When we think about geometry we generelly think about straight lines forming triangles, the angle between those lines and all the objects those yield. Rarely do we question those concepts, it is what we learned in school as geometry. In fact those ideas about straight lines and the objects they form is called euclidean geometry and was invented by the mathematician Euklid who lived around 300 a.c.
Is it not funny that we still have firm believes in concepts that are thousands of years old, even though all the other scientific concepts from that time be it medicin, chemistry or biology are considered more than outdated? You will argue that is due to the eternal nature of mathematics and philosophy and that is a valid point. But in fact mathematics and in particular geometry have undergone tramendous changes in the last couple of centuries, never appreciating those is, I would say, quite ignorant. Those new mathematical ideas are the corner stone of all new breakthroughts of modern physics, field theories in particular. When we venture into those fields, we can suddenly understand phenomena like black holes, expanding universes and concepts like time-travel, so they might be worth exploring.
But let’s quickly go back to Euklid. Euklidean geometry is defined as a set of objects and rules that talk about straight lines – if those lines are elongated at infinum and are parallel they will never ever meet. And if two lines cross vertically they intersect at a ninty degree angle. That’s it. This is the parallel postulat that defines Euklidean geometry. That is fine. That is what we understand. But what if we propose a new form of geometry, a set of objects and rules and we allow those lines to be curved and to actually cross at a certain point in space even if at another point in space they are aligned parallel. This idea is the core of non-euclidean geometry.
To give a very intuitive example, let’s have a look at a globe. A globe is described by longitudes and latitudes. Let’s think about two longitude lines on the globe, they start at the pole move down to the equator which they cross with an angle of ninty degrees and they meet again at the other pole. So they are parallel at the equator, but cross at the pole… In fact, look at the triangle that is formed with the two longitude lines and the equator line. They form a triangle made up of two angles of ninty degree and another angle at the pole. The sum is larger than 180 degree… but we learned in school (!!) that the sum of angles in a triangle is always 180 degree. How could the school be possible wrong?! Well in fact you did lean not the whole truth. Spherical geometry is defined as a geometry where the sum of the angles in a triangle is always more than 180 degrees. The geometry where the sum is 180 is a subset of all the other possible geometries in space.
We can also look at the other way and look at triangles where the sum is smaller than 180 degrees. This geometry is called hyperbolic geometry.
In hyperbolic geometry, everything is curved inwards. The saddle is often used as a visualisation or the leaves of a lattice. It is contracted space. It becomes harder to visualize, there are some nice tesselation patterns that are often used to depicture hyperbolic geometry. M.C. Escher was quite intrigued by those and draw them again and again.
Even more difficult to imagine are those geometries in three dimensional space. When we talk about the surface of a globe or a saddle we are really staying in two dimensions. But what about curved three dimensional space? Now it gets really interesting because this is something we cannot find in day to day life, we need vivid imagination or digital tools to help us. There are in fact digitally visualized curved spaces in 3d. Those have also be brought in VR. [Hart] et al. programmed a hyperbolic space in VR. Exploring those you can suddenly get a far better intuitive understanding of what curved space entails. The space seems almost dynamic. The ordinary rules of perspective do not apply anymore. In a “normal” space we are used to objects being far away to appear smaller and those who are closed to appear bigger. That is strinkingly not true anymore in hyperbolic space. Objects far away appear larger and, when going through the space, it appears as if the space would open up with every step. It is difficult to imagine and to describe but this can be explored in VR.
In fact some of the characteristics of hyperbolic space makes it enormously interesting for VR applications especially for its nature of contracting infinite geometry into a finite space. VR has the problem of being limited to the space of the users room, hyperbolic geometry could solve this issue. Programmers and artists could design virtual spaces that are hyperbolic in nature but look euclidean for the Eikonaut, so the Eikonaut thinks ze walks in a large room but the space is in fact far smaller [Hawthorne]. (There are a couple of interesting possible solutions for this problem another one is sarcadic supression [Sun]).
Now back to the question of what this has to do with modern physics and phenomenae like black holes. One of the biggest realizations in the last century in physics has been that space in the universe is in fact curved in similiar ways we just discussed and that this curvature gives rise to the phenomena which we call gravity. It was this realization that brought Albert Einstein the status he has today, describing this theory in General Relativity, his masterpiece besides his many other ingenious contributions to physics.
So gravity is curvature. Mass curves space. Time and space are dynamic moldable entities in modern physics. Every object with mass curves space, so our earth and our sun curve space and as the mass of a star exceed by far the one of a planet the gravity of a star is higher that of earth. The objects with the highest mass in the universe are the so called black holes – enormous objects that used to be stars. After the collabse of supermassive stars, black holes form by the inward pull and collapse of the outburning star. A small area in time and space is curved to infinity – infinite space in a tiny region of the universe.
Much has been said about this phenomena but not so much about the visual appearance of these objects. Here again we arrive at a wonderful opportunity to use VR to visualize something otherwise very abstract.
As space is curved so are the trajectories of light. This fact results in a very distorted image of the actual object. We can visualize the new trajectories in this image:
So the light that arrives in your eyes comes actually from a different point as opposed to where you see it. This has many visual consequences. You can for example see stars that are behind the black hole, which you would actually normally not see (this is also true about stars behind the sun – this was the first experimental verification of general relativity, as experimentalist observed those stars behind our sun in an eclipse taking place in 1919).
Sometimes these star trajectories are bended so extremely that they take multiple turns around the black hole until they arrive at your eyes, like this you would see the star multiple times. This is exactly what we observe around a black hole – stars are clustered in a spherical fashion around the black hole. The same thing happens with the accretion disk. An accretion disk is matter that was attracted by the black hole and took a form around the black hole very much similiar to the form of the rings of saturn. It just looks very different. This is again because you can see what is actually behind the black hole. The light that is emitted from the part of the disk that lies behind the black hole is visually distorted in a way that you see it as an arc above and below the centre of the black hole. It looks like an eye but has really just the ordinary form of a ring. What you perceive is not really what there actually is. This let’s us question our perception of reality. Our perception has evolved not in an objective vacuum but in the context of our day to day euclidean environment.
We need imagination, science and tools of art to get to an idea of the actual underlaying reality. Virtual Reality poses as a tool to visualize our limitations in perception and other perspectives on reality. This is true especially in the abstract realms of euclidean geometry where all other forms of visualizations fall short.
This brings me to my final thought. The german philosopher Wittgenstein once said – the boundaries of my language are the boundaries of my world. He had in fact a very broad idea about language, that included abstract mathematical ideas. I think it is fair to see art as a form of language as we use it to express ideas. Art, math and technology can help us understand new worlds, we can express new ideas in them. Like this art helps us to broaden our perception of the world, suspending the boundaries of our language and our world and thereby deepens our human experience on earth.
[James] Oliver James, Eugenie von Tunzelmann, Paul Franklin, Kip S. Thorne. Gravitational Lensing by Spinning Black Holes in Astrophysics, and in the Movie Interstellar. Classical and Quantum Gravity, 2015.
[Hart] Vi Hart, Andrea Hawksley, Elisabetta A. Matsumoto, Henry Segerman. Noneulcidean virtual reality I: explorations of H3. arXiv, 2017.
[Hawthorne] Nico Hawthrone, Vincent A. Pisani, Ocean Hurd, Sri Kurniawan. Navigation by Walking in Hyperbolic Space Using Virtual Reality. CHI PLAY´19, 2019.
[Sun] Qi Sun, A. Patney, L. Wei, O. Shapira, J. Lu, P. Asente, S. Zhu, M. McGuire, D. Luebke, A. Kaufman. Towards Virtual Reality Infinite Walking: Dynamic Saccadic Redirection. ACM Trans. Graph, 2018.