### Dr. Sarah's Geometry of the Universe

• Review perspective drawing and look at Excel program used to give the dots in a perspective drawing mathematically. Connect the dots to get a 3d object!
Perspective drawing using Excel We won't have time to make a complete colored house like on this web page, but we will go through some of the basics.
--After going to the main class web page, and clicking on the universe lab, click on this excel file from Netscape (under the apple, under internet). You will see some garbage character symbols. Under File, release on Save As... and then click Save to save the file as perspectivehouse.xls Under the apple, under Math 1010 Apps, release on Microsoft Excel. Under File, release on Open, and then click on benf.xls and Open it. You will see a chart that is partly filled in.
--We will use the perspective theorem to mathematically project the three dimensional house onto the mathematically precise perspetive image in the plane (where we can draw it). So, we want to transform x, y and z to new coordinates x' and y'. Hence x'=d x/(z+d) and y'=dy /(z+d) by the perspective theorem. We will make Excel do these formulas for us!.
--To transform x'=dx/(z+d), click on E2 in the Excel sheet. The corresponding Excel formula is =d2*a2/(c2+d2), so type this formula into E2 and hit return. You should now see -1.875. Notice that this is the correct formula to use for x'=dx/(z+d). At the bottom corner of E2 click until you get a square with arrows. Then fill down the Excel column and release in E18. The number you will see there is -2.7631579.
--To transform y'=dy/(z+d), click on F2 in the Excel sheet. The corresponding Excel formula is =d2*b2/(c2+d2), so type this formula into F2 and hit return. You should now see -2.8125. Notice that this is the correct formula to use for y'=dy/(z+d). At the bottom corner of F2 click until you get a square with arrows. Then fill down the Excel column and release in F18. The number you will see there is .39473684.
--To draw our house, click on the grey E box, so that that column is highlighted. Then hold down the shift key while you click on the grey F box, so that both the E and F columns are now highlighted. Under Insert, release on Chart. Then click on XY (Scatter) and then on Finish. Now we have our mathematical drawing! All we have to do is connect the dots!
--Under View, scroll to toolbars, and then release on drawing. Click on the line that is just a straight line (no arrow heads or anything).
---Your mouse should now be a thin cross when you take it to your picture.
---Take a look at page 2-6 from the Perspective handout by Marc Frantz. We want to connect the dots to make the picture represented in Figure 8.
---Click on one point in your picture, hold down the mouse, and release on another point in order to draw a line.
---To erase a line, click on the arrow in the drawing tools, and then go back to the picture. Hold the mouse over the line until it turns into a hand. Wait until it tells you Line (--) where (--) means some number. Then click down, and hit delete.
---Dr. Sarah will show you how to use the dots to make the line at the top left (first you make a complete line down to the corresponding dot, then make a line to the part that you desired, and then erase the longer line).
---After you are done, while you are waiting for Dr. Sarah to start up again, you might wish to find your original stock purchase price (from WebCT newsgroup, select the stock market forum, and then click on all messages, or get this info from your stock lab) since we will need the complete into later.
• Build a 3d cube from a 2d square by drawing it in the plane. Compare the distortion to a real cube.
• Watch the Rotating a cube movie from Davide Cervone again and discuss.
• Update on Jeff Weeks
• Build a 4d cube model from a 3d square in the room. Think about analogies from the cube to discuss distortions.
• Watch the Rotating a Hypercube movie from Davide Cervone again and discuss.
• Build a donut from a square!
• Use an analogy to get some intuition about a 4d object built from identifying the corresponding opposite sides of a cube.