Is It Possible To Visualize 4D?

This post is about 4 spatial dimensions, not 4D spacetime.

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Is it possible for humans to understand 4-dimensional geometry?

Is it possible for humans to visualize 4 spatial dimensions?

I see these questions a lot in geometry and hypercubing circles. A common response, which I believed for many years, is this:

No, it’s impossible for humans to visualize 4D because we are 3D creatures with 3D brains.

Or worse:

Our eyes are 3D so we cannot ever truly see in 4D.

These answers are utterly unhelpful and mostly wrong.

My answers are in the conclusion. The rest of this post is a deep dive into the semantics of those two questions, and answering them both using observable facts about human perception and capabilities.

What’s wrong with those answers? #

• Our computers are 3D, yet they can perform 4D computations. Given enough time and scratch paper, humans are able to perform any computation a computer can.
• Our retinas are 2D¹, yet we apparently visualize 3D.
• Blind people are able to operate in 3D environments, so vision is not necessary for many 3D reasoning tasks.

Take it down a notch #

When asking a question about higher dimensions, a very useful tool is dimensional analogy. Pondering the same question in a lower dimension does not always give the right answer, but it almost always yields insights that do apply in higher dimensions. We’ll break down these questions into smaller ones and use dimensional analogy to help us answer them.

Can humans understand 4D geometry? #

First, what does it mean to “understand 3D geometry”? The Wikipedia page for “spatial ability” lists many distinct tasks, which we’ll analyze one at a time.

Spatial perception #

Humans are generally able to take in sensory perceptual information (typically visual, but sometimes tactile, proprioceptive, or echolocational) to build a mental model of a 3D environment and objects relative to their own position. This same skill applies in 3D video games.

Anyone who has played 4D Miner or 4D Golf a sufficient amount can attest that it is possible to build a mental model of a (virtual) 4D environment and objects relative to their own position. 4D Miner displays a 3D cross-section of the space, while 4D Golf uses a cross-section with an optional orthographic projection overlay. Both games allow free 4D rotation with the constraint that the vertical axis remains fixed.

4D Blocks #

Oh, and by the way, the cute little trains are now truly four-dimensional […]. The train moves and is indescribably cute, and you can make it go forward and backward and control all the track switches.

I would be remiss if I did not mention 4D Blocks by John McIntosh. It is much older and less well-known than the more mainstream 4D games, but is notable for using a wireframe perspective projection with no cross-section view. While 3D cross-sections are easier for humans to parse using existing 3D visual processing, they limit how much of an environment or object can be perceived at once and make mental rotation more difficult.

This is much more difficult to understand at first than cross-sections, but is arguably more powerful. After spending some time in the program, the visualization starts to become intuitive—smaller objects near the center of the projection are farther away, and moving forward causes far-away objects in front to become closer, making them appear to grow.

4D Blocks and its predecessor, 4D Maze, are the only 4D games I’m aware of with first-person camera controls and an unconstrained vertical axis. Unfortunately they have no mouse controls and the keyboard controls are not particularly satisfying to use. I would love to see a modern rewrite².

Spatial visualization #

“Spatial visualization” seems to be defined rather vaguely in different places. As I’m writing this, Wikipedia says this in one place:

Spatial visualization is characterized as complicated multi-step manipulations of spatially presented information.

And this in another:

Spatial visualization ability or visual-spatial ability is the ability to mentally manipulate 2-dimensional and 3-dimensional figures. It is typically measured with simple cognitive tests and is predictive of user performance with some kinds of user interfaces.

I think hypercubing³ and playing 4D Sokoban qualify for both of these definitions, and humans are certainly capable of both.

Neither of these definitions mentions the subjective experience of visualizing something in the mind’s eye, but we’ll address that in Can humans visualize 4D geometry?.

Specific tasks #

Mental rotation, mental animation, and spatial working memory in higher dimensions are all used extensively in hypercubing:

• We use mental animation to predict where a piece will appear after doing some sequence of moves.
• We use mental rotation to compose the rotations that will be applied to a piece and predict whether it will match the surrounding pieces.
• We use spatial working memory to track which areas of the puzzle are solved and where we have placed partially-solved blocks of pieces.

Learning hypercubing is essentially the process of training these three skills in 4D and higher, and applying them using a handful of general techniques. This is true for solving 3D twisty puzzles as well, except for stages that are particularly algorithmic (such as the last layer in most 3D Rubik’s cube methods).

Wikipedia also lists mental folding (the ability to fold a 2D shape into a 3D solid) and visual penetrative ability (the ability to predict what the inside of something will look like based on external characteristics). While it’s very common to see 4D shapes described using their 3D nets, I haven’t seen any case of someone deducing properties of the 4D shape that aren’t obvious from the net. I’ll write off these two as “probably possible given some practice, but no one’s done it.”

Can humans visualize 4D geometry? #

Visualizing anything in your mind depends on two factors:

1. Are you aphantasic, hyperphantasic, or phantasia-typical?
2. Do you have a sufficient mental model of the thing to predict how it looks?

If someone is not aphantasic⁴, and they have enough understanding of 3D geometry to predict what a 3D environment or object will look like, then they should be able to visualize 3D geometry with no issues.

Similarly, if someone is not aphantasic, and they have enough understanding of 4D geometry to predict what a 4D environment or object will look like, then they should be able to visualize 4D geometry with no issues.

The question then remains: what does a 4D object look like? Let’s use dimensional analogy for that as well.

What does a 3D object look like? #

When you look at a 3D object in the world, in a book, or on a computer screen, you see a 2D projection of it. That’s what it looks like.

If some hypothetical 4D creature with a 3D retina were to look at a 3D object, they would instead see a 3D perspective distortion of it. But we are not 4D creatures, so to us it looks like its 2D projection.

What does a 4D object look like? #

When you look at a 4D object in a book or on a computer screen, you see a 2D projection of some 3D representation of the thing (typically either a perspective projection or a 3D cross-section, or perhaps multiple 3D cross-sections).

If some hypothetical 4D creature with a 3D retina were to look at a 4D object, they would instead see a 3D perspective projection of it. But we are not 4D creatures, so to us it looks like its 2D projection.

But 4D objects don’t exist! #

Neither do unicorns, yet if I show you a picture of a unicorn and ask what you see you’ll tell me you see a unicorn.

It’s useful to talk about things that do not exist in the real world as though they do. One of the most powerful capabilities of human thought is the ability to work with representations of an object rather than the object itself. You may insist on distinguishing between an object and its visual representations, but this is actively unhelpful most of the time.

But we only have 2D vision! #

This is true¹, so it’s quite difficult (perhaps impossible!) to imagine the sensation of 3D vision. 3D visual perception would be helpful for visualizing 4D space and 4D objects, but it is not necessary. 3D vision would also be helpful for visualizing 3D geometry, yet we make do without it.

Conclusion #

There is ample evidence to conclude that it is entirely possible for humans to understand and visualize 4D space, despite being 3D creatures with 2D vision. Puzzles and video games in 4D space require all manner of high-dimensional spatial reasoning skills and are enjoyed by a wide audience. There is typically a learning curve, but high-dimensional spatial reasoning is a skill that can be acquired and practiced like any other.

So please, stop saying it’s impossible.