Reaction to Chapter 7 of Beyond the
- In the discussion of dimensionality and dance, the concept
of the multiple dimensions of human arm positions made me think about the
complicated set of positions that a movable object with many points of
revolution could have. If every joint on the human arm can move freely
in the three spatial dimensions and there are four joints on the human
arm (shoulder, elbow, wrist and finger), there should be 12 dimensions
that determine possible movement. I found it interesting that there was
a hierarchy among the dimensions of movement of the arm (assuming that
the rest of one's body is stationary in space). Certain joints limit the
total possible set of postions of subordinate joints. For example, the
position of the shoulder determines the set of dimensions in which the
elbow or the finger can point. In this way, "superior"
dimensions dictate the ranges of "inferior" dimensions. There
is a hierarchy of dimensionality. This hierarchical order of dimensionality
reminded me of Abbott's social order of polygons in Flatland. In
the case of the joints in the arm, however, the hierarchy is based on real,
physical relationships based on degrees of freedom.
- I found the pictures of the colorful concentric circles
to create a peculiar optical illusion. The graphs on pages 147-8 seemed
to rotate when I moved the page back and forth or up and down. The combination
of the close circles and the vibrant colors produce a distict illusion
of rotation. There is also the perspective illusion of depth: the circles
look like slices of a tunnel that approach infinity. They can also appear
to be slices of a cone that ascends from the page towards the viewer. I
found these multiple dimensions of illusion quite fascinating.
- To form a hyperbolic paraboloid from strings along two
non-intersecting lines, do the lines have to be perpendicular (as represented
in the picture on p. 143), or can they have any direction as long as they
don't intersect? It would be interesting to see what three dimension object
would be made by a series of hyperbolic paraboloids along lines that shift
positions over time.
- Was the Flatland theme in the play Dimensions
purposely adapted from the story? How did the actors physically across
each others' paths when dancing? It seems extremely difficult to pass another
person when you cannot move laterally or even twist. The play must have
created a strong sense of physical and emotional claustrophobia that could
of had its own moral implications. How was the work received by critics?
Prof. Banchoff's Response