Below is the default output of the NDDEM visualisation software for a 3D simulation of spheres flowing down an inclined plane. You can manually move through time, or enable auto playing in the top right. The speed it is playing can be controlled with the "Rate" parameter. You can use a mouse to navigate in 3D space.
But what about dimensions higher than 3? For this, we need some way to project particles from higher dimensions down to 3D. In this example, we will project a 3D sphere down to a 2D circle by slicing it. Move the pink plane on the left around with the slider and see the resulting size of the circle on the right. The circle is largest when the sphere is sliced through its centre.
Now, lets try that same thing, but we are going to slice a 5D hypersphere with a 3D volume. For this, there are two coordinates for where we are slicing, which you can control with the sliders. The torus on the right hand side represents where we are in these two dimensions. Notice how as you move around, the small pink ball moves. This small pink ball represents the location of the hypersphere relative to these two coordinates. The sphere on the left is again largest when we slice through its centre — when the ball is at the centre of the black cross.
Let's now look at inclined plane flow in 4D. This looks pretty similar to the 3D example earlier, but now the particles don't seem to touch all the time. By moving around in the fourth dimension, can you find where they touch? Have a look at the torus, that marks out the position of every particle in the fourth dimension. Particles that you can see right now are close to the black cross.
To be able to see a particle rotate, we need to attach a texture to it. Here is how we visualise the earth rotating. By sliding the texture of the earth, the sphere appears to rotate.
In 3D, there are three directions that an object can rotate in. Here they are, layed out in the shape of the rotation velocity tensor.
In 4D, there are now six directions that an object can rotate in. Have a look at Supplmenetary Video 2, below, for more information.
The collision example from Figure 1 of the paper can be seen below.
We observed the formation of crystal lattices in higher dimensions. Here you can see one forming collision example from Figure 1 of the paper can be seen below collision example from Figure 1 of the paper can be seen below in 5D. The particles are coloured by their distance to the current viewpoint in D4.