Grassmann Algebra

GeometricInterpretations.nb 18
Geometric dependence
In Chapter 2 the notion of dependence was discussed for elements of a linear space. Non-zero 1elements
are said to be dependent if and only if their exterior product is zero.
If the elements concerned have been endowed with a geometric interpretation, the notion of
dependence takes on an additional geometric interpretation, as the following table shows.
x1 � x2 � 0
P1 � P2 � 0
x1 � x2 � x3 � 0
P1 � P2 � P3 � 0
x1 � � � xm � 0
P1 � � � Pm � 0
Geometric duality
x1, x2 �are �parallel ��coÐdirectional�
P1, P2 are �coincident
x1, x2, x3 are �coÐ2Ðdirectional ��or parallel�
P1, P2, P3 are �collinear ��or coincident�
x1, �, xm are �coÐkÐdirectional, k � m
P1, �, Pm are �coÐkÐplanar, k � m � 1
The concept of duality introduced in Chapter 3 is most striking when interpreted geometrically.
Suppose:
P defines a point
L defines a line
Π defines a plane
V defines a 3-plane
In what follows we tabulate the dual relationships of these entities to each other.
Duality in a plane
In a plane there are just three types of geometric entity: points, lines and planes. In the table
below we can see that in the plane, points and lines are 'dual' entities, and planes and scalars are
'dual' entities, because their definitions convert under the application of the Duality Principle.
L � P1 � P2
P � L1 � L2
Π�P1 � P2 � P3 1 � L1 � L2 � L3
Π�L � P 1 � P � L
Duality in a 3-plane
In the 3-plane there are just four types of geometric entity: points, lines, planes and 3-planes. In
the table below we can see that in the 3-plane, lines are self-dual, points and planes are now
dual, and scalars are now dual to 3-planes.
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