Visual Problem Sets

Projective due September 19

1.  Draw a tiled floor with square tiles in which neither side of the squares is parallel to the horizon line.  Make the ideal viewing distance equal to 6" (six inches, half a foot). 

2.  Problems invovling harmonic quadrads handed out in class. 

3.  State the dual of Desargues Theorem.  Draw a diagram illustrating the truth in one case.
 

Spherical due October 7 or 12

These questions are more essays than computations.  There should be explanations and pictures.  When I last taught this material (as part of a geometry course) the students submitted successive drafts to which I commented and we improved the work on each question.  This way the final product indicates the strongest understanding.  I strongly encourage each of you to hand in drafts as we progress in this subject.  You will be at a disadvantage if you do not take this opportunity.  Although it is difficult without the pictures, in the past students have submitted these drafts via email.  That way allows for easier comments on my part.  It is an approach worth considering.  Now we enter another phase of this course, and another way to learn. 

1.  Define a spherical triangle (your definition here will affect your results for #2).

2.  What conditions guarantee that two triangles on the sphere are congruent?  Write convincing arguments (i.e. proofs) for each that do, and provide counterexamples for those that do not. 

3.  Go to the library.  Look at the book Experiencing Geometry:  Euclidean and non-Euclidean with history, Henderson & Taimina, QA453.H497 2005.  Read pp. 98-101 regarding holonomy.  Complete Problem 7.4.

4.  Complete one of the following:

Space due October 31 November 2

1.  Describe what it would be like to live in a three-dimensional toroidal universe.  How could we tell?  What could we do?

2.  Describe what it would be like to live in a non-orientable universe.  How could we tell?  What could we do?

3.  [This question is intended to be easy.  That doesn't mean that it requires no answer, but it means that it has a direct answer that requires little creativity.  It does, however, require more justification than "because."]  Given that any surface can be written as #aT#bP (as a sum of tori and projective planes) prove using results discussed in class that any surface can be written as either #aT or #bP.  

4.  Select two of the four figures distributed in class handouts of views within flat 3-manifolds.  Describe the gluing necessary to produce the views presented. 

Fourth Dimension due November 21

1.  Explain how a four dimensional magician can separate two linked rings without breaking them.

2.  Our actual retinal images of the world are two-dimensional.  What sorts of visual experiences cause us to believe that our visible world is actually three-dimensional?  How do you think A. Square manages to translate his one-dimensional retinal images into a mental image of a two-dimensional world?

3.  Draw the following:  a cube, a hyper-cube (four dimensional, a tesseract), a five-dimensional cube, and a six-dimensional cube.  Please, oh please, make them large.  The final one ideally should use most of the page.  Draw each separately.  Do not claim that they are all in the picture of the last one.

4.  Make a set of Hinton cubes.  Ideally, they will be three-dimensional cubes with gluing directions.  If you are challenged to make three-dimensional cubes, then make two-dimensional foldouts of three-dimensional cubes.  The second option is a bit more challenging because you need to then include gluing directions for how to make the foldouts into cubes, and then how to glue the cubes together to make the boundary of a hypercube.