Multivector Mathematical Arcana
The following sections are somewhat mathematical and included for completeness. The reader interested primarily in exploting multivectors for standard Euclidean or Minkowski geometries can safely skip ahead to "Multivector Calculus".

Algebraic Equivalences
"Analytical geometry has never existed. There are only people who do linear geometry badly, by taking coordinates, and they call this analytical geometry. Out with them!" - Jean Dieudonné

N-D geometric algebras are essentially the algebras of N anticommuting scalar-squared generators. That a multivector is a linear combination of our particular generators choice is what makes it a 1-vector. Grades are "geometry dependant" and arise from a particular choice of N anticommuting generators deemed to be 1-vectors. This choice determines the restricted products and conjugations and also the signatures and associated metric.

Let (e1,e2,..,eN) be an orthogonal basis for geometric algebra Âp,q,r for p³1 with e12=1. Then {e1,e12,e13,..,e1N} provides an anticommuting <1;2>-grade generating basis for ÂN having all but the first signature negated so Âp,q,r @ Âq+1,p-1,r for p³1 .
For even N this has the disconcerting effect of making former (N-1)-blade e234..N the new "N-blade" pseudoscalar i', having opposite signature to the previous pseudoscalar i but (anti)commuting with the new "1-vector" basis in like manner to i=e12..N 's (anti)commutation with the original ej.
For any N, e1'ºe1 and e2'ºe12 must now both be regarded as orthogonal 1-blades with their geometric product e12'=e2 now regarded as a 2-blade arising from a new outter product Ù' with e1'e2'=e1(e12)=e1Ù'e12=-e12Ù'e1=e2. Former 1-blade e2 thus takes the role of a 2-blade e12' while 2-blade e12 assumes the role of a 1-blade e2'. The grade of a blade is thus ultimately a matter of interpretation. e1+e12 is mixed grade in Âp,q,r but a pure 1-vector in Âq+1,p-1,r.
As well as new restricted products Ù', ¿' etc. we aquire new conjugations §', #' etc. with e12§'=e2'§' = e2'=e12 and so forth, but since §# acts identically on 1 and 2 blades we have (§#)'=§# and can regard §# as geometrically invarient under the Âp,q,r @ Âq+1,p-1,r equivalence.

Adoption of such alternate geometric basies can be considered as differing "geometric interpretations" of the underlying multivector algebra.

Odd N

When N=p+q and ½N(N-1)+q are both odd (eg. Â4k+3 or Â4k,1) so that the pseudoscalar is central and negative squared then we can replace any even number of 1-vector generators whose signature we wish to change with their even (N-1)-vector duals. These have  opposite square and anticommutes with every other ei and ei*. [  If we dual an odd number we no longer have a generating basis (eg. { e23,e2,e3}  cannot generate e1) ]
For odd N>3, when an odd number of e1,e2, and e3 have negative signature so that e1232=1 then the anticommuting <1;4> generaters { e1,e2,e3,e1234,e1235,..,e123N} negates the signatures all but the first three elements with the same full product for odd N, giving Âp,q @ Âq+1,p-1 when p³2, q³1 and Âp,q @ Âq-3,p+3 when q³3 , with § invariant.
When N=4k+1 then starting from Â4k+1 we can judiciously replace pairs of 1-vectors with 4k-vectors to obtain any ÂN-2m,2m with § geometrically invarient. When N=4k+3 we can change 1-vectors to (4k+2)-vectors in pairs and it is Clifford conjugation #§ that is invarient.

When p³4 then {e1e1234,e2e1234,e3e1234,e4e1234,e5,e6,..,eN} provides anticommuting <1;3> generators with the first four positive signatures negated so Âp,q,r @ Âp-4,q+4,r for p³4. This works for odd and even N and keeps #§ invariant.

The distinct odd dimensioned geometric algebras of nonnull pseudoscalar for N£7 are as follows, with positive signature pseudoscalared listed first and geometrically invarient conjugations indicated in []..
Â1     ;; Â0,1 @ C1×1 .
Â2,1 ; Â0,3 @ C0,2 ;; Â3 @ Â1,2 @ C2 @ C2×2 [§#]
Â5 @ Â1,4  ; Â3,2   ;; Â4,1 @ Â2,3 @ Â0,5 @ C4 @ C4×4 [§] ;
Â0,7 @ Â4,3 ; Â6,1 @ Â2,5 @ C0,8  ;; Â7 @ Â1,6 @ Â5,2 @ Â3,4 @ C8 @ C8×8 [§#]

Even N
When N and ½N(N-1)+q are both even (eg Â4k-2l,2l) so that i has positive signature and anticommutes with 1-vectors we cannot selectivly dual the basis 1-vectors but we can dual them all to obtain Â4k-2l,2l @ Â2l,4k-2l and in particular Â4k @ Â0,4k.
Hence the distinct even dimensioned geometric algebras of nonnull pseudoscalar for N£8 are as follows, with invarient conjugations indicated in []. These do not divide according to pseudoscalar signature, which for even N is not geometrically invariant.
Â0,2 @ Q1×1 ; Â2 @ Â1,1 @ Â2×2 [§#]
Â4 @ Â0,4 @ Â1,3 @ Q2×2 ; Â3,1 @ Â2,2 @ Â4×4 [§#].
Â6 @ Â5,1 @ Â2,4 @ Â1,5 @  Q4×4     ; Â4,2 @ Â3,3 @   Â0,6 @ Â8×8[§#].
Â8 @ Â0,8 @ Â4,4 @ Â1,7 @ Â5,3 @ Â16×16 ; Â7,1 @ Â3,5 @ Â2,6 @ Â6,2 .

Thus we have Minksowski spacetime Â3,1 and timespace Â1,3 fundamentally distinct .

Lounesto [16.4] shows that Âp+8,q @ Âp,q+8 @ (Âp,q)16×16 , ie. the space of 16×16 matrices of Âp,q multivectors.

Algebraic Product Formulations
By considering generating sets some of which commute we can reduce Âp,q,r into a "product algebra" of two smaller geometric algebras. Suppose geometric algebra UN has a basis e1,e2,..eN for N > 2 with eN and eN-1 nonnull. We can express any a in UN not only as a real weighted sum of products of the ei (the ei "generate" UN) but alternatively as a real-wieghted sum of products of the N-2 3-blades eN-1eNe1 , eN-1eNe2 , ... , eN-1eNeN-2 and the two 1-vectors eN-1 and eN which commute with all the 3-blade generators but anticommute with eachother. The 3-blade generators all anticommute with eachother so we have UN @ UN-2 × U2   where U2 is a 2-D geometric algebra having signatures eN-12 and eN-22    and UN-2 is an (N-2)-D geometric algebra with signatures (eN-1eNei)2 = -eN-12eN2ei2   .
Symbol × here indicates the "product" algebra obtained by allowing the generators of seperate algebras to multiply commutatively.
If N³ 4 we can repeat the trick, "picking out" a pair of the nonnull 3-blades to produce an alternate set of N generators for UN consitisting of two 1-vectors, two 3-blades, and N-4 5-blades; all generators commuting with those of different grade and anticommuting with those of the same grade. If N³6 , we can replace all but two of the 5-blades with 7-blades, and so forth.
By such means, we can express any N-D multivector in terms of N odd-graded blade generators that each commute with all but at most one other generator (the unique generator sharing the same grade). The signatures of these "multigrade generators" depend on the signatures of the original 1-vector basis. We can express these results most succinctly as
Âp+2,q+0,r @ Â2,0 × Âq,p,r    @ Â2×2 × Âq,p,r
Âp+1,q+1,r @ Â1,1 × Âp,q,r    @ Â2×2 × Âp,q,r
Âp+0,q+2,r @ Â0,2 × Âq,p,r    @ Q × Âq,p,r

For p³q we thus have Âp,q @ (Â1,1)q × Âp-q @ (Â1,1)q × (Â2)2l × Âm,0,r where p-q = 4l+m with mÎ {0,1,2,3} .

"Factoring" geometric algebras in this way does not make for compacter storage since we still have N elements generating 2N distinct products amd so requiring 2N real coordinates for the genreral real-weighted sum. We merely obtain more commutative multiplication tables for the "multigrade basis" than for the 1-vectors one.

Complexification
For odd N, if the central pseudoscalar i has negative square (ie. if ½(N-1)N+q is odd) we can express any mutlivector a as b+ic for even b and c giving Âp,q @ (Cp,q)+ @ ((Â0,1)p,q)+ , associating i with i. The complex conjugation b-ic is provided by the main involuation #.
If ½(N-1)N+q is even we have Âp,q @ ((Â1,0)p,q)+ .

Minimal Geometric Algebras

Our geometric vocabulary is somewhat broad with various product symbols ¿,.,Ù,°...; operators +,/,,...;  conjugations §,#,... ; and so forth. Which symbols are fundamental in that they cannot be semantically defined using other fundamental symbols?
Let us first suppose just + and ¨ ; precedence indication brackets ( and ) ;  orthonormal N-frame symbols {e1,e2,...eN} ; and some multivector "variable" symbols a, b, ... together with an assigment statement = .
Unity symbols 1 and (-1) (and hence the left and right negation conjugation operators - and - ) are available from {e1e1 , e2e2, e1e2e1e2} depending on the signatures and we obtain, semantically at least, all the integer-coordinated multivectors in the given frame.
Exponmentiation can be defined using + and ¨ given some form of infinitite summation å symbolism and if we then suppose logarithm as a particular inverse of we obtain a-1 as (-(a)) which we can write as -1 = - whence a/b º a(b-1)   =   a(b-) and we attain the general real-coordinate multivector space. [under construction]

Bivectors

Left multiplication by a bivector ( a ® b2a ) casts scalars into b2. If b2 is a 2-blade It rotates directions within b2 (by b2), while casting directions perpendicular to b2 into trivectors. For N=3, it casts bivectors in b2 to scalars while bivectors normal to b2 are rotated by b2 and the pseudoscalar is cast to 1-vector b2*.
Right multiplication by a bivector has similar effects. Indeed a 2-blade commutes with all multivectors in its dual space , which is not the case for a 1-vector. Consequently, it is occasionally advantageous to represent 3D 1-vectors by their bivector duals.
Let b be multivector having only bivector and scalar components. a¿b = a.b while b¿a = b.a + b0a
b2×a sends ^(a,b2) to 0 and rotates ¯(a,b2) in b2 by ½p, scaling it by |b2|.
The operation a ® (ba).a for nonnull b is interesting, casting ¯(a,b) to 0 and ^(a,b) into
|^(a,b)|2b = |aÙb|2(b-2)b = |aÙb~|2b .

Expressed as sum of commuting 2-blades
Any Euclidean     N-D 2-vector b2 can be represented as the scalar-weighted sum of at most ½N orthogonal (by which we mean geometrically commuting) unit 2-blades. Such a decomposition is unique except in cases where two or more wieghts are identical. "Decomposing" a given bivector b2 in this way is nontrivial but a computational algorithm exists.
[ We seek 2-blades a1,A2..,am with aiaj=aiÙaj=ajai and b2=a1+A2+..+am .
Now (b2k)<2k> = b2Ù...Ùb2 = k!Sr<s<..<varÙas..Ùav = k!Sr<s<..<v aras..av     where there are k terms in each product and k suffices r,s,..v .
Hence (b2k-1)<2k-2> ¿ (b2k)<2k> = k!(k-1)! åi=1m  ai (    Sr<s<..<u ¹ i ar2as2..au2 )     for 1£k<m which, if the scalar ai2 are known and distinct, provides m linear NC2-dimensional equations , solvable for ai by conventional numerical methods.
The scalar ai2 can shown to be the m roots of the mth order scalar polynomial åk=0m  (-l)m-k((b2k)<2k> ¿ (b2k)<2k>)=0 . See Hestenes & Sobczyk (3-4) for a fuller treatment. ]

Spinors
 Recall that for puresquare a,     a↑ º ea = cos(|a|) + a~ sin(|a|) if a2 < 0 ; cosh(|a|) + a~ sinh(|a|) if a2 > 0 ; 1+a if a2 = 0.

Authors differ in precise meaning of the terms "spinor" and "rotor". [ Hestenes & Sobczyk define a spinor as a multivector a satisfying (aba§)<1>= aba§     " 1-vector b, which is equivalent to a being an even versor except when N=p+q divides 4, in which case we have a=(a+bi)v where v is an even versor and a and b are scalars ]
Here we use k-rotor to indicate a nonnull 2k-versor and pure/impure k-spinor to indicate an exponentiated unit or null pure/impure k-vector. ebkf for scalar f. Postive scalars are thus pure 0-spinors. If b is a nonnull 2-versor, then so is eb [   cos(f)+ sin(f)e12 = e1( cos(f)e1+ sin(f)e2) ] . Since any 2-blade is a 2-versor, any 2-spinor is a 1-rotor.

Under the geometric product, the pure spinors generate ÂN+ which for N=3 is isomorphic to quaternion space (the 2-spinor (inf) corresponding to normalised quaternion { cos(f/2), sin(f/2)n} ).

Whenever i squares to -1 and commutes with a we have the spinor adjustment rule
(1+a)eqi = (1+a)eqai = (1+a)e-qa* (when i=i)     provided a2=1.
(1-a)eqi = (1-a)e-qai = (1+a)eqa*     provided a2=-1 .
[ Proof : (1+a)(eqi - eqia) = (1+a) sinq(i - ia) = (1+a)(1-a)i sinq = 0 for a2=1  .]
In Â3 we have (1+a)eqi=(1+a)eqai which means that under a (1+a) mutiplier we can replace ai exponentiations by i exponentiations, which tend to be more commutative.

Alternate representations of spinors
Physicists tend to represent spinors in convoluted and confusing ways that obsfucate more than they reveal. In Quantum Mechanics , a "spinor" is typically defined via two complex numbers a0+b0i = r0eq0i and a1+b1i = r1eq1i = refi r0eq1i when r0¹0 as the singular (zero determinant) 2×2 complex matrix
 b= æ (a0+ib0)(a1+ib1) -(a0+ib0)2 ö = æ r0r1e(q0+q1)i -r02 e2q0i ö = r02e2q0i æ refi -1 ö è (a1+ib1)2 -(a0+ib0)(a1+ib1) ø è r12e2q1i -r0r1 e(q0+q1)i ø è r2e2fi -r efi ø

The spinor is considered normalised if the real scalar part ½(a02+b02+a12+b12) =½(r02+r12) of the singular matrix
 b(b§)= æ (a0+ib0)(a0-ib0) (a0+ib0)(a1-ib1) ö is equal to ½. è (a0-ib0)(a1+ib1) (a1+ib1)(a1-ib1) ø
This gives r0Î[0,1] , r1=(1-r02)½ , r0=(1+r2) , r0r1=r02r=r(1+r2)-1 .

Such 2×2 complex matrices are often represented via the si and 1 "Pauli" matrix representors for 1,e1,e2,e3 defined above using
 æ a b ö = ½(a+d)1 + ½(b+c) s1   + ½i(b-c) s2 + ½(a-d) s3 è c d ø

We can associate i with i=e123 and regard b as the null (b2=0) Â3 multivector
b= ½r02e2q0i (e1(r2e2fi-1) -e2i(r2e2fi+1) + e32refi )
= ½r02e2q0i(r2+1)(u+vi)     where
u = (e1(r2 cos2f-1) + e2r2 sin2f + e32r cosf)(r2+1)-1     ; v = (e1r2 sin2f - e2(r2 cos2f+1) +e32r sinf)(r2+1)-1     are orthonormal 1-vectors (u2=v2=1) ; uv = (r2+1)-1(e23 2r cosf + e31 2r sinf + e12 ( 1-r2) ) .
Letting w=-u-1v i= e1 2r cosf + e2 2r sinf + e3 (1-r2) = (r2+1)Riem(refi) we have
b= ½r02e2q0i(r2+1)u(1-w)       where w2=1 .

Riemann Sphere Representation

In Riemann sphere representation of C + ¥ we map complex "point" z=a+bi=refi to the real 3D unit 1-vector corresponding to the intersection of the line joining Â3 point ae1+be2 to the "south pole" -e3 with the unit sphere centred at 0. This point is given in spherical polar coordinates as
Riem(z) º [2 tan-1(r),f,1] = efe12) e-2( tan-1(r))e31 e3 = (1+r2)-1( e3(1-r2) + 2r( cosfe1 + sinfe2) ) .
We further define Riem(¥)=-e3 .
We have Riem(z-1) = [ p-2 tan-1(r),-f,1] ; Riem(-z) = [2 tan-1(r),f+p,1] ; Riem(efi) = e1efe12 .
[ If we site the argand plane tangent at the "north pole" e3  rather that at the equator we have Riem(z)º[ 2 tan-1r),f,1] ]
[ Proof : u2 = (r2 cos2f-1)2 + r4 sin2f2 + 4r2 cosf2 = r4 + 1 - 2r2 cos2f+ 4r2 cosf2 = r4 - 1 - 2r2( cosf2- sinf2)+ 4r2 cosf2 = r4 - 1 + 2r2( cosf2+ sinf2) = (r2 + 1)2
v 2 = r4 sin2f2 + (r2 cos2f+1)2 + 4r2 sinf2 = r4 + 2r2 cos2f +1 + 4r2 sinf2 = 1+r4 + 2r2
u¿v= (r2 cos2f-1)r2 sin2f -r2 sin2f(r2 cos2f+1) +2r2 cosf sinf = (r2 cos2f-1) sin2f - sin2f(r2 cos2f+1) + sin2f
uv = (e1(r2 cos2f-1) + e2r2 sin2f + e32r cosf) (e1r2 sin2f - e2(r2 cos2f+1) +e32r sinf)
= e23 (2r3 sin2f sinf+2(r2 cos2f+1)r cosf) + e31 (2r3 sin2f cosf-2(r2 cos2f-1)r sinf) + e12 ( -(r2 cos2f-1)(r2 cos2f+1) - r4 sin2f2)
= e23 2(r3( sin2f sinf+ cos2f cosf)+r cosf) + e31 (2r3( sin2f cosf- cos2f sinf)+r sinf) + e12 ( 1-r4)
= e23 2r(r2+1) cosf + e31 2r(r2+1) sinf + e12 ( 1-r4)
= (r2+1)(e23 2r cosf + e31 2r sinf + e12 (1-r2) )  .]

u = e1(r2 cos2f-1) + e2r2 sin2f + e32r cosf     ;
v = e1r2 sin2f - e2(r2 cos2f+1) +e32r sinf
w = e1 2r cosf + e2 2r sinf + e3 (1-r2)

For small r, u » -e1 + e32r cosf     ; v » - e2 +e32r sinf     ; w » 2r(e1 cosf + e2 sinf + e3 (1-r2)

For large r,
u » r2(e1 cos2f + e2 sin2f) v » r2(e1 sin2f - e2 cos2f) w » -r2e3 . Thus with r=0 we have u=e1, v=-e2, w=e3 . As r increases u aquires a cosf e3 component,
When r0=0 we have b= ½ (e1r12e2q1i - e2i(r12e2q1i) ) = ½ e2q1i r12 (e1 - e31 ) = ½ r12 e2q1i e1 (1 + e3 ) .

Let z = refi =r1(r0-1)eq1-q0 .

• z=1 Þ r=1, f=0 Þ be2q0ie3(1-e1)
[ Proof : ½ ½ e2q0i(-2e31 + e32 )    = ½ e2q0i(-e31 + e3) = ½ e2q0ie3(1-e1)  .]
• z=-1 Þ r=1, f=p Þ b= ½ e(2q0+p)i e3(1+e1)
[ Proof : (e1(e2fi-1) - e31(e2fi+1) + e32efi )   = ( - e31(2) + e3(-2) )  = -2e3(1+e1)  .]
• z=i Þ r=1, fp Þ b= ½ e(2q0+p)i e1(1-e2)
[ Proof : (e1(e2fi-1) - e31(e2fi+1) + e32efi )   =(e1(-2) + e32i )   =2(-e1 + e12 )   =-2e1(1-e2 )    .]
• z=-i Þ r=1, f=-½p Þ b= ½ e2q0i e1(1+e2)
[ Proof : (e1(e2fi-1) - e31(e2fi+1) + e32efi )   =(e1(-2) - e32i )   =2(-e1 - e12 )   =2e1(1+e2 )    .]
• z=0 Þ r0=1,r1=0 Þ b= =½ e2q0+pi   e1(1-e3)
[ Proof : e2q0i (e1(-1) - e31(+1)  )    =½ e2q0i   e1(-1+e3)      =½ e(2q0+p)i e1(1-e3)       .]
• z=¥ Þ r0=0,r1=1 Þ b= ½ e2q1i e1 (1 + e3 ) .

Multivector Kinematics
A natural way to represent a 3D "oriented point" conventionally represented by c, A=(i,j,k) for orthogonal _3x3 real matrix A in GHC space Â4,1 is with the unit plussquare 3-blade (ce¥)Ùk* a3 representing a unit radius 1-sphere (circle) of centre c, and axis k scaled in some manner by a measure of Eulerian angle f.

Rigid Body Kinematics
Let H be an N-D object moving through UN space and rotating about a central point G which traverse a path x0(t). Let A(t) be the orientation matrix of the object at time t.
We can represent the rotation with a unit rotor R(t) (satisfying  R(t)R(t)§ = 1) as A(t) = R(t)§(A(0)) º R(t)(A(0))R(t)§ .
For brevity we now drop explicit (t), write 0 for (0), and let ' denote differentiation with respect to t . We then have A = RA0R§ . ( we define matrix-multivector products by treating matrix columns as seperate 1-vectors b(a1,a2,..)=(ba1,ba2,..) ; (a1,a2,..)b=(a1b,a2b,.. ) . All other products (¿, Ù, ×, ...) are treated likewise.   )

A' = W.A where W º 2R'R§ is a pure bivector known as the angular veclocity bivector.
[ Proof :   RR§=1 Þ R'R§ + R(R§)' = 0 Þ R'R§ = -R(R§)' = -(R'R§)§ Þ W pure.
A' = (RA0R§)' = (R'A0R§ + RA0(R§)') = (R'R§)A - A(R'R§) = (R'R§)×A = (2R'R§).A = W.A  .]

For N=3 letting w=W* gives     A' = W.A = (w*).A = (wÙA)i = w×A     so w is the conventional 3D angular velocity 1-vector.

2R'R§ = W Þ R solves R' = ½WR .
Writing WL = R§WR = 2R§R' for the local angular-velocity bivector expressed in the frame of the object, we have R' = ½RWL .
These rotor equations are generally easier to solve than their matrix counterparts.
An aesthetic advantage in defining angular velocities and momenta as bivectors rather than vectors is the avoidance of invoking a third dimension to "hold" them in the case of otherwise planar kinematics.

Next : Conclusion

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Copyright (c) Ian C G Bell 1998
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Latest Edit: 18 May 2007.