Continuing on from D.E. Littlewood's Chapter XIII, (p. 101, Algebras) in his monograph The Skeleton Key Of Mathematics' we learn the existence of a non-commutative algebra in higher than 3 dimensions. Enter Grassmann algebra, also known as Exterior algebra, is an algebraic system created by 19th-century mathematician Hermann Grassmann. It concerns vector products but is applicable to any number of dimensions whereas quaternions (Part 1) are confined to 3 dimensions.
It also extends vector algebra to include geometric objects like directed planes and volumes. Its fundamental feature is the exterior product, which is anti-symmetric, meaning reversing the order of the terms flips the sign, e.g.
e ij = - e ji
As Littlewood notes (p. 103):
"In Grassman's space analysis a vector with components:
x 1, x 2,...... x n
is associated with:
e 1x 1 + e 2 x 2,.....+ e n x n
where the quantities:
e 1, e 2,...... e n
Satisfy the following:
e i 2 = 0, e ij = - e ji
In other words, displays the anti-symmetric feature. (e ij = - e ji )
In three dimensions a vector would be of the form:
ae 1 + be 2 + ce 3
An element of area would take the form:
ae 2 e 3 + be3 e1 + ce1 e 2
And would thus be distinguishable from the vector with which it would be identified in quaternions. An element of volume would have the single component: k e1 e 2 e 3
For:
e1 e 2 e 3 = e2 e 3 e 1 = e 3 e 1 e2 = - e1 e 3 e 2 = - e 2 e1 e 3
= - e 3 e2 e 1
The algebra immediately yields the formula for the area of a triangle as half of the product of the two sides. Or for the formula of a tetrahedron as one-sixth of the product of three concurrent edges.
Littlewood goes on (p. 104) to make a number of key distinctions to the previous algebra (for quaternions):
- With quaternion algebra there are no factors of zero, i.e. =xy = 0
IFF x= 0 or y = 0
- In Grassman algebra some power of every element is equal to zero.
- The square of a vector is zero but the square of every element of the algebra is not zero, i.e.
( e1 e 2 + e3 e 4 ) 2 = 2 e1 e 2 e3 e 4
Addendum note:
In Grassman algebra the fundamental volume element V is represented by the wedge product of three basis vectors, i.e.:
e1 e 2 e 3
Problems for budding mathematicians:
1) Consider the parallelopipped below spanned by the vectors u, v and w in 3-space.
Find the volume of this solid given that:
u = (1, 1, 3)
v = (1, 2, -1)
w = (1, 4, 1)
2)Show how the Grassman algebra formula for a single component of volume element k e1 e 2 e 3 can be used to find the volume of a tetrahedron.
Find that volume given for this tetrahedron:
a = (3, 0, 0)
b = (1, 4, 0)
c = (2, 1, 5)
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