Trying to Understand What a 4D Landscape Would Look Like

[Splat!]

For starters, here's a top-down orthographic view of a Trionian landscape:



This is how a top-down orthographic view of a Tetronian landscape would look like in a Tetronian's 3d retina (no, this is not green feta cheese ):



The empty areas would be bodies of water, which I left out for clarity.

Now, what I'm trying to wrap my head around is how the landscape would appear in the Tetronian's 3d retina if he/she/it was standing on the surface of the landscape (possibly the surface of a hyperspherical planet or the surface of an infinite cube [4d analogy to an infinite plane] with variations in it's surface for mountains and valleys) ather than looking down at it from far above.

I'm guessing you would have a sky sort of on the top half of the 3d retina, like you would with a 2d retina and things would be warped toward the center of the cube (3d retina) due to perspective distortion.

Would this be right, or am I way off base here?

[Keiji]

First off, a tetronian would be able to see all the points inside a cube (which their retina is bounded by) at the same time. They wouldn't see the image you generated, which is a perspective projection of said cube. That said, there's hardly any way to visualize it correctly.

surface of the landscape (possibly the surface of a hyperspherical planet or the surface of an infinite cube [4d analogy to an infinite plane]

I'll correct you a little before you go any further here. A glome is the 4D analog of a sphere, and a realm is the analog to a plane.

I'm guessing you would have a sky sort of on the top half of the 3d retina, like you would with a 2d retina and things would be warped toward the center of the cube (3d retina) due to perspective distortion.

Yeah, pretty much. I agree it's quite hard to visualize how a tetronian would see their world in perspective. I'm sure it could be done with a 4D->3D raytracer, though.

[Splat!]

They wouldn't see the image you generated, which is a perspective projection of said cube.

My thread title might have been mis-leading. It wasn't my intention to portray what the tetronian would actually be perceiving, but to illustrate how the top-down orthographic view of the landscape would be mapped out onto the surface of the 3d-retina, as a preliminary step toward understanding how the 4-dimensional landscape might appear.

POV-Ray has a way to generate a heightfield by using a grayscale 2-dimensional image to displace each point vertically on a square to make a landscape.

The 4-dimensional analogy to handle this in a 4-dimensional raytracer would be to generate a heightfield by using a grayscale 3-dimensional voxel-style image to displace each point on the w axis within a cube to make a Tetronian-style landscape.

Unfortunately, I don't think any 4-dimensional raytracers at the moment have this feature, so I'll either have to wait for someone to make one or to somehow program one on my own, someday.

[Keiji]

I'll either have to wait for someone to make one

someone = me http://fusion-global.org/wiki/HNET

[papernuke]

if a 4D being were standing on a 4D peice of land, then it would be like us standing on the ground, but when we preceive the ground as 2D (flat) the 4D being would see it as a 3D figure. just imagine yourself looking at the floor.. then adding another dimension to the floor.

[pat]

I'll either have to wait for someone to make one

someone = me http://fusion-global.org/wiki/HNET

And, me: http://www.nklein.com/products/rt

[pat]

I also started a 4-D hunt-and-chase game that has a bump-mapped landscape. I should upload what I have of it sometime. Hmmm... it doesn't seem to run for me at the moment... But, here are some old movies from it... The top view is x-forward, y-right, w-up. The bottom view is x-forward, z-right, w-up.

http://www.csh.rit.edu/~pat/lj/hunt4dPreview3.mov
http://www.csh.rit.edu/~pat/lj/hunt4dPreview2.mov
http://www.csh.rit.edu/~pat/lj/hunt4dPreview1.mov

[Keiji]

Do, do convert those to AVIs which we can actually watch.

Quicktime is a piece of shit.

[pat]

So, here they are as AVIs... (many comments about "shit" omitted):

http://www.csh.rit.edu/~pat/lj/hunt4dPreview3.avi
http://www.csh.rit.edu/~pat/lj/hunt4dPreview2.avi
http://www.csh.rit.edu/~pat/lj/hunt4dPreview1.avi

[pat]

So, it only took me a second or two to get it running again.
Here's a more current screen shot...

[Keiji]

Interesting... so what is the point of this game?

[pat]

Eh... as originally conceived, it was to be for a programming contest titled "Double Vision". It was just going to be a chase/evade game.... The main point was to start getting comfortable with 4-D by moving around in it.