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Stephan Tillmann
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TSP Showcase Project (SCTP1907) at the University of Sydney
Shapshots of the Weber-Seifert dodecahedral space from Jeff Weeks' Curved Spaces
What is the shape of our universe? No one knows the answer to this question, but we have an idea about the possible shapes it can have. An intuitive and visual understanding of these possible shapes is gained by looking at them from a fourth dimension — just as a cartographer obtains a better view of the land from the top of a mountain.
Space-time also has shape and geometry. Our perception of the physical world depends fundamentally on how we imagine the geometry of space-time. In how many essentially different ways is is it possible to visualise 4-dimensional space-time in 2-dimensional pictures or in the 3-dimensional setting of a moving image on a 2-dimensional screen?
We will visualise 3-dimensional spaces or 4-dimensional space-time with the aim to train our geometric imagination and to seek hidden patterns and beauty that were previously invisible.
Weekly meetings start in Week 5 (time and place TBA), and the project concludes in Week 12.
The following books have been put on reserve in the library, and I have indicated which Chapters may be relevant to this project. I do not expect you to work through the whole chapters or sections, but rather to look at them with your project in mind.
You can order the following book for less than AUD 8 (with free shipping) from http://www.bookdepository.co.uk.
These geometry games by Jeffrey Weeks can help to gain some intuition about spaces of dimensions 2 and 3:
You are especially encouraged to explore the Curved Spaces software and the hyperbolic games, and to go through the study questions, which are accessible through the games' help menu.
The heading indicates a main theme for the week. As stated above, I do not expect you to work through the whole chapters or sections, but rather to look at the concepts, results and techniques with your project in mind. Some key notions will be introduced in Week 5 in a friendly manner, so that you can use your imagination to formulate some aims and ideas of what you'd like to be able to "see" by the end of the project. These aims will be expanded and refined throughout the project, as you learn more mathematics around the theme as well as the techniques that you need to realise your aims.