"I do it for my past self"

First published in 1985, Weeks’ book "The Shape of Space" invites readers to expand their understanding of space without all the formalisms and technical terms, and instead through the medium of full-colour figures. Weeks is dedicated to teaching and constantly on the lookout for better ways to convey geometrical and topological ideas. This is what led him to start developing games that provide a fun and playful setting in which players are beamed into shapes of space like a 3-torus and asked to answer questions such as “What would you see if you lived in a universe like this?”. In the upcoming goMATH exhibition “The Shape of Space”, his games will be available for visitors to try out, helping them grasp new ideas of space.

杰弗里·周goMATH展览啊f a giant Möbius strip will be displayed in the Main Building of ETH Zurich. The Möbius strip is a very accessible phenomenon that is a source of fascination to every age group, including children. Do you remember when your own fascination for the shape of space started?

在高中，我们在现代术语中有五年或十名 - 学校的书呆子 - 谁将阅读关于数学和物理学，甚至哲学的普及的书籍，并通过它们。这是一个非常积极和有趣的经历。一本书，我读书是埃德文A. Abbott的“Flatlland”。有四维空间和一个超立方体的想法。一般而言，我可以看到它得到的东西，但我无法正确地敏感。我真的没有抓住它，我记得几周的课程，就像我走过我的生活一样，这个想法在我的脑海里转过身来。几周后，我终于明白了它，这只是如此快乐。我认为这就是卖我在数学上的原因：似乎困难的想法，你挣扎和斗争，最后你看到它非常简单和美丽。

You work as a freelancer, which is quite unusual for a mathematician. Why did you choose this path?

我在普通的大学职位讲授几年。然后我的妻子和我想要一个婴儿，我决定成为一个留在家里的爸爸几年，我绝对被爱。正如我们儿子年纪大了，我开始做一些兼职合同 - 例如，对于那个明尼阿波利斯的几何中心是什么。当我现在的项目完成后，我想我会回到教学，但是，正如你所知道的，你现在的项目从未完成 - 当一件事完成时，还有其他东西已经开始了。

到目前为止，我职业生涯的过程中已经有三个阶段。第一阶段是纯几何和拓扑研究。第二阶段开始在有关宇宙微波背景的有趣数据时出来，并提供了有机会看真正的宇宙是有限的。我在宇宙拓扑中工作了大约十年。当那个伤害下来时，我致力于全职努力促进公众的普及。这就是现在的事情。

Gomath是一个重要的计划，旨在吸引年轻一代和女性，并激起他们对数学的热情。有什么动力的是，您参与普及数学领域的展览和工作？

I wanted to make maths more accessible. Basically, to share cool ideas with a lot of people. By the time I got to graduate school, I realised that there were some gaps in the library of popularisations. Among them were these beautiful ideas about space, like multiconnected space or curved space. The only available books were weighty tomes filled with lots of formalisms and technical terms. If you want to do research mathematics on these topics, you need all those technical terms. But for someone who wants just to learn the basic ideas, they are more of a barrier. That inspired me to take these beautiful ideas and strip away the formalisms to write a book that, in effect, my past self and my friends could have read. I guess it all goes back to the thought about what I and my friends would have liked to see and read.

At goMATH, you exhibit the games you have developed yourself and give visitors a chance to play well-known games like tic-tac-toe or chess in other shapes of space such as a torus or a Klein bottle. Using virtual reality (VR), they can also immerse themselves in curved space for a round of billiards. Why did you start developing games like these?

I was trying to find progressively better ways to convey geometrical and topological ideas. Words alone are for sure the worst way to do it. Somewhat better are pictures – that is why my book is full of pictures. But better than pictures are animations where you can rotate things around and see them from different angles. Better still are interactive animations in which the viewer can decide where to go and what point to look at things from. Games on a touch-screen tablet, like the ones at goMATH, add a tactile component. But for something like curved space, the best thing is to put on VR goggles and really be there. That’s my motivation: to make these ideas come more alive. To create an experience where you can use your human senses to walk around and see different shapes of space.

How does a game come alive?

There are two elements you need for a successful game. Only two, but both are essential. First, it has to be fun to play, otherwise people won’t use it. Second, it needs to have some real substance, something of depth that people get some new insight from. I have a mental shelf full of all these interesting ideas and they kind of go in search of the right media to convey them. So, I look out for new technology and how I can make it an effective way of conveying an idea.

游戏是如何改变自己对空间的理解？

我曾与弯曲空间几十年来和thought I understood them well. But even for me, the first time I put on the VR goggles and was actually in the curved space, there were surprises. I am almost embarrassed to say that. Because had I thought of it, I should have known what to expect. For example, when you play billiards in the spherical space, the billiards table seems to slope uphill. In the hyperbolic space it seems to slope downhill. If you had asked me, I should have been able to figure it out, but I never thought about that until I was there with VR. My mental picture of space changed, and I got a better intuition from the games.

Moreover, it made me appreciate more what we do in our everyday Euclidian experience: we can look at a table in perspective and still understand that it has to be a rectangle with four 90-degree angles. After playing billiards in curved space for a while, your brain gradually starts to make sense of it, and you get an intuition for this new shape of space. Our brain and visual system are flexible enough to get used it. But without VR, I think I never would have gained that sort of understanding of how things look as you walk around and how everything behaves.

您在游戏中传达的想法不仅仅是思考实验，而且在宇宙学等领域应用。您的游戏如何与Research相关联？

The games on the iPads – the Torus Games – illustrate this idea that the universe could be finite but not have a boundary. It is still an open question. Even though the actual research on the shape of the universe has wound down now, there is still at least one group that is working on it. There were signs of a finite universe but there is no conclusive evidence. Because the speed of light is finite, we only see a certain distance, which might not be far enough to see repetitions that indicate a finite universe.

弯曲的空间台球游戏连接到空间是否弯曲的问题。实际观察表明，我们看到的部分接近平坦 - 所以它是平坦的，或者它略微弯曲。而且，它归结为一个规模问题。如果你走到一个冷冻湖上，例如，湖上的冰表面沿着地球的曲率遵循。表面是大球体的一部分。但是你看到了地球表面的一部分，看起来几乎是平坦的。它与空间相同：如果宇宙比我们看到的更大，那么 - 即使它是弯曲的 - 我们看到的部分似乎近乎净。

Does that mean we will never be able to figure out the actual shape of the universe?

I’m not willing to say never but probably not within our lifetime. The reason people thought the universe might be finite has to do with the fluctuations in the microwave background. Researchers expected broader-scale fluctuations than the ones you actually observe. One explanation for why that would be is if the universe is finite. That means you cannot have fluctuations broader than the width of the universe. There is still one group working on finding a conclusive way of looking at things. However, in the future, there could be a completely new way of looking at things that might bring us a step further.

So the visitors to goMATH may learn a completely new way of looking at things by playing your games. What should they expect while they’re learning about the shape of space? Will their world view change completely?

在iPad上的游戏的帮助下，他们可以期待才能理解宇宙不一定是无限的。它可以是有限的，没有任何边缘。但是，在VR上，他们的空间看法可能完全改变。我们的大脑只有经历过欧几里德的空间，我绝对希望进入其他一些空间可以成为一个醒目者，表明还有其他可能性我们所居住的欧几里多空间。