# Dimension in Dance

Performing and viewing dances in one, two, or three dimensions may seem like a purely gedank exercise since we live in a three dimensional world, but some aspects may actually manifest. For example, a single spotlight may give the viewer a one dimensional view, while projection onto a screen or writhing on the floor may provide two dimensional movement.

One dimesional viewer:
one dimensional dance motion: If the viewer's line and dance line do not intersect, nothing will be seen; if the two lines intersect in a point, the viewer will just see presence or absence of the dancer(s) depending on whether they occupy the point of intersection; If the viewers line and dancer's line intersect the viewer will be able to perceive distance, he will only see the closest dancer (which will never disappear.
two dimensional dance motion: If the viewer's line does not intersect the dance plane, nothing is seen; if the viewer's line intersects the dance plane in a point, only the presence or absence of a dancer at the point of intersection will be seen; if the viewer's line lies in the dance plane, only absence of any dancer in the vision line or distance to the nearest dancer will be seen, multiple dancers will be evidenced when the distance to the closest dancer changes abruptly.
three dimensional dance motion: the viewer's line will necessarily lie in the 3-space of dance, hence it will appear to the viewer the same as a two dimensional dance space contining the viewing line (dancer's can leave the viewing line and reappear at another point).

Interlude: What is possible when the dance line coincides with the viewer line that is not possible when the intersection is a single point? What is possible when the intersectin is a single point that is not possible when the dance line and viewer line intersect? What additional possibilities does a two dimensional dance space introduce? What further possibilities does a three dimensional dance space introduce?

Two dimensional viewer:
one dimensional motion: if the dance line does not intersect the viewers plane, nothing is seen; if the dance line intersects the viewer plane in a single point, the viewer will see that point as occupied or empty depending on whether a dancer occupies that point; if the dance line lies in the viewer plane but does not contain the viewer, the viewer willl be able to see all the dancers, their sizes (lengths) and distances from each other; if the dance line lies in ther viewer plane and comtains the viewer, the viewer will only be able to see the closest dancer on each side of him (and they will appear as points).
two dimensional motion: if the motion and viewing planes do not intersect (are parallel) nothing is seen; if the dance and viewiing planes intersect in a line not containing the viewer the viewer will be able to see the "lengths" of dancers intersecting the line of intersection, but the dancers will appear and disappear in accordance with their intersection with the line of intersection (and hence their "length" will change); if the dance plane and viewing plane intersect in a line containing the viewer, he will only be able to see (as points) the nearest dancer on each side (which dancers will change with their intersection with the line); if the dance plane and viewing plane coincide, the viewer will see all the dancers (unless some are blocked by others) with lengths and even "shape" to the extent that distance can be perceived.
three dimensional motion: the viewer will see all the dancers as they pass through the viewing plane including the length and shape (to the extent that distance can be perceived) of the intersection with the viewing plane.

Three dimensional viewer:
one dimensional dance: if the viewer lies in the line of dance, only the nearest dancer (as a point, the dancer may have length, but the viewer cannot perceive that) on each side can be seen; if the viewer does not lie in the dance line, all the dacers and their lengths can be seen (none can be blocked by another).
two dimensional dance: if the viewer lies in the dance plane, all the dancers and there lengths (and shapes to the extent that distance can be perceived) can be seen, except that some dancers may be blocked by others; if the viewer does not lie in the dance plane, all the dancers can be seen including their size and shape (a real two dimensional shape rather than just a hemi-perimeter which is the sense in which "shape" has been used above).
three dimensional dance: The viewer can see all the dancers (unless blocked by other dancers) and their three dimensional shape (to the extent that distance can be perceived).

Caveat: the above has assuned that all dance and viewing spaces are linear (lines, planes, and 3-space). If one dimensional motion is along a helix or two dimensional motion is along a warped plane (or the viewing space is bent or warped), there are far more possibioities.

February 2007

campbell@math.uni.edu