Space: Appearance, Imagination and the Fourth Dimension

TSP Showcase Project (SCTP1907) at the University of Sydney

Shapshots of the Weber-Seifert dodecahedral space from Jeff Weeks' Curved Spaces


3 April 2013: Page created and e-mail sent to all participants


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.

meeting times

Weekly meetings start in Week 5 (time and place TBA), and the project concludes in Week 12.


Written report (45%) due on 24 May at 17:00
Showcase presentation (35%) on 29 May, 17:30-20:00
Individual performance (15%)
Summary of project (5%) due on 30 May at 17:00


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.

The shape of space by Jeffrey Weeks (Marcel Dekker, New York, 2002) — Chapters: 1-3, 6, 7, 9, 10, 13-19.
Geometry and topology by Miles Reid and Balazs Szendroi (Cambridge University Press, 2005) — parts of Chapters 1, 3, 5 and 9.3, 9.4
Geometry and the Imagination by David Hilbert and Stefan Cohn-Vossen (Chelsea Publishing, New York, 1952) — §36

You can order the following book for less than AUD 8 (with free shipping) from

Geometry, Relativity, and the Fourth Dimension by Rudolf v.B. Rucker (Dover Publishing, 1977) — Chapters: 4, 5 and 7
(Chapters 1-3 have large overlap with The shape of space and use conventions from physics rather than mathematics)


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.

suggested reading/activities

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.

Week 5 "Flatland and time"
Weeks: Chapters 1, 2, 3 + play some geometry games; Rucker: Chapters 4, 5
Week 6 "Flat manifolds and space-time"
Weeks: Chapters 6, 7; Rucker: Chapter 7; Reid-Szendroi: Chapter 1 (much of this should be familiar from MATH1902; omit 1.11, 1.13, 1.16)
Week 7 "Spherical and hyperbolic geometry"
Weeks: Chapters 9 and 10 + the study questions from the hyperbolic games; Reid-Szendroi: Chapter 3 (omit proofs and focus on the concepts and techniques)
Week 8 "Four-dimensional spaces and projective geometry"
Weeks: Chapters 13 and 14; Reid-Szendroi: Sections 5.1-5.6
Week 9 and Week 10 "Focussed Research Period"
Tackle one or two main aims either as the whole group or in smaller teams.
Possibilities include visualisation of some of the Thurston geometries (see Weeks Chapters 17, 18) or space-time (see Weeks Chapter 19).
Week 11
Consolidate your findings, prepare the report (due on Fri) and the presentation.
Week 12
Polish and practice the presentation (given on Wed) and prepare the summary (due on Thu).