### Project 5: Polyhedral and Metric Considerations

Prepare to turn in work that follows the guidelines. I will post on ASULearn answers to select questions I receive via messaging there or in office hours! You may work alone or in a group of at most 3 people, and turn in one project per group.
Part 1: Polyhedral Perspectives

### Problem 1

Use Euler's Formula to prove that a polyhedron with three triangular faces meeting at each vertex must have a total of four faces (without assuming that you already know what the 5 Platonic solids are).

### Problem 2

The Archimidean solids are solids whose faces are regular polygons (but not all the same) such that every vertex is symmetric with every other vertex.
• Part A: Make a model (paper nets, or toothpicks and gumdrops, or any model you like!) of one of the Archimedean solids and bring it to class. You will present the solid to the class.
• Part B: On your project writeup, list at least 2 different names of the solid you made and describe how you made it.
• Part C: List what types and how many of each type of polygon faces there are.
• Part D: Specify the total number of vertices, edges, and faces and show that V - E + F = 2.
• Part E As Susan Goldstine explains [What the Origami Means] A symmetry of a polyhedron is a way of moving the polyhedron so that it occupies the same physical space as before it was moved. As an example, consider the symmetries of the cube. A cube can be rotated 90 degrees, 180 degrees, or 270 degrees around an axis passing through the centers of two opposite faces of the cube,

or it can be rotated 120 degrees or 240 degrees around an axis passing through two opposite vertices of the cube,

or it can be rotated 180 degrees around an axis passing through the midpoints of two opposite edges of the cube.

In fact, these comprise all of the symmetries of the cube except for the mundane but mathematically important move of leaving the cube where it is. The symmetries of a polyhedron reflect its structure and regularity.
Discuss some of the different symmetries of your Archimedean solid.

### Problem 3

• Part A: This figure shows a cube decomposed along the diagonal into 3 square pyramids (the bottom square of the cube is the base of one pyramid, the square face towards us is another base, and the square face on the right side is the third), which would explain why the volume of a square pyramid is one-third the height times the base. Are they congruent pyramids though?

To answer this question, use this pyramid net. Make 3 copies. Do they fit together along the diagonal?
• Part B: Can you find a similar decomposition, using the diagonal, relating a rectangular box and a rectangular pyramid? Explain why or why not and what you tried. If so, are the pyramids the same shape or different shapes? Explain.
• Part C: Can you find a similar decomposition, using the diagonal, relating a rectangular box and a right triangular prism? Explain why or why not and what you tried.

Net for a right triangular prism

Part 2: Metric Perspectives (Taxicab and Euclidean)
For the following problems, you may do by-hand work and/or use Sketchpad. Be sure to show work and explain/annotate (if you use Sketchpad then print out your Sketchpad work and annotate it).

### Problem 4

Look at the triangle whose vertices are A=(2,2), B=(-1,1), and C=(1,-1).
• Part A: Show that this triangle is equilateral under the taxicab metric.
• Part B: Is the triangle equilateral or isosceles under the Euclidean metric?

### Problem 5

In order to show that the Pythagorean Theorem sometimes but not always holds in taxicab geometry...
• Part A: Look at the triangle whose vertices are A=(-3,9), B=(12,4) and C=(0,0). Notice that the slope of AC = -3 while the slope of BC = 1/3 and so AC is perpendicular to BC. Hence this is a right triangle. Let c be the hypotenuse (opposite vertex C) and let a be the side opposite vertex A and b be the side opposite vertex B. Compute a, b and c under the taxicab metric. Compare a2 + b2 with c2. Does the Pythagorean theorem hold for this right triangle in taxicab geometry?
• Part B: Look at the triangle whose vertices are A=(0,0), B=(4,3) and C=(4,0). Notice that this is a right triangle. Compare a2 + b2 with c2. Does the Pythagorean theorem hold for this right triangle in taxicab geometry?

### Problem 6

A man has a newspaper stand at W=(1,0), eats regularly at a cafeteria located at E=(8,3), and does his laundry at a laundromat at L=(7,2)
• Part A: If he wants to find a room R using the taxicab metric so as to be at the same walking distance from each of these points, where could R be located? (Give the coordinates).
• Part B Is there more than one possible answer for Part A?
• Part C: How many blocks does he have to walk from his room to each of the three points, assuming that he finds a shortest distance room that has the same walking distance to each of the points.
• Part D: If all conceivable shortcuts are possible (ie, using the Euclidean metric), where should the man's room be so that it is equidistant? (Hint: use the perpendicular bisectors).
• Part E: What is the Euclidean measurement of the distance that he has to walk from his room in Part C to each of the three points?