Reaction to Chapter 3 of * *

Andrew Miller

`Summary: Chapter 4 of Beyond the Third Dimension
describes how slices of a three dimensional object that pass through a
plane can provide insight into the shape of the object. Time becomes a
significant dimension in such a scenario because, if the velocity of the
object is known, the slices of the object can be pieced together to give
a description of the object’s shape. Similarly, one could examine three
dimensional slices of four dimensional objects that change with time to
understand the shapes of such objects.`

Exercises

- For various three dimensional shapes, such as a sphere, a tilted cylinder and a torus, draw a series of pictures that represent the passage of the objects through a plane. This perspective corresponds to that of A Square when he sees the sphere pass through Flatland. Put the images into a booklet so they can be flipped from page to page. This flip-book should provide a good representation of a three dimensional object in two dimensions that change with time.
- Draw a two dimensional picture of a mountain range as seen perpendicularly from the north (or south). Draw the same mountain range as seen from the west (or east). Now use the properties of width, length and height of these drawings to construct a contour map of the mountain range that provides all of this information.

Questions

- How can one visualize a four dimensional object moving perpendicularly through three dimensions? It's easy to conceptualize a three dimensional object moving through a two dimensional plane, but what direction is perpendicular to space? Is it simply the direction of the fourth dimension?
- Do the lines on a contour map represent quantitative increments in height? For example, does each line of a map of a hill represent 100ft., 200ft., 300ft., etc.? If there are no numbers to correspond to the lines, is it possible to confuse height and depth? It seems that one uses past knowledge of geology to supplement the lines on a contour map to understand its characteristics.

Links