Reaction to Chapter 7 of Beyond the Third Dimension

Andrew Miller

• 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.

Questions

• 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