**NOTE:** Not about coding today.

For no good reason, I started thinking about 4 dimensions. While I’ve seen a number of explanations of what a 4-dimensional object would look like from 3-dimensional space, I’ve never seen an explanation of what a 3-dimensional object would look like from 4-dimensional space. It occurred to me that in a universe with 4 spatial dimensions every point (inside and out) of a 3-dimensional object^{1} would visible to an observer. That is, every single point inside my 3D body would be perfectly visible viewed by an observer from 4 dimensional space.

Let me explain using an analogy to 2-dimensional space.^{2}

First, look at the drawing below of a circle with dots on the inside, a triangle with lines inside, and the two-eyed observer. What can the observer see?

Since she’s two-eyed, she can perceive depth and therefore can see two distinct objects and can see that one is curved and can see that the other has one flat surface facing her, and that its surface is partially obscured by the object with the curved surface. But, she cannot see what is inside the objects, nor can she see behind the objects. But we, in 3-dimensional space can see the entire shape of each object, and we can also see literally every single point inside the entirety of each object, in a glance.

I understand this in the following way. In 2-dimensional space, observation is constrained by straight-lined trajectories within the x-y axes, but in 3-dimensional space, every point of a 2-dimensional object is viewable along an axis that is orthogonal to the x-y axes.

Now think of the analogous 3-dimensional objects. If I look at two people, one standing partially in front of the other, it’s just like the 2-d situation, I can see height, width, and depth of each object along a trajectory defined x-y-z axes, so I can’t the part of the person behind the other, and I can’t see the shape of the rear side of the people, and I definitely cannot see inside the people. But a 4 dimensional observer could see all of those things perfectly, in a glance along the orthogonal 4th axis. Whoa.

If it helps, you can make the further analogy of a 1-dimensional object being placed in 2-dimensional space: look at this picture, showing a straight line that merely varies in color along its one axis. The two dimensional observer can see the entire object, inside and out. There is no “behind” or “inside” for the 2-d observer.

I don’t have a “point” to this write-up, but I’ve always struggled to think of what higher dimensional spaces/shapes would look like, and this analysis doesn’t really answer that, but it does help me understand a little bit how our 3 dimensions might be perceived in 4-dimensional space.

- To be clear, in 4 dimensional space, it would actually be “a rendering of a 3 dimensional object”, just like a drawing on paper is a rendering of 2 dimensions, not actually 2 dimensions. ↩︎
- I’m just figuring this out on my own with inspiration from the idea of Flatland. I haven’t read it, and I am certain that others have fleshed these ideas out more fully and correctly elsewhere. But I haven’t personally come across this take before. ↩︎