Duality in Spatial Algebra

15 Sep 2017
by ignat

Let’s take a scalar unit 1 and multiply by a tri-vector i, we obtain the trivector i for the dual unit. We multiply p by i and obtain the dual trivector -1.

Vectors are dual to bivectors, and vectors to vectors for bivectors. The transition from the p-number A to the dual p-number iA = Ai is not an involution, since in the chain of maps A -> iA -> -A-> -iA -> A, the initial p-number A occurs only in 4 steps.

Thus, the space of vectors and bi-vectors has a duality.


UV = 1 (U1V1 + U2V2 + U2V2)
+ i[(U2V3 – U3V2)k1 + (U3V1 – U1V3)k2 + (U1V2 – U2V1)k3]

In the first line, the scalar product of two vectors.
The terms in the square brackets determine the vector product of two vectors.
In the second line is the outer product of two vectors. It is a bivector, the dual vector of the vector product of vectors.

The exterior product of vectors is denoted by the sign of V-inverted and analogous to the multiplication of differential forms.
In the general case, the Clifford product of two vectors is the sum of a scalar and a bivector.


The scalar product of bivectors is equal to the scalar product of the vectors to which these bivectors are dual.

The product of bivectors is simply the product of their dual vectors taken with the opposite sign.


The norm of the outer product of two vectors is equal to the area of the parallelogram constructed on these vectors.

The angle between the bi-vectors B = iU and C = iV is defined as the angle between the vectors U and V, to which these bivectors are dual.


Euler’s formula relating exponential and trigonometric functions in the algebra of complex numbers and generating phase transformations, as well as the Hamilton-Cayley formula, which describes spatial rotations in the algebra of quaternions.

Scalars and Trivectors

As we discussed earlier for scalars and trivectors, the cycle is multiplied by i.

1 -> i -> -1 -> -i -> 1.

If the length of the circle in radians is 2pi, and we will put in correspondence a segment of the circle for each transformation. Then we get that the phase pi / 2 transforms the scalar 1 into a three-vector (volume) i.

Note that when converting to phase pi, scalars and trivectors change sign to the opposite.

Vectors and Bivectors

On the Ender Crystal model, the inner rotating cube can be considered rotated to the phase pi / 2 with respect to the outer cube.
In this case, the edges of the outer cube (vector) correspond to the faces of the inner (bivector).

With the phase transformation pi / 2, the vectors become bivectors, and bivectors into vectors. Or in other words, the edges enter the faces, and the faces turn into edges.

In the presence of a geometric imagination, such a morph can be represented on one single cube.

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