And understand the universe in the process
Can we perceive higher dimensions? [1]
I have figured out how to see in four dimensions.
All you need to do is watch TV.
You probably already know the three dimensions of space, where motion can be represented in three directions: left/right, forward/back, up/down. A fourth dimension would be another ‘invisible’ direction you can move in. Before saying this is impossible, consider a squashed person living in a two-dimensional ‘flatland’ who can only see cross-sections of 3D objects. The up/down direction is completely invisible to them, yet it still exists. What if you are that person in the flatland, and there exists a whole new dimension you have just never come across? What if you are only seeing 3D ‘cross-sections’ of a four-dimensional world?
To arrive at 4D you need to somehow collapse a dimension. You do this every time you watch TV, draw an object on paper, or enjoy an artwork (humans could have seen 4D ever since perspective painting was used in the Renaissance).
Imagine a piece of paper that contains within it an entire three-dimensional world (like a very thin TV screen). People living in that world can move in three directions, yet from your outside vantage point it looks like a flat page; call it a ‘hyperplane’.
Stack several of these ‘hyperplane’ pages into a book. Keep inserting more pages until there are infinitely many inside. The book now in front of you is a four-dimensional ‘hyperspace’. Each page is a separate 3D world, a cross-section of the entire hyperspace, yet there remains a sense of continuity between them.
Let’s pause and take a quick detour to our squashed person in two-dimensional flatland. What would they see when they come across 3D objects? When they pass through a sphere, they can only see it in cross-sections; they would see a point grow into a small circle, increase to a maximal circular size, then decrease and disappear back into a point. When they ‘flip’ or rotate 90˚ in space, their entire world of flat images would change (or ‘morph’) except for a constant line; the further images are away from the line, the more quickly they would change.
Now let’s return to our hyperspace book. As you flip its pages, you see objects morph (just like the flatland person did). As you pass through a hypersphere (the 4D extension of a sphere), you see a point continuously grow into a small sphere, increase to a maximal spherical size, then decrease and disappear back into a point.
What would it look like to move around in four dimensions?
Imagine you are a hyperspace explorer, and you hold in your hand a remote control with four buttons (w, x, y, z) and a switch that can be pushed forwards, backwards, or neutral. Each of the buttons correspond to an axis in hyperspace. Currently the ‘w’ button is pressed, and you are standing inside an x-y-z hyperplane. ‘x’ and ‘y’ are two perpendicular directions along the ground, and ‘z’ is the up/down direction. Push the switch forward, and objects will begin to morph. The conservation of mass doesn’t seem to hold as objects grow, shrink, or completely disappear and reappear. When the switch is neutral, objects will stop morphing.
Push the switch backwards, and objects morph in the opposite direction. The switch makes you move back and forth (translate) along the invisible w-axis.
You now press the ‘z’ button, and the world around you changes dramatically. The floor remains unchanged; if you previously drew flat pictures on the floor they would still be there. However, everything above and below the floor begins to morph. The further away objects are from the floor, the more rapidly they change. The transformation slows to a halt, and you find yourself in a strange, alien world which, apart from the floor, bears no resemblance to what you saw before. You have undergone a 4D ‘flip’ (90˚ rotation) around the x-y plane, and ‘w’ is the new up/down direction. When you push the switch back and forth, objects will morph as you translate along the now invisible z-axis.
4D rotations of external objects look the same as in 3D. If you want to rotate an object around the w-z plane, make sure the ‘w’ button is pressed and the switch is on neutral. Within your hyperplane, the 4D rotation looks just like a regular rotation around the z-axis.
4D reflections are simpler still. A reflection about the x-y-z hyperplane simply means that your switch is now reversed. When you push the switch forwards, objects will instead morph in the opposite direction along the w-axis.
Why do we care about seeing in four dimensions?
Firstly, it’s pretty darn cool. Secondly, all the world’s a four-dimensional stage. Modern physics has shown us that we can no longer separate time from space, since they are intimately connected in a 4D spacetime. When we can see four dimensions, we can see the very fabric of the cosmos.
APPENDIX – EXPLANATION OF 4D TRANSFORMATIONS
The transformations of 4D rotation and reflection can be understood by continuing patterns from lower dimensions (a point is 0D, a line is 1D, a plane is 2D, a space or hyperplane is 3D, a hyperspace is 4D):
A plane rotates about a point. A space rotates about a line. Hence, a hyperspace rotates about a plane.
A line reflects about a point. A plane reflects about a line. A space reflects about a plane. Hence, a hyperspace reflects about a hyperplane. [2]
An explorer can only directly see a single hyperplane cross-section of the entire hyperspace. We can map the entire hyperspace by monitoring four explorers each with a different button pressed out of w, x, y, z (they are inside different hyperplane orientations), and combining their observations.
A single explorer can reach any point in hyperspace by traversing each of the four hyperplanes and ‘flipping’ (rotating 90˚) at the right moment, analogous to a 2D being reaching any point in space by traversing the three plane orientations x-y, x-z, y-z in a ‘zig-zag’ method.
By Peter Lavilles
Article References/Further Reading:
[1] Scene from Christopher Nolan’s Interstellar (2014)
[2] It is possible for a space to reflect about a line or a point, hence it is also possible for a hyperspace to reflect about a plane, line or point. In this article I am considering the ‘maximal’ type of reflection, i.e. reflection of an n-space about an (n-1)-space.