Hello marcm200.

I've had a bit of experience with the concept of rendering higher dimensions, due to working with 4D fractals a lot.

In my experience, there are several different ways to generalize how to render objects at any dimension.

1- We can simply render a perspective image of the 4D shape directly, just like we do for 3D shapes - but there is a slight catch I'll get to below.

2- We can "slice" the 4D shape into 3D shapes, and then render those individually.

3- We can swap the 4th dimension for one of the other 3, and get a new 3D image.

4- Or we can apply the 4th dimension to some visual cue other than a spacial dimension.

**1) 4D Perspective Rendering**If you've worked with 3D perspective rendering, then expanding that to another dimension isn't really that difficult. It basically comes down to adding another number to every spacial point or vector.

But there is one part that is often overlooked - and that is that each time you do perspective render, it (only) cuts down the dimensions of the image by 1 dimension.

So if you have a 1D world, with 1D objects, then a "perspective" image would just be taken one direction or the other down the 1D line, and the result would be a "0"-Dimensional "image" (which is just a single colored point).

If you have a 2D world, with 2D objects (like circles, squares, or even solid 2D fractals), then a perspective image would be taken by starting at a 2D point, and getting the colors from the closest "surface" of the objects in front of the camera inside a triangle that is the viewing angle. The resulting "image" would be a 1D colored line.

3D is easy to imagine, because we do it all the time - but just to recap: a 3D image would be taken by starting at a 3D point, and getting the colors of the surfaces of objects closest to the camera, and in front of the camera within a pyramid (frustum) that is the viewing angle of the camera. The result is a 2D image, once again colored based on the surfaces seen in the 3D world.

So then - if we take this one step further, we will see that a 4D world can also be rendered by starting with a 4D point, and getting the colors of the 4D surfaces of objects within a 4D pyramid (frustum) in front of the camera.

And so - what we have established through the previous examples, but what we tend to overlook - is that the resulting perspective "image" of the 4D shape will actually be a 3D "image".

That means if you want to do a pure perspective render of a 4D shape, and display it on a 2D screen - you actually need to do 2 stages of rendering.

First you need to do a 4D perspective render, which will result in a 3D shape (a 3D "image").

Then you need to use any traditional 3D rendering technique to render the 3D image onto a 2D screen.

So you'd be capturing a 2D perspective image of a 3D perspective image of a 4D shape

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**2) Slicing**"Slicing" a 4D object is a much more simple and clean way to convert it to 3D, rather than using 4D perspective rendering. But just using a single slice doesn't give you as complete of a picture of what the 4D shape looks like.

The process of slicing is exactly what it may sound like: just cutting a single cross-section of the object in order to view the cross-section as an image that is 1 Dimension less than the original object.

We're used to this concept in 3D, as we've often seen 3D objects that are cut open to show what their cross-section looks like.

But again - the one thing to remember is that this method always only reduces the dimensions by 1.

So a cross-section of a 4D object is a 3D object.

We can then render this 3D object down to 2D by either using perspective rendering, or even just taking another cross-section (which would be a 2D image now).

Getting a feel for the shape of the full 4D object using the slicing method can be done quite effectively if you take a bunch of slices, and then make an animation that shows how the 3D cross-section of the 4D shape changes as you move the slice's position along the object in the 4th dimension.

**3) Dimension Swapping**This method is pretty straight-forward, but doesn't always give you the best overall feel for the 4-dimensionality of the shape.

Essentially this is the same concept as the "slicing" method above, but instead of slicing along the 4th-dimension, we actually slice along different dimension axes.

So if we call the different axes "x", "y", "z", and "w", then what we can do is take 3D slices of the 4D object with each combination of 3 dimensions.

So, we'd take a slice along the "w" axis, resulting in a 3D shape that is using the "x", "y" and "z" axes.

Then we'd take another slice along the "x" axis, with the result using the "y", "z", and "w" axis.

And so on.

So the result is 4 individual images of 3D slices of the 4D object.

This is kind of like creating one of the quick cardboard sculptures where it's just flat cutouts of cardboard that are fitted together at right angles to approximate a 3D shape.

**4) Other Visual Cues**Lastly - it's also possible to view a 4D object all at once in a way that makes quite a bit of sense intuitively, allowing you to (at least partially) mentally wrap your head around the 4-Dimensionality of the shape. And this is by applying the 4th dimension to some other visual cue.

This is typically done by taking 3D slices of the 4D shape along the 4th axis (like I mentioned before), and then combining those 3D slices in different ways.

There are multiple ways this can be done, but here are some I've found effective:

- Setting the 4th dimension to Time: This is what I mentioned above in the "slicing" section. Creating some sort of animation where the 4th dimension changes over time is a good way to show how the 4th dimension affects the shape.

- Setting the 4th dimension as Density: If you have all the 3D slices, and put them int the same space so they overlap eachother - then you can color the space based on how many slices overlap each point, creating a sort of "cloudy" or "glowy" shape where denser or brighter parts represent where the shape exists more across the 4th dimension.

- Setting the 4th dimension as Color: Similar to the Density one, if you overlap each slice, but color each slice differently (probably based on some gradient like a rainbow), then the resulting image will use that color change to express the 4th dimension, which makes some intuitive sense, and also often looks pretty

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Ok - that was everything for now.

Sorry about the long reply.

I definitely got carried away

But visualizing higher dimensions is a passion of mine, so I couldn't help it