Part 2 of a presentation delivered in Gainesville, FL, during the IMAGING PLACE conference in February 2007.

1. Second Life: Imaging Virtual Place (part 2) Richard Smyth, Ph.D. Invent-L Conference 2007: Imaging Place University of Florida 24 February 2007

10. Joining (i.e. Identifying) Edges CYLINDER MOBIUS STRIP TORUS KLEIN BOTTLE

11. The Cylinder and the Torus A torus is topologically equivalent to a rectangle: gluing the opposite edges of a rectangle turns it into a cylinder; joining the free pair of edges turns the cylinder into a torus CYLINDER TORUS

12. Creating a Torus from a flat surface A “flat” torus and the doughnut surface both have the same topology

13. The Mobius Strip and the Klein Bottle MOBIUS STRIP KLEIN BOTTLE Cutting a Klein Bottle along a curve yields two Mobius bands. That is, a Klein Bottle can be made from joining two Mobius bands along their boundaries.

14. Creating a Klein Bottle from a cylinder KLEIN BOTTLE

15. Torus Tic-Tac-Toe “ These four positions are equivalent in torus tic-tac-toe” (Weeks 16). “ The second position is obtained from the first by moving every- thing ‘up’ one notch (when the top row moves ‘up’ it naturally reappears at the bottom). Similarly, the third positionis the result of moving everything in the second position onenotch to the right. The fourth position is obtained a little differently: it results from rotating the third position one quarterturn clockwise” (Weeks 15-16).

16. Klein Bottle Chess The diagram above represents a chess position on a board that is a Klein bottle. It is White's turn to play. Find a move for White that checkmates Black. How many such moves are there? http://mathforum.org/wagon/fall06/p1058.html

18. Hidden Dimensions in Real Space The hose appears to be a line, but up close we see that it’s a hose with a hidden circular dimension. http://people.cs.uchicago.edu/~mbw/astro18200/dimensions.html

19. Hidden Dimensions in Real Space Two normal dimensions with two extra “curled up” dimensions represented by spheres. Now, instead of a one-dimensional line, imagine a plane in two dimensions. Just as the hose has an extra dimension at each point, we can see an extra dimension in the form of a loop or a circle at each point on the plane. If we use spheres, we now have two curled up dimensions at each point on the plane. Remember that even though the image shows spheres only at intersections of grid lines, there would be spheres at every point on the plane.

20. Hidden Dimensions in Real Space Two normal dimensions with six extra dimensions curled up in Calabi-Yau spaces

22. Hidden Dimensions in Cyberspace “ Data available at the intersection of the three ‘crosshairs’ opens into a subspace of three dimensions” (Benedikt 118).

23. Hidden Dimensions in Cyberspace “ Two surfaces of the subspace continue to display navigation data…while the third surface is beginning to show destination data, that is, the sought images” (Benedikt 118).

24. Hidden Dimensions in Cyberspace “ The user has moved in to inspect the images more closely” (Benedikt 118).

25. Hidden Dimensions in Cyberspace “ The intrinsic dimensions of the six-dimensional data object p , located at ( x,y,z ) in the (extrinsic) dimensional space of XYZ , are unfolded into the space of ABC and have the values given by the location ( a,b,c )” (Benedikt 145).