Math 127

Homework 6

due on Monday, 3/20, by 4:00PM.
You may work in either West Hall lab or Angell Hall Classroom B, or you may download the Surface Explorer software on your own computer (works only with windows).

If you choose the first option, once you log in, go to Start -- Programs -- Mathematics -- Surface Explorer. Make sure that you save your work as you go along-- you will be turning these files in.
In case you decided to go with the second option, go to http://www.geometrygames.org/SoS/ and click on Surface Explorer (under Chapter 11).

To turn this homework in proceed as follows:

Save this file as myname.html and modify in any text editor
Erase all the text before the beginning of the listed exercises and replace it by your name. Answer all the questions given within each exercise where the question was asked.
When you are done with (I) save that file as mysampleA.txt. Do the same after you have finished with (II) and (III).
In 1 -- 4 record your answers on or instead of empty underlined spaces.
When you are done with each 2, 3 and 4 save the last modified file as samplea.txt, sampleb.txt and samplec.txt respectively.
Submit by email myname.html, mysampleA.txt, mysampleB.txt, mysampleC.txt, samplea.txt, sampleb.txt and samplec.txt.
In class submit the maps of your three spaces (ones you drew on the handed out triangles).

To start the program double click on SurfaceExplorer.exe

Here are some instructions on how to use the program:

Navigation
Mouse
   Press and hold the left mouse button.
   Drag the cursor towards the top/bottom of the window to go forward/backward.
   Drag the cursor towards the left/right side of the window to rotate left/right.
Creating a World
   You begin in an empty world. You may add features using the right mouse button. The effect of a right mouse click depends on which tool is selected.
Terrain Height
   Click on the terrain and drag upwards. The nearest vertex in the terrain mesh will move upwards accordingly.
Ground Cover
   Choose a Ground Cover, and then right click on the terrain. You can drag the cursor across large areas to quickly change them all to the selected ground cover.
Buildings
   Choose House or Castle Turret, and right click where you want it to appear. Buildings always align with the triangular faces in the terrain mesh.
Walls
   Choose a type of wall, fence or hedge, and right click where you want it to appear. It will automatically align with the edges between triangles.
Doors
   Choose Add/Remove Door and click on any existing wall (or fence) to add a door (or gate). Click again to remove it. Open/Close Door does what you'd expect.
Remove Element
   Choose Remove Element and right click on a object you want to get rid of. Building walls may be deleted individually so that you may, for example, construct two Castle Turrets next to each other, and remove the intervening wall to create a single large room.
Options
   Ground View lets you explore and edit a world from the ground. Bird's Eye View shows the same scene from the air. If you enable Water and make a depression (choose Terrain Height and lower part of the terrain) you will discover the ground water. The water level can be raised and lowered just as your raise and lower the terrain itself.
Saving Your Work
   The commands on the File Menu work as you would expect.
        New creates an empty world of a given type.
        Save As saves your world and its contents in a file of your choice.
        Open opens a world you have previously created and saved.

Here is what you need to do:

(I) Start up the Surface Explorer and open the file SampleA from the Worlds folder. You'll find a world divided into triangular plots of land. Create some scenery in each plot. How many plots are there? When you are done save the modified file as mysampleA.txt. Using copies of the triangle template provided, draw a map of each triangular plot of land in this world. Then cut out each triangle, fold the tabs along each edge, and join one side of one triangle to the corresponding side of another triangle. Look how the triangles' remaining sides correspond. What is the topology of this world?

(II) Repeat the above for the world SampleB.

(III) Repeat the above for the world SampleC.

Begin with your map for SampleA (with mysampleA.txt) Write the number of "faces" (that is, the number of plots of land) here:
f = ___________
Write the number of "edges" (that is, the number of fences) here:
e = ___________
Be sure you didn't count any fences twice. If you're unsure, go back to the Surface Explorer, open SampleA, make sure no fences have gates, and then create gates one at a time until every fence has one. [That is, every long edge, not every atomic fence segment.] How many gates did you create?

Write the number of "vertices" (that is, the number of corners where fences meet) here:
v = ___________
To check your vertex count in Surface Explorer, switch into Bird's Eye View and put a different type of ground cover around each vertex. For example, you might surround the first vertex with a bed of yellow flowers, the second with purple flowers, and so on. How many different types of ground cover did you use before all vertices had been marked?
Open a fresh copy of SampleA, and divide one of the triangular plots in two by building a fence across it. Repeat Step 1 for this new land division, and write the number of vertices, edges and faces here:
v = ____________ e = ____________ f = ____________
Continue building new fences to further subdivide the land, and at each step record the number of vertices, edges and faces:
v = ____________ e = ____________ f = ____________ v = ____________ e = ____________ f = ____________ v = ____________ e = ____________ f = ____________ v = ____________ e = ____________ f = ____________
What is the relationship between the number of vertices, the number of edges, and the number of faces?
Save this file as samplea.txt.
Repeat Step 2 for SampleB.
v = ____________ e = ____________ f = ____________ v = ____________ e = ____________ f = ____________ v = ____________ e = ____________ f = ____________ v = ____________ e = ____________ f = ____________
What is the relationship between the number of vertices, the number of edges, and the number of faces?
Save this file as sampleb.txt.
Repeat Step 2 for SampleC.
v = ____________ e = ____________ f = ____________ v = ____________ e = ____________ f = ____________ v = ____________ e = ____________ f = ____________ v = ____________ e = ____________ f = ____________
What is the relationship between the number of vertices, the number of edges, and the number of faces?