View inside the shadow of the 120-cell.
March the 24th is the birthday of Ada King, Countess of Lovelace. An icon for mathematicians and computer scientists as the first programmer. She developed algorithms that could be run on Babbage‘s analytical engine. For more on the history and relationship between Babbage and Lovelace and some exciting comic adventures twisted round it take a look at 2d goggles.
Ada is also one of the most iconic women in the history of science, so to celibrate that and all the other women who have helped develop science and technology we have “Finding Ada” today. For that I would like to remember a woman whose contribution was not just ahead of its time, it was out of this world: Alicia Boole Stott.
Alicia’s contribution was an intuition that few others have achieved. Though we claim to be able to see in 3d our eyes actually only see two dimensional images. It takes some clever processing in the brain to build a 3d world out of that. In fact even quite simple questions in 3d can be very hard to imagine. For example take the intersection of three cylinders at right angles. What shape is created? I recommend thinking about this for a while before clicking the link! Another problem, that I will not give the answer to, concerns cubes: How can I pass one cube through a small cube without touching or crossing the sides?
I hope that this convinces that seeing 3d is already a hard problem. Alicia did one better. She could almost see 4d. This is not the idea of 4d being 3d space and time. That is really 3+1 d, although it can be useful it also has some problems. For example I cannot have intuition of a rotation that puls the time dimension into the 3 spacial dimensions. Alicia’s intution was for four spacial dimensions. She was able to show that just as there are only 5 regular polyhedra in 3d (tetrahedron, cube, octahedron, dodecahedron and icosahedron) there are 6 regular shapes in 4d. She named them polytopes, a name we still use today. The shapes are:
- The 5-cell, this has 5 tetrahedra as its 3d faces.
- The Tesseract or 8-cell, the 4d cube, with 8 cubes as faces. The 3d net of which was used by Salvador Dali in Corpus Hypercubus.
- The 16-cell, which has 16 tetrahedra as faces.
- The 24-cell, with its 24 octahedra
- The 120-cell with 120 dodecahedra
- The 600-cell, containing a massive 600 tetrahedra.
It is very hard to imagine these shapes but there are two ways that we can bring them back to 3d. The first worked on by Stott-Boole is to take 3d cuts. Without using a computer (it is a lot easier to use an abstract method, once a shape is described) she was able to construct models of 3d sections of all six regular polytopes.
A 3d slice through a 4d stellated polytope made from wood. There are some animations here.
The second method, that in some ways can be more instructive is to take the 3d shadow. Just a the sun casts a 2d shadow of an object in 3d, a light in will cast a shadow of a 4d object on a 3d surface. If you want to glimpse the fourth dimension for yourself. The best way to do it is to build one of these for yourself. This is quite easy thanks to the magic of zometool (and shown at the start of this post). If you want to know more about the life of Boole-Stott, there is an excellent short biography in Theory and history of geometric models by Irene Polo-Blanco. There is also lots more on 4d geometry at How do shapes fill space?. Finally you can play with the shapes themselves and their rotations in Jenn3d and SceneScreen, both open source projects.
Posted by gelada 
