zenorogue

For an analogy: an Abelian group is a structure in which four axioms hold. A non-Abelian group is a structure in which three of these axioms hold, and the fourth does not. It is not a structure in which some random proper subset of axioms holds, because such a notion would be useless. While structures where specific subsets of axioms hold (non-Abelian groups, semigroups, monoids, etc.) are useful.

The hyperbolic paraboloid is not an isometric embedding of the hyperbolic plane H2 in Euclidean space E3, because it does not have constant negative Gaussian curvature.

In general it is impossible to have an isometric embedding of H2 in E3, although it is possible to have isometric embeddings of fragments of H2.

No, you are confusing geometry and topology here. Topology does not change when you stretch the space but changes when you cut/glue it. Geometry does not change when you cut/glue but changes when you stretch. So portals change the topology but the geometry is Euclidean (you get a Euclidean manifold). This are the meanings used by mathematicians working in these areas.

(The original meaning is of course about breaking 5th postulate while all the others hold, showing that it was possible was a celebrated result in mathematics, while it is trivial to break the postulates in some arbitrary way.)

Yeah, Jeff calls it "non-Cartesian". He changes the topology but the geometry is Euclidean. There are also lots of roguelikes doing other experiments with topology, tilings, dimensions (Verdagon 7DRLs, geometry episode of Roguelike Radio, etc.)

AppStore only accepts games compiled in a new version of XCode, and I have only a very old Mac on which it does not seem possible to install such a new version.

Google Play is not very friendly to small developers who do not want to share their home address (not their fault, it is some recently introduced EU law), but the Android version can be downloaded from itch.io

For Epic, I am using Heroic Games Launcher, I believe it is much better than the official Epic app.

There is a fascinating genre of games where space does not function as in does in our real world. Such games are celebrated, among others, for their mind-bending puzzles, psychedelic 3D visuals, and surreal exploration.

Sure, CDDA is a roguelike -- the word was supposed to mean primarily a specific way of controlling the character, introduced in Rogue. Everybody in the roguelike community calls it a roguelike.

"Stone Soup" refers to the Stone Soup folk story, where people contribute their supplies to make a great soup for everyone (it is developed by many people who contribute to the open-source project started by Linley Henzell as Linley's Dungeon Crawl).

Still a better name than NetHack (which is not a game about hacking networks... "Net" just refers to collaboration over net to improve the game Hack). There is also BrogueCE ("Brogue Community Edition").

But how much have you played it?

When I started working on HyperRogue, I expected it to be a "gimmick or tech demo" as you say. But after implementing the basic roguelike gameplay, it felt surprisingly good, better than what I had actually planned. So released it, and other people liked it too.

So it is not surprising that people expect it to be a gimmick (given that I did expect that hyperbolic geometry would make basic roguelike gameplay feel better myself). Some people do not get it on their first try, but love it on their second try.

Might depend on the interpretation of "roguelike" -- Rogue combined single-character adventure with grid tactics, so to many people, "roguelike" means single-character grid tactics, usually with some other less unique features (such as randomness), just like it did in the 80s. This was also the most important roguelike feature for the design of HyperRogue. While more commonly "roguelike" is used for marketing anything that takes whatever inspiration from a game marketed as roguelike. In the roguelike communities, mentioning The Binding of Isaac (in a way suggesting that roguelike means a game like it) happens regularly and yields very negative reactions, because it is not tactical and thus not interesting to the community. Similar for the other ones you mention.

Still, there are many very polished roguelikes on Steam (Dungeons of Dredmor, Crown Trick, Sproggiwood, Moonring, etc.).

It is possible to run the game in a Euclidean hex grid and compare. For example:

* In Euclidean open space, if you are attacked by two adjacent monsters at once, you cannot escape, because the monsters would just move in parallel lines. In hyperbolic, you can, because parallel lines do not work.

* In Euclidean open space, if you are ambushed by many (say, 12) monsters on all sides, you cannot escape. In hyperbolic, you can. There is always more directions than it seems.

There is a land Hunting Grounds which teaches the tactics above. It simply does not work in Euclidean mode. There is also this article: http://zenorogue.blogspot.com/2012/03/hyperbolic-geometry-in...

If you have some static scene in Euclidean geometry, you can possibly also have a static scene in hyperbolic geometry which looks the same*. But the parallax effects when you move would be different, and also that hyperbolic scene likely would not be natural for hyperbolic geometry (as you say, you would have to make the distant objects larger; if you approach them, you learn that they are in fact large, while the objects close to your original POV are tiny).

* depending on what you mean by "looks the same". How do you perceive depth? In a natural model of depth perception based on binocular vision or parallax, the hyperbolic space looks like a bounded ellipsoid (more precisely, stretched Beltrami-Klein model), so it could not look like an infinite Euclidean scene.

What do you mean by "weird combos of those"? AFAIK Hyperbolica only has H3 and S3 (and a bit of E3), no combos. (While HyperRogue has all eight Thurston geometries and some more.)

Well, there is "casual mode" which makes Orb of Safety saves permanent, and also the "Orb strategy mode", where a single instance of carelessness of bad luck should no longer end the run (but it makes the game harder in some other aspects). People complain about the menus, although it seems that there are just too many options to make all of them easily discoverable...

Hyperbolic plane has tree-like structure. If you tried to draw a binary tree of depth 10 on Euclidean paper, you would run out of space. In hyperbolic plane, the straight lines are diverging and new space is created exponentially (very roughly speaking), so such a tree would fit perfectly. This makes hyperbolic embeddings better than Euclidean ones for all kinds of hierarchical data. I would also like to mention that while the uses in machine learning get lots of hype, the ML researchers seem to not be aware of the earlier impressive results on hyperbolic embeddings obtained in other communities (social network analysis/algorithms).

If you want to get intuitions about how this tree-like structure works, it is the best to play HyperRogue. For formal math, I guess it is the best to read the relevant papers.

(I guess the hierarchical structure of transportation networks, from hub airports -> ordinary airports -> major roads -> minor roads, could be interpreted as hyperbolic geometry, in the same way as Internet has hyperbolic geometry. A very liberal and abstract interpretation though. I do not see how making sharper turns makes sense.)

What would you consider a severe flaw? I see severe flaws even in very popular games :)

Regarding making geometry a hidden component: it, kind of, is a hidden component in HyperRogue. I mean, yeah, it is in the name. But people play HyperRogue without understanding what hyperbolic geometry is. They think it is a game taking place on a sphere, or a normal Euclidean map with some kind of fish-eye projection. On the other hand, I do not think it is really possible to hide it from someone who knows how these things work, as the perspective works very different in hyperbolic geometry.

Which is actually quite cool: while so-called "non-Euclidean" games do all they can to show how weird they are, the actual non-Euclidean geometry pretends to be normal (but actually it is way more weird).

More precisely -- you can have x,y coordinates, or (easier to work with) x,y,z coordinates, but you would quickly run into numerical precision issues, and that binary representation prevents that. (Probably it was HyperRogue, other hyperbolic games are wrapped or small enough to work without it. David Madore's hyperbolic maze has a wrapped world so it uses a totally different system. I think Sokyokuban also has tree-based representation of the map, even if the world is small. It could also be Hypermine by Ralith, it is also open world and open source.)

Thanks for sharing! I thought that Circull was a game that once existed for iOS and then was lost, did not know about the web version. (Esfera Chess was another iOS game that was non-Euclidean in some sense and is lost.)

Yeah, it is updated from time to time. Mostly small changes, but the version from 2014 was much less detailed, so better to list it as newer.

Currently working on a more detailed "book" on this :)

Addressing hierarchically constructed tiling is quite similar to addressing hyperbolic tilings. Both the "hat" tiling and "spectre" tiling already work in HyperRogue. (Spectre is not yet released but pushed to GitHub.)

There is a brief explanation here: https://twitter.com/ZenoRogue/status/1638997769141604360

Our article on various tile-based game maps might give some inspiration: http://roguetemple.com/z/hyper/geoms.php

See also the catalog https://zenorogue.github.io/tes-catalog/

Antichamber has lots of weird, cool stuff in it, but nothing to do with non-Euclidean geometry. (They just use the term incorrectly.)