Visualising 4D

danny-4@newsletter.paragraph.com (Danny) — Sat, 26 Nov 2022 17:57:52 GMT

It is mathematically provable that 4D shapes exist, but the more I tried to think about this concept, the more frustrating it became. Take the example of a tesseract. A square has two edges at each corner, a cube has three, similarly a tesseract has four. We know that in the same way that a 3D carton can be unfolded into a 2D net with 6 sides, we would be able to perceive an unfolded 4D tesseract as 8 cubes in net with the same 2D footprint. Trying to visualise putting it back together is a fruitless task - we cannot perceive this fourth positional dimension.

Unfolded in a way we can easily perceive in 3D space

Example of how we can most accurately depict a tesseract

Just as we find it hard to conceive a fourth dimension, there are problems when we are making spreadsheets or databases in 2D space and need to introduce a third dimension - in this case not a spatial or positional one. In the below example we can see the quantity of burgers sold as the week progresses:

What if we want to introduce a third data point - the profit made along with the quantity sold? You could make another table or include profit alongside each quantity, for example (58 - 183), but this is messy and potentially confusing.

Colour could be used as the third dimension; a simple way of doing this would be to have each number in green, amber or red depending on if a burger made profit, broke even or made a loss on any given day. With the inclusion of a key this becomes simple to read.

Alternatively, we could just introduce a Z axis and create a 3D bubble chart or similar in a Cartesian plane, adding a sense of depth. As we can see below, you could denote a 3D shape using a series of coordinates:

Problems arise when trying to add a fourth axis. Now depth has been included all vectors are accounted for. We can display data with four data points in 3D by combining the two aforementioned solutions and have a colour spectrum alongside X, Y and Z axes. This approach is commonly used in programming and graphing.

Since we cannot visualise 4D shapes, it is necessary to substitute the ‘Q’ axis for a different kind of dimension that our senses, ideally sight, can process. What other alternatives are there beyond colour? In terms of what we can see on a chart, none that I can think of.

Apple’s iPhone timer app uses tactile feedback and sound to add a degree of skeuomorphism, even if visually their design has moved in the opposite direction. Without looking at the screen, you can feel and hear the scroll wheel spinning and losing momentum, clicking less frequently as it comes to rest like a roulette ball. A similar design could be added to a colorless 3D heatmap - when the cursor is moved over ‘warmer’ areas there is increased haptic or audio feedback.

Ultimately, we will never be able to visually comprehend a 4D space in the same way Yoshi could not escape his square prison. Using a different kind of ‘dimension’ that we can process, such as colour, sound or texture, presents some solutions when working with data.