TITLE: How to visualize $1$-forms and $p$-forms?
QUESTION [14 upvotes]: I am having trouble understanding the common way of visualizing one-forms.
Example of the visualization: On Wikipedia and in several math and physics texts books, I have come across visualizations such as this one:

What The Image Seems to Represent: These images involve a series of planes with a vector "piercing" them. The number of planes pierced corresponds to number assigned to the vector. It is thus a map from $\mathbb{R}^{n}\rightarrow \mathbb{R}$ such that it inputs a vector and outputs a number.
What I don't understand: vectors are often described by their direction and magnitude. So I don't understand how the notion of orientation fits in here when we are talking about a series of planes. A vector is a single object, so having direction and magnitude makes sense. But how can a series of planes have a direction?
Remark: I have recently been reading about equivalence relations, equivalence classes, and orientability. Are these concepts useful in defining one forms and $p$-forms?

REPLY [7 votes]: This picture for $1$-forms is closely related to picturing $0$-forms -- that is, ordinary functions -- in terms of their level curves; it's essentially the infinitesimal version of a contour plot.
When following a curve along a contour plot, we can work out how much a function changes by counting how many contour lines it crosses over, and in which direction (i.e. from the small to the large side or vice versa).
The picture you've drawn for a $1$-form is essentially the same: the planes are the level curves, and their action on a vector is determined by how many planes it passes through and in which direction.
IMO, trying to push this picture to higher forms in three dimensions means that $2$-forms should be pictured as lines rather than planes. This idea should also be familiar, by the name of "flux line". Similarly, a $2$-vector should be depicted as a two-dimensional region, and the value of combining the $2$-form with the $2$-vector is something like counting up how many flux lines pass through the surface.
And similarly, $3$-forms would be just points, which nicely corresponds with the idea that $3$-forms represent density. Combining a $3$-form with a $3$-vector should resemble counting up how many points are inside the volume, and choosing the sign based on the orientation of the volume and the orientation on the points. (an orientation on a point is simply choosing to label it positive or negative)