Chapter 4 : Shadows and Structures

Study Questions and Projects

Consider objects with various symmetries, such as a box, a ball, a cone, a regular pyramid, a book, an egg, etc. How would the shadows of these objects appear? For each object, determine if there is a direction in which it may be turned so that the shadow stays the same.
What is the difference in the world described by Plato in the Republic and that described by Abbott in Flatland? How does the dimensionality of the creatures living in these worlds affect their perception of them?
How do we know we are not four-dimensional creatures viewing the three-dimensional shadows of four-dimensional objects, as in the cave allegory?

What is the main property of shadows that allows us to use them to learn about the structures that create them? Why is this so important?
What happens when a shadow is cast on a surface other than a plane (for instance, a ball)?

If you are given a partial drawing of a cube that you must complete to draw a unique cube, what is the minimum you must be given?
What would be the analogy in four-space? How much of a hypercube must you see to know it as a unique hypercube?

What property of two-dimensional images of objects allows us to draw a two-dimensional image of an object of n dimensions?

What defines a simplex? How do you construct one?
What is the formula for the number of vertices of an n-dimensional simplex? For the number of edges? Of triangles? Of k-dimensional simplexes?

What determines a group?
How do you determine the number of squares in a hypercube, and what is this number?
What does Q(k, n) signify, and how is it calculated?