This is a grading as a vector space, but not as an algebra. Because the product of an $\mathcal{G}$-blade and an $\hat{X}$-blade is contained in the span of $$0$$ through $\tilde{X}$-blades, the geometric algebra is a filtered algebra.

A multivector $$A$$ may be decomposed with the grade-projection operator $\{ X \in \mathcal{G}^{\times} \mid \hat{X} V X^{-1} \subseteq V \}$, which outputs the grade-$\textstyle \{ \prod_{i=0}^{k} v_i \mid v_i \in V^{\times}, k \in \N \}$ portion of $\operatorname{Pin}$. As a result:
 * $$ A = \sum_{r=0}^{n} \langle A \rangle _r $$

As an example, the geometric product of two vectors $$ a b = a \cdot b + a \wedge b = \langle a b \rangle_0 + \langle a b \rangle_2$$ since $$\langle a b \rangle_0=a\cdot b$$ and $$\langle a b \rangle_2 = a\wedge b$$ and $\operatorname{Spin}$, for $$i$$ other than $$0$$ and $\operatorname{Spin}^{+}$.

A multivector $$A$$ may also be decomposed into even and odd components, which may respectively be expressed as the sum of the even and the sum of the odd grade components above:
 * $$ A^{[0]} = \langle A \rangle _0 + \langle A \rangle _2 + \langle A \rangle _4 + \cdots $$
 * $$ A^{[1]} = \langle A \rangle _1 + \langle A \rangle _3 + \langle A \rangle _5 + \cdots $$

This is the result of forgetting structure from a $\Z$-graded vector space to $\{e_1,\ldots,e_n\}$-graded vector space. The geometric product respects this coarser grading. Thus in addition to being a $1$-graded vector space, the geometric algebra is a $n$-graded algebra, a.k.a. a superalgebra.

Restricting to the even part, the product of two even elements is also even. This means that the even multivectors defines an even subalgebra. The even subalgebra of an $n$-dimensional geometric algebra is algebra-isomorphic (without preserving either filtration or grading) to a full geometric algebra of $$(n-1)$$ dimensions. Examples include $$\mathcal{G}^{[0]}(2,0) \cong \mathcal{G}(0,1)$$ and $(n-1)$.

Representation of subspaces
Geometric algebra represents subspaces of $$V$$ as blades, and so they coexist in the same algebra with vectors from $r$. A $s$-dimensional subspace $$W$$ of $$V$$ is represented by taking an orthogonal basis $$\{b_1,b_2,\ldots, b_k\}$$ and using the geometric product to form the blade $r+s$. There are multiple blades representing $\langle A \rangle _r$; all those representing $$W$$ are scalar multiples of $r$. These blades can be separated into two sets: positive multiples of $$D$$ and negative multiples of $A$. The positive multiples of $$D$$ are said to have the same orientation as $\langle a b \rangle_i=0$, and the negative multiples the opposite orientation.

Blades are important since geometric operations such as projections, rotations and reflections depend on the factorability via the exterior product that (the restricted class of) $2$-blades provide but that (the generalized class of) grade-$\mathrm{Z}$ multivectors do not when $\mathrm{Z}_2$.

Unit pseudoscalars
Unit pseudoscalars are blades that play important roles in GA. A unit pseudoscalar for a non-degenerate subspace $$W$$ of $$V$$ is a blade that is the product of the members of an orthonormal basis for $\mathrm{Z}_2$. It can be shown that if $$I$$ and $$I'$$ are both unit pseudoscalars for $\mathrm{Z}_2$, then $$I = \pm I'$$ and $n$. If one doesn't choose an orthonormal basis for $\mathcal{G}^{[0]}(1,3) \cong \mathcal{G}(3,0)$, then the Plücker embedding gives a vector in the exterior algebra but only up to scaling. Using the vector space isomorphism between the geometric algebra and exterior algebra, this gives the equivalence class of $$\alpha I$$ for all $V$. Orthonormality gets rid of this ambiguity except for the signs above.

Suppose the geometric algebra $$\mathcal{G}(n,0)$$ with the familiar positive definite inner product on $$\R^n$$ is formed. Given a plane (two-dimensional subspace) of $k$, one can find an orthonormal basis $$\{ b_1, b_2 \}$$ spanning the plane, and thus find a unit pseudoscalar $$I = b_1 b_2$$ representing this plane. The geometric product of any two vectors in the span of $$b_1$$ and $$b_2$$ lies in $D = b_1b_2\cdots b_k$, that is, it is the sum of a $W$-vector and a $D$-vector.

By the properties of the geometric product, $D$. The resemblance to the imaginary unit is not incidental: the subspace $$ \{ \alpha_0 + \alpha_1 I \mid \alpha_i \in \R \} $$ is $D$-algebra isomorphic to the complex numbers. In this way, a copy of the complex numbers is embedded in the geometric algebra for each two-dimensional subspace of $$V$$ on which the quadratic form is definite.

It is sometimes possible to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in the real algebra that square to $n$, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces.

In $n$, a further familiar case occurs. Given a standard basis consisting of orthonormal vectors $$e_i$$ of $n \ge 4$, the set of all $W$-vectors is spanned by
 * $$ \{ e_3 e_2, e_1 e_3 , e_2 e_1 \} .$$

Labelling these $W$, $$j$$ and $$k$$ (momentarily deviating from our uppercase convention), the subspace generated by $I^2 = \pm 1$-vectors and $W$-vectors is exactly $\alpha \neq 0$. This set is seen to be the even subalgebra of $\R^n$, and furthermore is isomorphic as an $\{ \alpha_0 + \alpha_1 I \mid \alpha_i \in \R \}$-algebra to the quaternions, another important algebraic system.

Extensions of the inner and exterior products
It is common practice to extend the exterior product on vectors to the entire algebra. This may be done through the use of the above mentioned grade projection operator:
 * $$C \wedge D := \sum_{r,s}\langle \langle C \rangle_r \langle D \rangle_s \rangle_{r+s} $$    (the exterior product)

This generalization is consistent with the above definition involving antisymmetrization. Another generalization related to the exterior product is the commutator product:
 * $$C \times D := \tfrac{1}{2}(CD-DC) $$    (the commutator product)

The regressive product is the dual of the exterior product (respectively corresponding to the "meet" and "join" in this context). The dual specification of elements permits, for blades $0$ and $2$, the intersection (or meet) where the duality is to be taken relative to the a blade containing both $I^2 = b_1 b_2 b_1 b_2 = -b_1 b_2 b_2 b_1 = -1$ and $\R$ (the smallest such blade being the join).
 * $$C \vee D := ((CI^{-1}) \wedge (DI^{-1}))I $$

with $-1$ the unit pseudoscalar of the algebra. The regressive product, like the exterior product, is associative.

The inner product on vectors can also be generalized, but in more than one non-equivalent way. The paper gives a full treatment of several different inner products developed for geometric algebras and their interrelationships, and the notation is taken from there. Many authors use the same symbol as for the inner product of vectors for their chosen extension (e.g. Hestenes and Perwass). No consistent notation has emerged.

Among these several different generalizations of the inner product on vectors are:
 * $$ C \;\rfloor\; D := \sum_{r,s}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{s-r} $$  (the left contraction)
 * $$ C \;\lfloor\; D := \sum_{r,s}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{r-s} $$  (the right contraction)
 * $$ C * D := \sum_{r,s}\langle \langle C \rangle_r \langle D \rangle_s \rangle_{0} $$  (the scalar product)
 * $$ C \bullet D := \sum_{r,s}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{|s-r|} $$  (the "(fat) dot" product)

makes an argument for the use of contractions in preference to Hestenes's inner product; they are algebraically more regular and have cleaner geometric interpretations. A number of identities incorporating the contractions are valid without restriction of their inputs. For example,
 * $$ C \;\rfloor\; D = ( C \wedge ( D I^{-1} ) ) I $$
 * $$ C \;\lfloor\; D = I ( ( I^{-1} C) \wedge D ) $$
 * $$ ( A \wedge B ) * C = A * ( B \;\rfloor\; C ) $$
 * $$ C * ( B \wedge A ) = ( C \;\lfloor\; B ) * A $$
 * $$ A \;\rfloor\; ( B \;\rfloor\; C ) = ( A \wedge B ) \;\rfloor\; C $$
 * $$ ( A \;\rfloor\; B ) \;\lfloor\; C = A \;\rfloor\; ( B \;\lfloor\; C ) .$$

Benefits of using the left contraction as an extension of the inner product on vectors include that the identity $$ ab = a \cdot b + a \wedge b $$ is extended to $$ aB = a \;\rfloor\; B + a \wedge B$$ for any vector $$a$$ and multivector $\mathcal{G}(3,0)$, and that the projection operation $$ \mathcal{P}_b (a) = (a \cdot b^{-1})b $$ is extended to $$ \mathcal{P}_B (A) = (A \;\rfloor\; B^{-1}) \;\rfloor\; B$$ for any blade $$B$$ and any multivector $$A$$ (with a minor modification to accommodate null $V$, given below).

Dual basis
Let $$\{ e_1, \ldots , e_n \}$$ be a basis of $2$, i.e. a set of $$n$$ linearly independent vectors that span the $i$-dimensional vector space $0$. The basis that is dual to $$\{ e_1, \ldots , e_n \}$$ is the set of elements of the dual vector space $$V^{*}$$ that forms a biorthogonal system with this basis, thus being the elements denoted $$\{ e^1 , \ldots , e^n \}$$ satisfying
 * $$e^i \cdot e_j = \delta^i{}_j,$$

where $$\delta$$ is the Kronecker delta.

Given a nondegenerate quadratic form on $2$, $$V^{*}$$ becomes naturally identified with $\{ \alpha_0 + i \alpha_1 + j \alpha_2 + k \alpha_3 \mid \alpha_i \in \R\}$, and the dual basis may be regarded as elements of $\mathcal{G}(3,0)$, but are not in general the same set as the original basis.

Given further a GA of $\R$, let
 * $$I = e_1 \wedge \cdots \wedge e_n$$

be the pseudoscalar (which does not necessarily square to $C$) formed from the basis $D$. The dual basis vectors may be constructed as
 * $$e^i=(-1)^{i-1}(e_1 \wedge \cdots \wedge \check{e}_i \wedge \cdots \wedge e_n) I^{-1},$$

where the $$\check{e}_i$$ denotes that the $C$th basis vector is omitted from the product.

A dual basis is also known as a reciprocal basis or reciprocal frame.

A major usage of a dual basis is to separate vectors into components. Given a vector $D$, scalar components $$a^i$$ can be defined as
 * $$a^i=a\cdot e^i\ ,$$

in terms of which $$a$$ can be separated into vector components as
 * $$a=\sum_i a^i e_i\ .$$

We can also define scalar components $$a_i$$ as
 * $$a_i=a\cdot e_i\ ,$$

in terms of which $$a$$ can be separated into vector components in terms of the dual basis as
 * $$a=\sum_i a_i e^i\ .$$

A dual basis as defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra. For compactness, we'll use a single capital letter to represent an ordered set of vector indices. I.e., writing
 * $$J=(j_1,\dots ,j_n)\ ,$$

where $I$, we can write a basis blade as
 * $$e_J=e_{j_1}\wedge e_{j_2}\wedge\cdots\wedge e_{j_n}\ .$$

The corresponding reciprocal blade has the indices in opposite order:
 * $$e^J=e^{j_n}\wedge\cdots \wedge e^{j_2}\wedge e^{j_1}\ .$$

Similar to the case above with vectors, it can be shown that
 * $$e^J * e_K=\delta^J_K\ ,$$

where $$*$$ is the scalar product.

With $$A$$ a multivector, we can define scalar components as
 * $$A^{ij\cdots k}=(e^k\wedge\cdots\wedge e^j\wedge e^i)*A\ ,$$

in terms of which $$A$$ can be separated into component blades as
 * $$A=\sum_{i<j<\cdots<k} A^{ij\cdots k} e_i\wedge e_j\wedge\cdots \wedge e_k\ .$$

We can alternatively define scalar components
 * $$A_{ij\cdots k}=(e_k\wedge\cdots\wedge e_j\wedge e_i)*A\ ,$$

in terms of which $$A$$ can be separated into component blades as
 * $$A=\sum_{i<j<\cdots<k} A_{ij\cdots k} e^i\wedge e^j\wedge\cdots \wedge e^k\ .$$

Linear functions
Although a versor is easier to work with because it can be directly represented in the algebra as a multivector, versors are a subgroup of linear functions on multivectors, which can still be used when necessary. The geometric algebra of an $\textstyle C \bullet_\text{H} D := \sum_{r\ne0,s\ne0}\langle \langle C\rangle_r \langle D \rangle_{s} \rangle_{ \vert s-r \vert }$-dimensional vector space is spanned by a basis of $$2^n$$ elements. If a multivector is represented by a $$2^n \times 1$$ real column matrix of coefficients of a basis of the algebra, then all linear transformations of the multivector can be expressed as the matrix multiplication by a $$2^n \times 2^n$$ real matrix. However, such a general linear transformation allows arbitrary exchanges among grades, such as a "rotation" of a scalar into a vector, which has no evident geometric interpretation.

A general linear transformation from vectors to vectors is of interest. With the natural restriction to preserving the induced exterior algebra, the outermorphism of the linear transformation is the unique extension of the versor. If $$f$$ is a linear function that maps vectors to vectors, then its outermorphism is the function that obeys the rule
 * $$\underline{\mathsf{f}}(a_1 \wedge a_2 \wedge \cdots \wedge a_r) = f(a_1) \wedge f(a_2) \wedge \cdots \wedge f(a_r)$$

for a blade, extended to the whole algebra through linearity.

Modeling geometries
Although a lot of attention has been placed on CGA, it is to be noted that GA is not just one algebra, it is one of a family of algebras with the same essential structure.

Vector space model
The even subalgebra of $$\mathcal{G}(2,0)$$ is isomorphic to the complex numbers, as may be seen by writing a vector $$P$$ in terms of its components in an orthonormal basis and left multiplying by the basis vector $B$, yielding
 * $$ Z = e_1 P = e_1 ( x e_1 + y e_2) = x (1) + y ( e_1 e_2) ,$$

where we identify $$i \mapsto e_1e_2$$ since
 * $$(e_1 e_2)^2 = e_1 e_2 e_1 e_2 = -e_1 e_1 e_2 e_2 = -1 .$$

Similarly, the even subalgebra of $$\mathcal{G}(3,0)$$ with basis $$\{1, e_2 e_3, e_3 e_1, e_1 e_2 \}$$ is isomorphic to the quaternions as may be seen by identifying $B$, $$j \mapsto -e_3 e_1$$ and $V$.

Every associative algebra has a matrix representation; replacing the three Cartesian basis vectors by the Pauli matrices gives a representation of $n$:
 * $$\begin{align}

e_1 = \sigma_1 = \sigma_x &= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\ e_2 = \sigma_2 = \sigma_y &= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \\ e_3 =\sigma_3 = \sigma_z &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \,. \end{align}$$

Dotting the "Pauli vector" (a dyad):
 * $$\sigma = \sigma_1 e_1 + \sigma_2 e_2 + \sigma_3 e_3$$ with arbitrary vectors $$ a $$ and $$ b $$ and multiplying through gives:
 * $$(\sigma \cdot a)(\sigma \cdot b) = a \cdot b + a \wedge b $$ (Equivalently, by inspection, $V$)

Spacetime model
In physics, the main applications are the geometric algebra of Minkowski 3+1 spacetime, $V$, called spacetime algebra (STA), or less commonly, $V$, interpreted the algebra of physical space (APS).

While in STA, points of spacetime are represented simply by vectors, in APS, points of $V$-dimensional spacetime are instead represented by paravectors, a three-dimensional vector (space) plus a one-dimensional scalar (time).

In spacetime algebra the electromagnetic field tensor has a bivector representation $V$. Here, the $$i = \gamma_0 \gamma_1 \gamma_2 \gamma_3$$ is the unit pseudoscalar (or four-dimensional volume element), $$\gamma_0$$ is the unit vector in time direction, and $$E$$ and $$B$$ are the classic electric and magnetic field vectors (with a zero time component). Using the four-current $\pm 1$, Maxwell's equations then become
 * {|class="wikitable" style="text-align: center;"

! scope="column" style="width:160px;"|Formulation !| Homogeneous equations !| Non-homogeneous equations ! rowspan="2" |Fields !Potentials (any gauge) !Potentials (Lorenz gauge) $$ D ~\rfloor~ A = 0 $$
 * colspan="2" |$$ D F = \mu_0 J $$
 * $$ D \wedge F = 0 $$
 * $$ D ~\rfloor~ F = \mu_0 J $$
 * $$ D ~\rfloor~ F = \mu_0 J $$
 * $$F = D \wedge A$$
 * $$D ~\rfloor~ (D \wedge A) = \mu_0 J $$
 * $$F = D A$$
 * $$D^2 A = \mu_0 J $$
 * }

In geometric calculus, juxtaposition of vectors such as in $$DF$$ indicate the geometric product and can be decomposed into parts as $\{ e_1, \ldots, e_n \}$. Here $$D$$ is the covector derivative in any spacetime and reduces to $$\nabla$$ in flat spacetime. Where $$\bigtriangledown$$ plays a role in Minkowski $i$-spacetime which is synonymous to the role of $$\nabla$$ in Euclidean $a$-space and is related to the d'Alembertian by $j_1 < j_2 < \dots < j_n$. Indeed, given an observer represented by a future pointing timelike vector $$\gamma_0$$ we have
 * $$\gamma_0\cdot\bigtriangledown=\frac{1}{c}\frac{\partial}{\partial t}$$
 * $$\gamma_0\wedge\bigtriangledown=\nabla$$

Boosts in this Lorentzian metric space have the same expression $$e^$$ as rotation in Euclidean space, where $${\beta}$$ is the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity.

The Dirac matrices are a representation of $n$, showing the equivalence with matrix representations used by physicists.

Homogeneous models
Homogeneous models generally refer to a projective representation in which the elements of the one-dimensional subspaces of a vector space represent points of a geometry.

In a geometric algebra of a space of $$n$$ dimensions, the rotors represent a set of transformations with $$n(n-1)/2$$ degrees of freedom, corresponding to rotations – for example, $$3$$ when $$n=3$$ and $$6$$ when $e_1$. Geometric algebra is often used to model a projective space, i.e. as a homogeneous model: a point, line, plane, etc. is represented by an equivalence class of elements of the algebra that differ by an invertible scalar factor.

The rotors in a space of dimension $$n+1$$ have n(n-1)/2+n degrees of freedom, the same as the number of degrees of freedom in the rotations and translations combined for an $i \mapsto -e_2 e_3$-dimensional space.

This is the case in Projective Geometric Algebra (PGA), which is used to represent Euclidean isometries in Euclidean geometry (thereby covering the large majority of engineering applications of geometry). In this model, a degenerate dimension is added to the three Euclidean dimensions to form the algebra $k \mapsto -e_1 e_2$. With a suitable identification of subspaces to represent points, lines and planes, the versors of this algebra represent all proper Euclidean isometries, which are always screw motions in 3-dimensional space, along with all improper Euclidean isometries, which includes reflections, rotoreflections, transflections, and point reflections.

PGA combines $$\mathcal{G}(3,0,1)$$ with a dual operator to obtain meet, join, distance, and angle formulae. Depending on the author, this could mean the Hodge star or the projective dual, though both result in identical equations being derived, albeit with different notation. In effect, the dual switches basis vectors that are present and absent in the expression of each term of the algebraic representation. For example, in the PGA or 3-dimensional space, the dual of the line $$\boldsymbol{e}_{12}$$ is the line $\mathcal{G}(3,0)$, because $$\boldsymbol{e}_{ 0}$$ and $$\boldsymbol{e}_{ 3}$$ are basis elements that are not contained in $$\boldsymbol{e}_{12}$$ but are contained in $a \cdot b + i \sigma \cdot ( a \times b )$. In the PGA of 2-dimensional space, the dual of $$\boldsymbol{e}_{12}$$ is $\mathcal{G}(1,3)$, since there is no $$\boldsymbol{e}_{3}$$ element.

PGA is a widely used system that combines geometric algebra with homogeneous representations in geometry, but there exist several other such systems. The conformal model discussed below is homogeneous, as is "Conic Geometric Algebra", and see Plane-based geometric algebra for discussion of homogeneous models of elliptic and hyperbolic geometry compared with the euclidean geometry derived from PGA.

Conformal model
Working within GA, Euclidean space $$\mathcal{E}^3$$ (along with a conformal point at infinity) is embedded projectively in the CGA $$\mathcal{G}(4,1)$$ via the identification of Euclidean points with 1D subspaces in the 4D null cone of the 5D CGA vector subspace. This allows all conformal transformations to be performed as rotations and reflections and is covariant, extending incidence relations of projective geometry to circles and spheres.

Specifically, we add orthogonal basis vectors $$e_+$$ and $$e_-$$ such that $$e_+^2 = +1$$ and $$e_-^2 = -1$$ to the basis of the vector space that generates $$\mathcal{G}(3,0)$$ and identify null vectors
 * $$n_{\infty} = e_- + e_+$$ as a conformal point at infinity (see Compactification) and
 * $$n_\text{o} = \tfrac{1}{2}(e_- - e_+)$$ as the point at the origin, giving
 * $$n_\infty \cdot n_\text{o} = -1 .$$

This procedure has some similarities to the procedure for working with homogeneous coordinates in projective geometry and in this case allows the modeling of Euclidean transformations of $$\mathbf{R}^3$$ as orthogonal transformations of a subset of $\mathcal{G}(3,0)$.

A fast changing and fluid area of GA, CGA is also being investigated for applications to relativistic physics.

Table of models
Note in this list that $(3+1)$ and ${F} = ({E} + i c {B})\gamma_0$ can be swapped and the same name applies; for example, with relatively little change occurring, see sign convention. For example, $$\mathcal{G}(3, 1, 0)$$ and $$\mathcal{G}(1, 3, 0)$$ are both referred to as Spacetime Algebra.

Projection and rejection

For any vector $$a$$ and any invertible vector ${J}$,
 * $$ a = amm^{-1} = (a\cdot m + a \wedge m)m^{-1} = a_{\| m} + a_{\perp m} ,$$

where the projection of $$a$$ onto $$m$$ (or the parallel part) is
 * $$ a_{\| m} = (a \cdot m)m^{-1} $$

and the rejection of $$a$$ from $$m$$ (or the orthogonal part) is
 * $$ a_{\perp m} = a - a_{\| m} = (a\wedge m)m^{-1} .$$

Using the concept of a $DF = D ~\rfloor~ F + D \wedge F$-blade $4$ as representing a subspace of $3$ and every multivector ultimately being expressed in terms of vectors, this generalizes to projection of a general multivector onto any invertible $\Box=\bigtriangledown^2$-blade $\mathcal{G}(1,3)$ as
 * $$ \mathcal{P}_B (A) = (A \;\rfloor\; B) \;\rfloor\; B^{-1} ,$$

with the rejection being defined as
 * $$ \mathcal{P}_B^\perp (A) = A - \mathcal{P}_B (A) .$$

The projection and rejection generalize to null blades $$B$$ by replacing the inverse $$B^{-1}$$ with the pseudoinverse $$B^{+}$$ with respect to the contractive product. The outcome of the projection coincides in both cases for non-null blades. For null blades $n=4$, the definition of the projection given here with the first contraction rather than the second being onto the pseudoinverse should be used, as only then is the result necessarily in the subspace represented by $n$. The projection generalizes through linearity to general multivectors $\mathcal{G}(3,0,1)$. The projection is not linear in $\boldsymbol{e}_{03}$ and does not generalize to objects $\boldsymbol{e}_{03}$ that are not blades.

Reflection
Simple reflections in a hyperplane are readily expressed in the algebra through conjugation with a single vector. These serve to generate the group of general rotoreflections and rotations.

The reflection $$c'$$ of a vector $$c$$ along a vector $\boldsymbol{e}_{0}$, or equivalently in the hyperplane orthogonal to $\mathbf{R}^{4,1}$, is the same as negating the component of a vector parallel to $p$. The result of the reflection will be
 * $$ c' = {-c_{\| m} + c_{\perp m}} = {-(c \cdot m)m^{-1} + (c \wedge m)m^{-1}}

= {(-m \cdot c - m \wedge c)m^{-1}} = -mcm^{-1} $$

This is not the most general operation that may be regarded as a reflection when the dimension $q$. A general reflection may be expressed as the composite of any odd number of single-axis reflections. Thus, a general reflection $$a'$$ of a vector $$a$$ may be written
 * $$ a \mapsto a' = -MaM^{-1} ,$$

where
 * $$ M = pq \cdots r$$ and $$ M^{-1} = (pq \cdots r)^{-1} = r^{-1} \cdots q^{-1}p^{-1} .$$

If we define the reflection along a non-null vector $$m$$ of the product of vectors as the reflection of every vector in the product along the same vector, we get for any product of an odd number of vectors that, by way of example,
 * $$ (abc)' = a'b'c' = (-mam^{-1})(-mbm^{-1})(-mcm^{-1}) = -ma(m^{-1}m)b(m^{-1}m)cm^{-1} = -mabcm^{-1} \,$$

and for the product of an even number of vectors that
 * $$ (abcd)' = a'b'c'd' = (-mam^{-1})(-mbm^{-1})(-mcm^{-1})(-mdm^{-1}) = mabcdm^{-1} .$$

Using the concept of every multivector ultimately being expressed in terms of vectors, the reflection of a general multivector $$A$$ using any reflection versor $$M$$ may be written
 * $$ A \mapsto M\alpha(A)M^{-1} ,$$

where $$\alpha$$ is the automorphism of reflection through the origin of the vector space ($\mathrm{SO}(3,1)$) extended through linearity to the whole algebra.

Rotations

If we have a product of vectors $$R = a_1a_2 \cdots a_r$$ then we denote the reverse as
 * $$\tilde R = a_r\cdots a_2 a_1.$$

As an example, assume that $$ R = ab $$ we get
 * $$R\tilde R = abba = ab^2a = a^2b^2 = ba^2b = baab = \tilde RR.$$

Scaling $$R$$ so that $$R\tilde R = 1$$ then
 * $$(Rv\tilde R)^2 = Rv^{2}\tilde R = v^2R\tilde R = v^2 $$

so $$Rv\tilde R$$ leaves the length of $$v$$ unchanged. We can also show that
 * $$(Rv_1\tilde R) \cdot (Rv_2\tilde R) = v_1 \cdot v_2$$

so the transformation $$Rv\tilde R$$ preserves both length and angle. It therefore can be identified as a rotation or rotoreflection; $$R$$ is called a rotor if it is a proper rotation (as it is if it can be expressed as a product of an even number of vectors) and is an instance of what is known in GA as a versor.

There is a general method for rotating a vector involving the formation of a multivector of the form $$ R = e^{-B \theta / 2} $$ that produces a rotation $$ \theta $$ in the plane and with the orientation defined by a $\mathrm{SO}(4,1)$-blade $m$.

Rotors are a generalization of quaternions to $k$-dimensional spaces.

Hypervolume of a parallelotope spanned by vectors
For vectors $B$ and $V$ spanning a parallelogram we have
 * $$ a \wedge b = ((a \wedge b) b^{-1}) b = a_{\perp b} b $$

with the result that $k$ is linear in the product of the "altitude" and the "base" of the parallelogram, that is, its area.

Similar interpretations are true for any number of vectors spanning an $B$-dimensional parallelotope; the exterior product of vectors $A$, that is $B$, has a magnitude equal to the volume of the $B \;\rfloor\; B^{+} = 1$-parallelotope. An $B^{+} = B^{\dagger}/(B \;\rfloor\; B^{\dagger})$-vector does not necessarily have a shape of a parallelotope – this is a convenient visualization. It could be any shape, although the volume equals that of the parallelotope.

Intersection of a line and a plane

We may define the line parametrically by $B$, where $B$ and $\mathcal{P}_{B}(A) = (A \;\rfloor\; B^{+}) \;\rfloor\; B$ are position vectors for points P and T and $(A \;\rfloor\; B) \;\rfloor\; B^{+}$ is the direction vector for the line.

Then
 * $$B \wedge (p-q) = 0$$ and $$B \wedge (t + \alpha v - q) = 0$$

so
 * $$\alpha = \frac{B \wedge(q-t)}{B \wedge v} $$

and
 * $$p = t + \left(\frac{B \wedge (q-t)}{B \wedge v}\right) v. $$

Rotating systems
A rotational quantity such as torque or angular momentum is described in geometric algebra as a bivector. Suppose a circular path in an arbitrary plane containing orthonormal vectors $B$ and $B$ is parameterized by angle.
 * $$\mathbf{r} = r(\hat{u} \cos \theta + \hat{v} \sin \theta) = r \hat{u}(\cos \theta + \hat{u} \hat{v} \sin \theta)$$

By designating the unit bivector of this plane as the imaginary number
 * $${i} = \hat{u} \hat{v} = \hat{u} \wedge \hat{v}$$
 * $$i^2 = -1 $$

this path vector can be conveniently written in complex exponential form
 * $$ \mathbf{r} = r \hat{ u} e^{i\theta} $$

and the derivative with respect to angle is
 * $$ \frac{d \mathbf{r}}{d\theta} = r \hat{u} i e^{i\theta} = \mathbf{r} i .$$

For example, torque is generally defined as the magnitude of the perpendicular force component times distance, or work per unit angle. Thus the torque, the rate of change of work $A$ with respect to angle, due to a force $A$, is
 * $$ \tau = \frac{dW}{d\theta} = F \cdot \frac{dr}{d\theta} = F \cdot (\mathbf{r} i) .$$

Rotational quantities are represented in vector calculus in three dimensions using the cross product. Together with a choice of an oriented volume form $B$, these can be related to the exterior product with its more natural geometric interpretation of such quantities as a bivectors by using the dual relationship
 * $$a \times b = -I (a \wedge b) .$$

Unlike the cross product description of torque, $B$, the geometric algebra description does not introduce a vector in the normal direction; a vector that does not exist in two and that is not unique in greater than three dimensions. The unit bivector describes the plane and the orientation of the rotation, and the sense of the rotation is relative to the angle between the vectors $m$ and $m$.

Geometric calculus
Geometric calculus extends the formalism to include differentiation and integration including differential geometry and differential forms.

Essentially, the vector derivative is defined so that the GA version of Green's theorem is true,
 * $$\int_A dA \,\nabla f = \oint_{\partial A} dx \, f$$

and then one can write
 * $$\nabla f = \nabla \cdot f + \nabla \wedge f$$

as a geometric product, effectively generalizing Stokes' theorem (including the differential form version of it).

In 1D when $m$ is a curve with endpoints $m$ and $n \ge 4$, then
 * $$\int_A dA \,\nabla f = \oint_{\partial A} dx \, f$$

reduces to
 * $$\int_a^b dx \, \nabla f = \int_a^b dx \cdot \nabla f = \int_a^b df = f(b) -f(a)$$

or the fundamental theorem of integral calculus.

Also developed are the concept of vector manifold and geometric integration theory (which generalizes differential forms).

Before the 20th century
Although the connection of geometry with algebra dates as far back at least to Euclid's Elements in the third century B.C. (see Greek geometric algebra), GA in the sense used in this article was not developed until 1844, when it was used in a systematic way to describe the geometrical properties and transformations of a space. In that year, Hermann Grassmann introduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to the propositional calculus) that encoded all of the geometrical information of a space. Grassmann's algebraic system could be applied to a number of different kinds of spaces, the chief among them being Euclidean space, affine space, and projective space. Following Grassmann, in 1878 William Kingdon Clifford examined Grassmann's algebraic system alongside the quaternions of William Rowan Hamilton in. From his point of view, the quaternions described certain transformations (which he called rotors), whereas Grassmann's algebra described certain properties (or Strecken such as length, area, and volume). His contribution was to define a new product – the geometric product – on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently, Rudolf Lipschitz in 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations in $v \mapsto -v$ dimensions. Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra.

Nevertheless, another revolutionary development of the 19th-century would completely overshadow the geometric algebras: that of vector analysis, developed independently by Josiah Willard Gibbs and Oliver Heaviside. Vector analysis was motivated by James Clerk Maxwell's studies of electromagnetism, and specifically the need to express and manipulate conveniently certain differential equations. Vector analysis had a certain intuitive appeal compared to the rigors of the new algebras. Physicists and mathematicians alike readily adopted it as their geometrical toolkit of choice, particularly following the influential 1901 textbook Vector Analysis by Edwin Bidwell Wilson, following lectures of Gibbs.

In more detail, there have been three approaches to geometric algebra: quaternionic analysis, initiated by Hamilton in 1843 and geometrized as rotors by Clifford in 1878; geometric algebra, initiated by Grassmann in 1844; and vector analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis can be seen in the use of $\theta$, $\theta$, $\theta/2$ to indicate the basis vectors of $x$: it is being thought of as the purely imaginary quaternions. From the perspective of geometric algebra, the even subalgebra of the Space Time Algebra is isomorphic to the GA of 3D Euclidean space and quaternions are isomorphic to the even subalgebra of the GA of 3D Euclidean space, which unifies the three approaches.

20th century and present
Progress on the study of Clifford algebras quietly advanced through the twentieth century, although largely due to the work of abstract algebraists such as Élie Cartan, Hermann Weyl and Claude Chevalley. The geometrical approach to geometric algebras has seen a number of 20th-century revivals. In mathematics, Emil Artin's Geometric Algebra discusses the algebra associated with each of a number of geometries, including affine geometry, projective geometry, symplectic geometry, and orthogonal geometry. In physics, geometric algebras have been revived as a "new" way to do classical mechanics and electromagnetism, together with more advanced topics such as quantum mechanics and gauge theory. David Hestenes reinterpreted the Pauli and Dirac matrices as vectors in ordinary space and spacetime, respectively, and has been a primary contemporary advocate for the use of geometric algebra.

In computer graphics and robotics, geometric algebras have been revived in order to efficiently represent rotations and other transformations. For applications of GA in robotics (screw theory, kinematics and dynamics using versors), computer vision, control and neural computing (geometric learning) see Bayro (2010).

References and further reading

 * Arranged chronologically

 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * . Chapter 1 as PDF
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
 * Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory

title: Geometric algebra


text:
Genetic

Genetic may refer to:
 * Genetics, in biology, the science of genes, heredity, and the variation of organisms
 * Genetic, used as an adjective, refers to genes
 * Genetic disorder, any disorder caused by a genetic mutation, whether inherited or de novo
 * Genetic mutation, a change in a gene
 * Heredity, genes and their mutations being passed from parents to offspring
 * Genetic recombination, refers to the recombining of alleles resulting in a new molecule of DNA
 * Genetic relationship (linguistics), in linguistics, a relationship between two languages with a common ancestor language
 * Genetic algorithm, in computer science, a kind of search technique modeled on evolutionary biology

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Genetic


text:
George Benson

George Washington Benson (born March 22, 1943) is an American jazz fusion guitarist, singer, and songwriter. He began his professional career at the age of 19 as a jazz guitarist.

A former child prodigy, Benson first came to prominence in the 1960s, playing soul jazz with Jack McDuff and others. He then launched a successful solo career, alternating between jazz, pop, R&B singing, and scat singing. His album Breezin' was certified triple-platinum, hitting no. 1 on the Billboard album chart in 1976. His concerts were well attended through the 1980s, and he still has a large following. Benson has won ten Grammy Awards and has been honored with a star on the Hollywood Walk of Fame.

Early career
Benson was born and raised in the Hill District of Pittsburgh, Pennsylvania. At the age of seven, he first played the ukulele in a corner drug store, for which he was paid a few dollars. At age eight, he played guitar in an unlicensed nightclub on Friday and Saturday nights, but the police soon closed the club down. At age nine, he started to record. Out of the four sides he cut, two were released: "She Makes Me Mad" backed with "It Should Have Been Me", with RCA Victor in New York. Although one source indicates this record was released under the name "Little Georgie", the 45rpm label is printed with the name George Benson. The single was produced by Leroy Kirkland for RCA's rhythm and blues label, Groove Records.

Benson attended and graduated from Schenley High School. As a youth he learned how to play straight-ahead instrumental jazz during a relationship performing for several years with organist Jack McDuff. One of his many early guitar heroes was country-jazz guitarist Hank Garland.

At the age of 21, he recorded his first album as leader, The New Boss Guitar, featuring McDuff. Benson's next recording was It's Uptown with the George Benson Quartet, including Lonnie Smith on organ and Ronnie Cuber on baritone saxophone. Benson followed it up with The George Benson Cookbook, also with Lonnie Smith and Ronnie Cuber on baritone and drummer Marion Booker. Miles Davis employed Benson in the mid-1960s, featuring his guitar on "Paraphernalia" on his 1968 Columbia release, Miles in the Sky before Benson went to Verve Records. Benson then signed with Creed Taylor's jazz label CTI Records, where he recorded several albums, with jazz heavyweights guesting, to some success, mainly in the jazz field. In addition to his own albums and performances, during this time Benson was a core member of the CTI All-Stars collective both touring and recording. As well as the collective's live albums, he also played and recorded on a number of the collective's members individual albums. Benson played on Freddie Hubbard 15th Annual Grammy Awards winning "Best Jazz Performance by a Group" album First Light as well as five other Hubbard studio albums.

His 1974 release, Bad Benson, climbed to the top spot in the Billboard jazz chart, while the follow-ups, Good King Bad (#51 Pop album) and Benson & Farrell (with Joe Farrell), both reached the jazz top-three sellers. Benson also did a version of The Beatles's 1969 album Abbey Road called The Other Side of Abbey Road, also released in 1969, and a version of "White Rabbit", originally written and recorded by San Francisco rock group Great Society, and made famous by Jefferson Airplane. Benson played on numerous sessions for other CTI artists during this time, including Freddie Hubbard and Stanley Turrentine, notably on the latter's acclaimed album Sugar.

1970s and 1980s
By the mid-to-late 1970s, as he recorded for Warner Bros. Records, a whole new audience began to discover Benson. On 1976's Breezin', Benson sang a lead vocal on the track "This Masquerade", a song written by Leon Russell. Benson's version (notable also for the lush, romantic piano intro and solo by Jorge Dalto), became a huge pop hit and won a Grammy Award for Record of the Year. (He had sung vocals infrequently on albums earlier in his career, notably his rendition of "Here Comes the Sun" on The Other Side of Abbey Road album.) The rest of the album is instrumental, including his rendition of the 1975 José Feliciano composition "Affirmation".

In 1976, Benson embarked on a tour called George & Minnie Live! with soul singer Minnie Riperton; she had recently been diagnosed with terminal breast cancer and would die in 1979. In addition, Benson appeared as a guitarist and backup vocalist on Stevie Wonder's song "Another Star" from Wonder's album Songs in the Key of Life.

He also recorded the original version of "The Greatest Love of All" for the 1977 Muhammad Ali bio-pic, The Greatest, which was later covered by Whitney Houston as "Greatest Love of All." During this time Benson recorded with the German conductor Claus Ogerman. The live take of "On Broadway," recorded a few months later from the 1978 release Weekend in L.A., also won a Grammy. The Qwest record label (a subsidiary of Warner Bros., run by Quincy Jones) released Benson's breakthrough pop album Give Me The Night, produced by Jones. Benson made it into the pop and R&B top ten with the song "Give Me the Night" (written by former Heatwave keyboardist Rod Temperton). He had many hit singles such as "Love All the Hurt Away," "Turn Your Love Around," "Inside Love," "Lady Love Me," "20/20," "Shiver," "Kisses in the Moonlight." More importantly, Quincy Jones encouraged Benson to search his roots for further vocal inspiration, and he rediscovered his love for Nat King Cole, Ray Charles and Donny Hathaway in the process, influencing a string of further vocal albums into the 1990s. Despite returning to his jazz and guitar playing most recently, this theme was reflected again much later in Benson's 2000 release Absolute Benson, featuring a cover of one of Hathaway's most notable songs, "The Ghetto." Benson accumulated three other platinum LPs and two gold albums.

1990s to present
In 1990, Benson was awarded an Honorary Doctorate of Music from the Berklee College of Music.

To commemorate the long relationship between Benson and Ibanez and to celebrate 30 years of collaboration on the GB Signature Models, Ibanez created the GB30TH, a limited-edition model with a gold-foil finish inspired by the traditional Japanese Garahaku art form. In 2009, Benson was recognized by the National Endowment of the Arts as a Jazz Master, the United States highest honor in jazz. Benson performed at the 49th issue of the Ohrid Summer Festival in North Macedonia on July 25, 2009, and his tribute show to Nat King Cole An Unforgettable Tribute to Nat King Cole as part of the Istanbul International Jazz Festival in Turkey on July 27.

In the fall of 2009, Benson finished recording an album entitled Songs and Stories with Marcus Miller, producer John Burk, and session musicians David Paich and Steve Lukather. As a part of the promotion for his album Songs and Stories, Benson has appeared or performed on The Tavis Smiley Show, Jimmy Kimmel Live! and Late Night with Jimmy Fallon.

He performed at the Java Jazz Festival March 4–6, 2011. In 2011, Benson released the album Guitar Man, revisiting his 1960s/early-1970s guitar-playing roots with a 12-song collection of covers of both jazz and pop standards produced by John Burk.

In June 2013, Benson released his fourth album for Concord, Inspiration: A Tribute to Nat King Cole, which included Wynton Marsalis, Idina Menzel, Till Brönner, and Judith Hill. In September, he returned to perform at Rock in Rio festival, in Rio de Janeiro, 35 years after his first performance at this festival, which was then the inaugural one.

In July 2016, Benson participated as a mentor in the Sky Arts programme Guitar Star in the search for the UK and Republic of Ireland's most talented guitarist.

In May 2018, Benson was featured on the Gorillaz single "Humility".

On July 12, 2018, it was announced that Benson had signed to Mascot Label Group.

Benson stopped touring internationally at the start of 2024 due to ill health - cancelling a series of UK concerts that summer.

Personal life
Benson has been married to Johnnie Lee since 1965 and has seven children. Benson describes his music as focusing more on love and romance, due to his commitment to his family and religious practices, with Benson being a Jehovah's Witness. Benson has been a resident of Englewood, New Jersey.

Grammy Awards
List of Grammy Awards received by George Benson

title: George Benson


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Grigory Barenblatt

Grigory Isaakovich Barenblatt (Григо́рий Исаа́кович Баренблат; 10 July 1927 – 22 June 2018) was a Russian mathematician.

Education
Barenblatt graduated in 1950 from Moscow State University, Department of Mechanics and Mathematics. He received his Ph.D. in 1953 from Moscow State University under the supervision of A. N. Kolmogorov.

Career and research
Barenblatt also received a D.Sc. from Moscow State University in 1957. He was an emeritus Professor in Residence at the Department of Mathematics of the University of California, Berkeley and Mathematician at Department of Mathematics, Lawrence Berkeley National Laboratory. He was G. I. Taylor Professor of Fluid Mechanics at the University of Cambridge from 1992 to 1994 and he was Emeritus G. I. Taylor Professor of Fluid Mechanics. His areas of research were:

 * 1) Fracture mechanics
 * 2) The theory of fluid and gas flows in porous media
 * 3) The mechanics of a non-classical deformable solids
 * 4) Turbulence
 * 5) Self-similarities, nonlinear waves and intermediate asymptotics.

Awards and honors

 * 1975 – Foreign Honorary Member, American Academy of Arts and Sciences
 * 1984 – Foreign Member, Danish Center of Applied Mathematics & Mechanics
 * 1988 – Foreign Member, Polish Society of Theoretical & Applied Mechanics
 * 1989 – Doctor of Technology Honoris Causa at the Royal Institute of Technology, Stockholm, Sweden
 * 1992 – Foreign Associate, U.S. National Academy of Engineering
 * 1993 – Fellow, Cambridge Philosophical Society
 * 1993 – Member, Academia Europaea
 * 1994 – Fellow, Gonville and Caius College, Cambridge; (since 1999, Honorary Fellow)
 * 1995 – Lagrange Medal, Accademia Nazionale dei Lincei
 * 1995 – Modesto Panetti Prize and Medal
 * 1996 - Visiting Miller Professorship - University of California Berkeley
 * 1997 – Foreign Associate, U.S. National Academy of Sciences
 * 1999 – G. I. Taylor Medal, U.S. Society of Engineering Science
 * 1999 – J. C. Maxwell Medal and Prize, International Congress for Industrial and Applied Mathematics
 * 2000 – Foreign Member, Royal Society of London
 * 2005 – Timoshenko Medal, American Society of Mechanical Engineers, "for seminal contributions to nearly every area of solid and fluid mechanics, including fracture mechanics, turbulence, stratified flows, flames, flow in porous media, and the theory and application of intermediate asymptotics. "

title: Grigory Barenblatt


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Grammatical tense

In grammar, tense is a category that expresses time reference. Tenses are usually manifested by the use of specific forms of verbs, particularly in their conjugation patterns.

The main tenses found in many languages include the past, present, and future. Some languages have only two distinct tenses, such as past and nonpast, or future and nonfuture. There are also tenseless languages, like most of the Chinese languages, though they can possess a future and nonfuture system typical of Sino-Tibetan languages. In recent work Maria Bittner and Judith Tonhauser have described the different ways in which tenseless languages nonetheless mark time. On the other hand, some languages make finer tense distinctions, such as remote vs recent past, or near vs remote future.

Tenses generally express time relative to the moment of speaking. In some contexts, however, their meaning may be relativized to a point in the past or future which is established in the discourse (the moment being spoken about). This is called relative (as opposed to absolute) tense. Some languages have different verb forms or constructions which manifest relative tense, such as pluperfect ("past-in-the-past") and "future-in-the-past".

Expressions of tense are often closely connected with expressions of the category of aspect; sometimes what are traditionally called tenses (in languages such as Latin) may in modern analysis be regarded as combinations of tense with aspect. Verbs are also often conjugated for mood, and since in many cases the three categories are not manifested separately, some languages may be described in terms of a combined tense–aspect–mood (TAM) system.

Etymology
The English noun tense comes from Old French tens "time" (spelled temps in modern French through deliberate archaization), from Latin tempus, "time". It is not related to the adjective tense, which comes from Latin tensus, the perfect passive participle of tendere, "stretch".

Uses of the term
In modern linguistic theory, tense is understood as a category that expresses (grammaticalizes) time reference; namely one which, using grammatical means, places a state or action in time. Nonetheless, in many descriptions of languages, particularly in traditional European grammar, the term "tense" is applied to verb forms or constructions that express not merely position in time, but also additional properties of the state or action – particularly aspectual or modal properties.

The category of aspect expresses how a state or action relates to time – whether it is seen as a complete event, an ongoing or repeated situation, etc. Many languages make a distinction between perfective aspect (denoting complete events) and imperfective aspect (denoting ongoing or repeated situations); some also have other aspects, such as a perfect aspect, denoting a state following a prior event. Some of the traditional "tenses" express time reference together with aspectual information. In Latin and French, for example, the imperfect denotes past time in combination with imperfective aspect, while other verb forms (the Latin perfect, and the French passé composé or passé simple) are used for past time reference with perfective aspect.

The category of mood is used to express modality, which includes such properties as uncertainty, evidentiality, and obligation. Commonly encountered moods include the indicative, subjunctive, and conditional. Mood can be bound up with tense, aspect, or both, in particular verb forms. Hence, certain languages are sometimes analysed as having a single tense–aspect–mood (TAM) system, without separate manifestation of the three categories. The term tense, then, particularly in less formal contexts, is sometimes used to denote any combination of tense proper, aspect, and mood. As regards English, there are many verb forms and constructions which combine time reference with continuous and/or perfect aspect, and with indicative, subjunctive or conditional mood. Particularly in some English language teaching materials, some or all of these forms can be referred to simply as tenses (see below).

Particular tense forms need not always carry their basic time-referential meaning in every case. For instance, the historical present is a use of the present tense to refer to past events. The phenomenon of fake tense is common crosslinguistically as a means of marking counterfactuality in conditionals and wishes.

Possible tenses
Not all languages have tense: tenseless languages include Chinese and Dyirbal. Some languages have all three basic tenses (the past, present, and future), while others have only two: some have past and nonpast tenses, the latter covering both present and future times (as in Arabic, Japanese, and, in some analyses, English), whereas others such as Greenlandic, Quechua, and Nivkh have future and nonfuture. Some languages have four or more tenses, making finer distinctions either in the past (e.g. remote vs. recent past) or in the future (e.g. near vs. remote future). The six-tense language Kalaw Lagaw Ya of Australia has the remote past, the recent past, the today past, the present, the today/near future and the remote future. Some languages, like the Amazonian Cubeo language, have a historical past tense, used for events perceived as historical.

Tenses that refer specifically to "today" are called hodiernal tenses; these can be either past or future. Apart from Kalaw Lagaw Ya, another language which features such tenses is Mwera, a Bantu language of Tanzania. It is also suggested that in 17th-century French, the passé composé served as a hodiernal past. Tenses that contrast with hodiernals, by referring to the past before today or the future after today, are called pre-hodiernal and post-hodiernal respectively. Some languages also have a crastinal tense, a future tense referring specifically to tomorrow (found in some Bantu languages); or a hesternal tense, a past tense referring specifically to yesterday (although this name is also sometimes used to mean pre-hodiernal). A tense for after tomorrow is thus called post-crastinal, and one for before yesterday is called pre-hesternal.

Another tense found in some languages, including Luganda, is the persistive tense, used to indicate that a state or ongoing action is still the case (or, in the negative, is no longer the case). Luganda also has tenses meaning "so far" and "not yet".

Some languages have special tense forms that are used to express relative tense. Tenses that refer to the past relative to the time under consideration are called anterior; these include the pluperfect (for the past relative to a past time) and the future perfect (for the past relative to a future time). Similarly, posterior tenses refer to the future relative to the time under consideration, as with the English "future-in-the-past": (he said that) he would go. Relative tense forms are also sometimes analysed as combinations of tense with aspect: the perfect aspect in the anterior case, or the prospective aspect in the posterior case.

Some languages have cyclic tense systems. This is a form of temporal marking where tense is given relative to a reference point or reference span. In Burarra, for example, events that occurred earlier on the day of speaking are marked with the same verb forms as events that happened in the far past, while events that happened yesterday (compared to the moment of speech) are marked with the same forms as events in the present. This can be thought of as a system where events are marked as prior or contemporaneous to points of reference on a time line.

Morphology of tense
Tense is normally indicated by the use of a particular verb form – either an inflected form of the main verb, or a multi-word construction, or both in combination. Inflection may involve the use of affixes, such as the -ed ending that marks the past tense of English regular verbs, but can also entail stem modifications, such as ablaut, as found as in the strong verbs in English and other Germanic languages, or reduplication. Multi-word tense constructions often involve auxiliary verbs or clitics. Examples which combine both types of tense marking include the French passé composé, which has an auxiliary verb together with the inflected past participle form of the main verb; and the Irish past tense, where the proclitic do (in various surface forms) appears in conjunction with the affixed or ablaut-modified past tense form of the main verb.

As has already been mentioned, indications of tense are often bound up with indications of other verbal categories, such as aspect and mood. The conjugation patterns of verbs often also reflect agreement with categories pertaining to the subject, such as person, number and gender. It is consequently not always possible to identify elements that mark any specific category, such as tense, separately from the others.

Languages that do not have grammatical tense, such as most Sinitic languages, express time reference chiefly by lexical means – through adverbials, time phrases, and so on. (The same is done in tensed languages, to supplement or reinforce the time information conveyed by the choice of tense.) Time information is also sometimes conveyed as a secondary feature by markers of other categories, as with the aspect markers 了 le and 過 guò, which in most cases place an action in past time. However, much time information is conveyed implicitly by context – it is therefore not always necessary, when translating from a tensed to a tenseless language, say, to express explicitly in the target language all of the information conveyed by the tenses in the source.

Nominal Tense
A few languages have been shown to mark tense information (as well as aspect and mood) on nouns. This may be called nominal tense, or more broadly nominal TAM which includes nominal marking of aspect and mood as well.

Syntax of tense
The syntactic properties of tense have figured prominently in formal analyses of how tense-marking interacts with word order. Some languages (such as French) allow an adverb (Adv) to intervene between a tense-marked verb (V) and its direct object (O); in other words, they permit [Verb-Adverb-Object] ordering. In contrast, other languages (such as English) do not allow the adverb to intervene between the verb and its direct object, and require [Adverb-Verb-Object] ordering.

Tense in syntax is represented by the category label T, which is the head of a TP (tense phrase).

Tenseless language
In linguistics, a tenseless language is a language that does not have a grammatical category of tense. Tenseless languages can and do refer to time, but they do so using lexical items such as adverbs or verbs, or by using combinations of aspect, mood, and words that establish time reference. Examples of tenseless languages are Burmese, Dyirbal, most varieties of Chinese, Malay (including Indonesian), Thai, Maya (linguistic nomenclature: "Yukatek Maya"), Vietnamese and in some analyses Greenlandic (Kalaallisut) and Guaraní.

In particular languages
The study of modern languages has been greatly influenced by the grammar of the Classical languages, since early grammarians, often monks, had no other reference point to describe their language. Latin terminology is often used to describe modern languages, sometimes with a change of meaning, as with the application of "perfect" to forms in English that do not necessarily have perfective meaning, or the words Imperfekt and Perfekt to German past tense forms that mostly lack any relationship to the aspects implied by those terms.

Latin
Latin is traditionally described as having six verb paradigms for tense (the Latin for "tense" being tempus, plural tempora):
 * Present (praesēns)
 * Imperfect (praeteritum imperfectum)
 * Perfect (praesēns perfectum)
 * Future (futūrum)
 * Pluperfect (plūs quam perfectum, praeteritum perfectum)
 * Future perfect (futūrum perfectum)

Latin has two past paradigms: imperfect and perfect. Imperfect verbs represent a past process combined with so called imperfective aspect, that is, they often stand for an ongoing past action or state at a past point in time (see secondary present) or represent habitual actions (see Latin tenses with modality). In contrast, perfect verbs usually represent past actions or present resulting states: for instance, mortuus est may mean either 'he died' or 'he is dead'. Like the imperfect, the pluperfect, the perfect and the future perfect may also realise relative tenses, standing for events that are past at the time of another event (see secondary past): for instance, mortuus erat, mortuus est, mortuus erit may stand for respectively 'he had died', 'he has died' and 'he will have died'.

Latin verbs are inflected for tense and aspect together with mood (indicative, subjunctive, infinitive, and imperative) and voice (active or passive). Most verbs can be built by selecting a verb stem and adapting them to endings. Endings may vary according to the speech role, the number and the gender of the subject or an object. Sometimes, verb groups function as a unit and supplement inflection for tense (see Latin periphrases). For details on verb structure, see Latin tenses and Latin conjugation.

Ancient Greek
The paradigms for tenses in Ancient Greek are similar to the ones in Latin, but with a three-way aspect contrast in the past: the aorist, the perfect and the imperfect. Both aorist and imperfect verbs can represent a past event: through contrast, the imperfect verb implies a longer duration ('ate' vs 'ate for a long time'). The aorist participle represents the first event of a two-event sequence and the present participle represents an ongoing event at the time of another event. Perfect verbs stood for past affecting actions if the result was still present (e.g. "I have found it") or for present states resulting from a past event (e.g. "I remember"). For further details see Ancient Greek verbs.

English
English has only two morphological tenses: the present (or non-past), as in he goes, and the past (or preterite), as in he went. The non-past usually references the present, but sometimes references the future (as in the bus leaves tomorrow). In special uses such as the historical present it can talk about the past as well. These morphological tenses are marked either with a suffix (walk(s) ~ walked) or with ablaut (sing(s) ~ sang).

In some contexts, particularly in English language teaching, various tense–aspect combinations are referred to loosely as tenses. Similarly, the term "future tense" is sometimes loosely applied to cases where modals such as will are used to talk about future points in time.

Other Indo-European languages
Proto-Indo-European verbs had present, perfect (stative), imperfect and aorist forms – these can be considered as representing two tenses (present and past) with different aspects. Most languages in the Indo-European family have developed systems either with two morphological tenses (present or "non-past", and past) or with three (present, past and future). The tenses often form part of entangled tense–aspect–mood conjugation systems. Additional tenses, tense–aspect combinations, etc. can be provided by compound constructions containing auxiliary verbs.

The Germanic languages (which include English) have present (non-past) and past tenses formed morphologically, with future and other additional forms made using auxiliaries. In standard German, the compound past (Perfekt) has replaced the simple morphological past in most contexts.

The Romance languages (descendants of Latin) have past, present and future morphological tenses, with additional aspectual distinction in the past. French is an example of a language where, as in German, the simple morphological perfective past (passé simple) has mostly given way to a compound form (passé composé).

Irish, a Celtic language, has past, present and future tenses (see Irish conjugation). The past contrasts perfective and imperfective aspect, and some verbs retain such a contrast in the present. Classical Irish had a three-way aspectual contrast of simple–perfective–imperfective in the past and present tenses. Modern Scottish Gaelic on the other hand only has past, non-past and 'indefinite', and, in the case of the verb 'be' (including its use as an auxiliary), also present tense.

Persian, an Indo-Iranian language, has past and non-past forms, with additional aspectual distinctions. Future can be expressed using an auxiliary, but almost never in non-formal context. Colloquially the perfect suffix -e can be added to past tenses to indicate that an action is speculative or reported (e.g. "it seems that he was doing", "they say that he was doing"). A similar feature is found in Turkish. (For details, see Persian verbs.)

Hindustani (Hindi and Urdu), an Indo-Aryan language, has indicative perfect past and indicative future forms, while the indicative present and indicative imperfect past conjugations exist only for the verb honā (to be). The indicative future is constructed using the future subjunctive conjugations (which used to be the indicative present conjugations in older forms of Hind-Urdu) by adding a future future suffix -gā that declines for gender and the number of the noun that the pronoun refes to. The forms of gā are derived from the perfective participle forms of the verb "to go," jāna. The conjugations of the indicative perfect past and the indicative imperfect past are derived from participles (just like the past tense formation in Slavic languages) and hence they agree with the grammatical number and the gender of noun which the pronoun refers to and not the pronoun itself. The perfect past doubles as the perfective aspect participle and the imperfect past conjugations act as the copula to mark imperfect past when used with the aspectual participles. Hindi-Urdu has an overtly marked tense-aspect-mood system. Periphrastic Hindi-Urdu verb forms (aspectual verb forms) consist of two elements, the first of these two elements is the aspect marker and the second element (the copula) is the common tense-mood marker. Hindi-Urdu has 3 grammatical aspectsː Habitual, Perfective, and Progressive; and 5 grammatical moodsː Indicative, Presumptive, Subjunctive, Contrafactual, and Imperative. (Seeː Hindi verbs)

In the Slavic languages, verbs are intrinsically perfective or imperfective. In Russian and some other languages in the group, perfective verbs have past and "future tenses", while imperfective verbs have past, present and "future", the imperfective "future" being a compound tense in most cases. The "future tense" of perfective verbs is formed in the same way as the present tense of imperfective verbs. However, in South Slavic languages, there may be a greater variety of forms – Bulgarian, for example, has present, past (both "imperfect" and "aorist") and "future tenses", for both perfective and imperfective verbs, as well as perfect forms made with an auxiliary (see Bulgarian verbs). However it doesn't have real future tense, because the future tense is formed by the shortened version of the present of the verb hteti (ще) and it just adds present tense forms of person suffixes: -m (I), -š (you), -ø (he,she,it), -me (we), -te (you, plural), -t (they).

Other languages
Finnish and Hungarian, both members of the Uralic language family, have morphological present (non-past) and past tenses. The Hungarian verb van ("to be") also has a future form.

Turkish verbs conjugate for past, present and future, with a variety of aspects and moods.

Arabic verbs have past and non-past; future can be indicated by a prefix.

Korean verbs have a variety of affixed forms which can be described as representing present, past and future tenses, although they can alternatively be considered to be aspectual. Similarly, Japanese verbs are described as having present and past tenses, although they may be analysed as aspects. Some Wu Chinese languages, such as Shanghainese, use grammatical particles to mark some tenses. Other Chinese languages and many other East Asian languages generally lack inflection and are considered to be tenseless languages, although they often have aspect markers which convey certain information about time reference.

For examples of languages with a greater variety of tenses, see the section on possible tenses, above. Fuller information on tense formation and usage in particular languages can be found in the articles on those languages and their grammars.

Rapa
Rapa is the French Polynesian language of the island of Rapa Iti. Verbs in the indigenous Old Rapa occur with a marker known as TAM which stands for tense, aspect, or mood which can be followed by directional particles or deictic particles. Of the markers there are three tense markers called: Imperfective, Progressive, and Perfective. Which simply mean, Before, Currently, and After. However, specific TAM markers and the type of deictic or directional particle that follows determine and denote different types of meanings in terms of tenses.

Imperfective: denotes actions that have not occurred yet but will occur and expressed by TAM e.

e naku mai te 'āikete anana'i

IPFV come DIR INDEF teacher tomorrow

'The teacher is coming tomorrow.'

e mānea tō pē'ā ra

IPFV pretty DEF woman DEIC

'That woman is beautiful.'

Progressive: Also expressed by TAM e and denotes actions that are currently happening when used with deictic na, and denotes actions that was just witnessed but still currently happening when used with deictic ra.

e 'āikete na 'ōna i te tamariki

IPFV learn DEIC 3S ACC INDEF child/children

'He is teaching some children.'

e kai na ou i kota'i kororio eika

IPFV eat DEIC 1S ACC one small fish

'I am eating a small fish.'

e tunu na ou i te mīkaka tonga te pōpongi

IPFV cook DEIC 1S ACC INDEF taro all INDEF morning

'I cook taro every morning.'

e kaikai ra te kurī i te moa

IPFV eat.continuously DEIC INDEF dog ACC INDEF chicken

'The dog is eating a chicken.'

e mate atu ra 'ōna

IPFV die DIR DEIC 3S

'She has just died.'

Perfective: denotes actions that have already occurred or have finished and is marked by TAM ka.

ka ngurunguru te kurī

PFV growl INDEF dog

'A dog growled.'

ka tākave tō tangata i te mango

PFV kill DEF man ACC INDEF shark

'The man killed the shark.'

ka tunu na ou i te mīkaka tonga te pōpongi

PFV cook DEIC 1S ACC INDEF taro all INDEF morning

'I used to cook taro every morning'

In Old Rapa there are also other types of tense markers known as Past, Imperative, and Subjunctive.

Past

TAM i marks past action. It is rarely used as a matrix TAM and is more frequently observed in past embedded clauses

i komo mātou

PST sleep 1PL.EXCL

'We slept.'

e a'a koe i 'aka-ineine

IPFV what 2S PST CAUS-ready

'What did you prepare?'

Imperative

The imperative is marked in Old Rapa by TAM a. A second person subject is implied by the direct command of the imperative.

a naku mai

IMP come DIR

'Come here.'

a kai tā-koe eika

IMP eat INDEF.PossA-2S fish

'Eat your fish.'

For a more polite form rather than a straightforward command imperative TAM a is used with adverbial kānei. Kānei is only shown to be used in imperative structures and was translated by the french as "please".

a rave mai kānei tō mea

IMP take DIR PREC DEF thing

'Please take the thing.'

a omono kānei koe tō ka'u ra

IMP dress PREC 2S DEF clothing DEIC

'Please dress yourself in those clothes.'

It is also used in a more impersonal form. For example, how you would speak toward a pesky neighbor.

a naku kānei

IMP go PREC

'Please leave now!' Subjunctive

The subjunctive in Old Rapa is marked by kia and can also be used in expressions of desire

kia naku ou i te 'are e kaikai ou

SBJV come 1S PREP INDEF house IPFV eat.continuously 1S

'When I get to the house, I will eat.'

kia rekareka kōrua

SBJV happy 2DU

'May you two be happy.'

Tokelau
The Tokelauan language is a tenseless language. The language uses the same words for all three tenses; the phrase E liliu mai au i te Aho Tōnai literally translates to Come back / me / on Saturday, but the translation becomes 'I am coming back on Saturday'.

Wuvulu-Aua
Wuvulu-Aua does not have an explicit tense, but rather tense is conveyed by mood, aspect markers, and time phrases. Wuvulu speakers use a realis mood to convey past tense as speakers can be certain about events that have occurred. In some cases, realis mood is used to convey present tense — often to indicate a state of being. Wuvulu speakers use an irrealis mood to convey future tense. Tense in Wuvulu-Aua may also be implied by using time adverbials and aspectual markings. Wuvulu contains three verbal markers to indicate sequence of events. The preverbal adverbial loʔo 'first' indicates the verb occurs before any other. The postverbal morpheme liai and linia are the respective intransitive and transitive suffixes indicating a repeated action. The postverbal morpheme li and liria are the respective intransitive and transitive suffixes indicating a completed action.

Mortlockese
Mortlockese uses tense markers such as mii and to denote the present tense state of a subject, aa to denote a present tense state that an object has changed to from a different, past state, kɞ to describe something that has already been completed, pɞ and lɛ to denote future tense, pʷapʷ to denote a possible action or state in future tense, and sæn/mwo for something that has not happened yet. Each of these markers is used in conjunction with the subject proclitics except for the markers aa and mii. Additionally, the marker mii can be used with any type of intransitive verb.

title: Grammatical tense