The project element of MA3101 (Euclidean and non-Euclidean Geometry) is *optional*. It carries up to 15 marks (same as each of the homework assignments).

The submission deadline is the end of Week 11 (Friday November 22).

Students are welcome to make an individual submission or to work in teams of up to three (if you would like to propose a bigger team than that, please talk to me about it).

### What to do?

You can do anything at all, as long as it explores or demonstrates some principles of geometry or some means of communicating geometric ideas. The submission can be anything – a physical object, a video, an image, a live performance, a game, anything at all. The emphasis is on creativity and thinking about ways of perceiving or explaining that could make geometry more accessible or more engaging for someone.

The project will be unsupervised, but I will respond to suggestions of topics and approaches.

Don’t worry too much about the level of sophistication in the finished product – homemade objects made from ordinary things are very welcome. Methods of communication that are not necessarily conventional in academic mathematics are encouraged (but not required). Arguments based on intuition or familiarity are fine if they convey an idea – there is no expectation of a formal approach, but your project should include mathematical content of substance. If the mathematical content is not evident from the object itself, please include a note of a couple of pages, to explain it.

If there are many physical objects to admire, we will organise an exhibition in Week 12.

### Assessment

Marks will be awarded for evidence of original thinking and creativity, and for clarity of explanation. The project should be self-contained, in the sense that someone who has participated in this module has the knowledge needed to understand it, without having to do any background reading. The most important thing is that it’s your work and your ideas. It’s fine to submit something that does not necessarily stand up to rigourous mathematical scrutiny, as long as you explain what it does and how it helps you (and potentially others) to understand or appreciate something about geometry. An essay or article is certainly acceptable as a project, but the advice is to think carefully about original content if you are submitting an essay. Something that “could have been” written by Chat GPT will not get a high mark – for example something that’s verbose and full of generalities but low on specific geometric ideas. Even though the style might be more informal than some types of academic work, it is as important as ever to cite all sources that have been used (no matter what the format is). And in all cases, the authors should be clear about what the mathematical content is and what can be learned from it.

### Possible Topics

These are just a few suggestions, meant as examples.

- Something about perspective. For instance, you could do a more elaborate version of the window-taping experiment if you find a location that works really well for it, and spend some time on a careful and detailed approach. This is something that might work well for a team of two or three people, and it might be suited to a video as (part of) the exposition.
- A physical model of a hyperbolic plane (maybe crochet or something else), maybe showing some geodesics.
- An exploration of geodesics on a variety of surfaces. Again this is something that might be good for a video (and a team).
- Some spherical geometry with physical models.
- A study of some theme in cartography, that deals with the mathematical challenges and solutions in this subject.
- An account of some classical theorems about Euclidean triangles, and their spherical analogues, that builds on what we discussed in lectures.