Is the rounded head of a tack in geometrical vectors intended?

Question

It seems as though the thumbtack in geometrical vectors is equivalent to the wedge product of two vectors $A \times B=C:$

$$A^\mu e_\mu \wedge B^\nu e_\nu=C^{\mu\nu}e_\mu\wedge_\nu$$

which would return an oriented parallelogram in $\mathbb R^3:$

Weinreich then considers the “cross product”, which leads him to further kinds of vectors. The cross product of two arrow vectors turns out to be a “thumbtack”, a kind of oriented area.

Probably the pin part in the tack makes reference to the right-hand rule, while the round head is a surface. However, its circular shape is as unsuggestive of a parallelogram as it gets. Very respectfully, I would presume something like a patio umbrella to be more fitting, but, again, the question is not "would it be better to ..." but rather, "is there a reason for":

My question is whether the circular shape of the head of the tack is intended to make reference to a mathematical point, or it is not the best aspect of an otherwise helpful analogy.

Answer 1

Stacks and tacks is very catchy, and the thumbtack analogy definitely conveys the idea of an oriented surface seen from a topological standpoint.

Quoting the free online available pertinent passages of Gabriel Weinreich's Geometrical Vectors:

In the traditional approach, the cross product of two vectors - that is, arrows - is defined as a new arrow whose direction is perpendicular to both of the original arrows, and whose magnitude is equal to the area of the parallelogram subtended by the two of them. It is immediately clear, however, that an object defined in this way is not invariant to the general transformations: even in the simple change of scale, for example, the new quantity will vary with the square of the factor by which distances are compressed. And, of course, we know that the property of perpendicularity is not topologically invariant either.

What we do instead is to take the bull by the horns and define a third type of vector, which is called a thumbtack. A thumbtack, whose algebraic symbol we will take to be a boldface letter with a small circle under it like this: $\underset{\circ}T,$ is a single finite piece of plane, with a sense indicated by a loose arrowhead:

The direction type of the thumbtack is obviously that of a stack, and not that of an arrow. In other words, its meaningful to say that a thumbstack is parallel to a stack, but not to an arrow; it can, however, be contained in an arrow, but not in a stack.

We define the magnitude of the thumbstack to be its area, the shape being immaterial. Two thumbstacks are, in other words, considered identical if they are parallel to each other and their area and senses are the same, even though one may be, say, circular and the other square.

It now becomes simple to define the cross product of two arrows as a thumbtack determined by the parallelogram which those two arrows subtend:

It is, in other words, the thumbtack which is contained in each of those arrows, with a magnitude equal to the area of the parallelogram subtended by them.

So the shape of a tack is immaterial, and a pair of vectors define a tack corresponding to the parallelogram they subtend.

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A thumbtack is a $2$-vector, eating up two $1$-forms. If, instead, only one $1$-form is provided (a stack), the result will be an arrow - i.e. a $1$-vector:

$$\begin{align} &\underset{-}B \times \underset{\circ}T\\ &=T^{\alpha\beta}\, \mathrm e_\alpha \wedge \mathrm e_\beta\; \left( B^\mu\mathrm e_\mu,\cdot\right)=C^\beta\,\mathrm e_\beta \end{align}$$

This can be visualized as a deforming thumbtack keeping its area constant, and defining an oriented segment (vector) as it reaches across one of the gaps in the stack: