A paper model of a part of the world of HyperRogue

[zeno]

http://zenorogue.blogspot.com/2015/10/a-paper-model-of-hyperrogues-world.html

[tricosahedron]

I wonder how hard it'd be to write a program that could compute such a 3D model from a given net. The tricky part is that each polygon has to be warped, and I don't even know how many different models you can derive from a net, so it's probably a difficult task.

If you warp a sufficiently large central part of the model in a specific way, will there be a unique, "meaningful" way to extend it, or will there be different, "equally meaningful" results?

Is it possible to create a model with very simple symmetry?

Anyway, that was very inspiring! If I find the time, I'll try to create a net for my 20-cell tiling, that should be simple with Tyler[www.superliminal.com] in the Euclidean mode.

[nezghul]

Originally posted by tricosahedron:

I wonder how hard it'd be to write a program that could compute such a 3D model from a given net. The tricky part is that each polygon has to be warped, and I don't even know how many different models you can derive from a net, so it's probably a difficult task.

If you warp a sufficiently large central part of the model in a specific way, will there be a unique, "meaningful" way to extend it, or will there be different, "equally meaningful" results?

Is it possible to create a model with very simple symmetry?

Anyway, that was very inspiring! If I find the time, I'll try to create a net for my 20-cell tiling, that should be simple with Tyler[www.superliminal.com] in the Euclidean mode.

Look at this https://www.youtube.com/watch?v=V9WZqL8zCSI

[tricosahedron]

Thanks, that's exactly what I was looking for!

More about the author of the video: https://rantonels.github.io/
(apparently, he's also working on a 4D puzzle game)

Did anyone identify the tiling? To me it looks like an irregular tiling where sometimes, there are 6 triangles per vertex, and sometimes 7, but it's hard to recognize in a wireframe model.


On another note:

It might be nice if Hyperrogue scenes could be projected onto a {7,3} or {3,7} tiling before creating a paper model, which means less polygons have to be glued or stapled together to get a scene of the same size. There could be an option whether the circular area is based on the (6,6,7) Hyperrogue distance metric, or the distance metric of the exported net tiling.

Also, it could be interesting to use segments of equidistant horocyclic segments as a basis, instead of a circular area.
(gets warped more towards away from the horocyclic "center")

[phenomist]

It appears to be irregular yeah, some times there are triangles and sometimes there are squares.

[tricosahedron]

Originally posted by tricosahedron:

Did anyone identify the tiling? To me it looks like an irregular tiling where sometimes, there are 6 triangles per vertex, and sometimes 7, but it's hard to recognize in a wireframe model.

If my observations here were accurate, it might be the dual to Hyperrogue's (6,6,7) tiling, which can have both 6 and 7 triangles around vertices.
(you can display it by pressing ctrl+w in cheat mode)

[zeno]

Very nice! Creating such a 3D model and rendering HyperRogue's world on it could be a very nice idea for the future versions...

tricosahedron: But there are crossing edges (see eg. 0:04), they would not be there in the dual tiling. The description says: "Just a simple experiment/proof of concept: I progressively add rings to the surface with a number of points proportional to sinh(distance from center)." So I think they are not using any specific regular tiling, just adding that number of points in each ring and connecting them somehow.

Regarding other tilings for the paper model: it is currently possible to create an "unsynchronized" paper model, where heptagonal faces on the model are not heptagonal cells on the map :) Using another tesselation should be possible too, but I am not sure whether it would be a good idea -- curvature would be less uniform (as it is concentrated on vertices, and there are less vertices), and the net created by HyperRogue reduces the required work by having as many as possible polygons already adjacent on the printout, and with larger faces, there would be less adjacent polygons possible -- so, I agre that probably less stapling would be required, but I think that the difference would not be that big. (And also I am not sure whether anyone would actually try :)

[tricosahedron]

Originally posted by zeno:

Very nice! Creating such a 3D model and rendering HyperRogue's world on it could be a very nice idea for the future versions...

That sounds like an awesome addition! :)

However, if you want to use it during actual play, you'd have to morph the 3D model every time the player moves. I suppose you could pre-generate one model centered on a heptagon, and one for a hexagon. Before moving to an adjacent space, the orientation (and maybe variant) of the model for the target space has to be determined, and then the coordinates of the model for the current space are simply morphed into the coordinates of the model for the target space (so the models need to have a radius of 8 if sight radius is 7). Teleporting would be a bit more complicated (either assume the player travels in a straight line 1 cell at a time, or use a more flexible algorithm).


You're probably right about the video, and there's no specific tiling.

You may have a point about other tilings like {7,3} for the net, I don't have any experience with it. However, I think creating a model from horocyclic segments could be a bit easier, since each segment is similar, and you could add new segments at the concave end of the net where the model is less warped, while it will become more warped automatically at the convex end. When horocycles orthogonal to a hex line are used, we'd only need 2 (mirror-symmetric) types of horocyclic segments, so nets of any size could be easily generated. :)

[Fulgur14]

Maybe the 3D model could be centered on a vertex?

[tricosahedron]

Originally posted by Fulgur14:

Maybe the 3D model could be centered on a vertex?

I wonder: If we use a vertex-centered 3D model as a basis, we could derive a polygon-centered 3D model by considering the 3D models centered on the polygon's vertices, and calculating the mean coordinate for each vertex in the visible part of the Hyperrogue plane. The question is: Will the result just be a warped version of our (6,6,7) tiling, or will there be some distortion (i.e. different edge lengths or angles)?