Hyper Rogue

[Marek14]

Just discovered this:

http://www.roguetemple.com/z/hyper/

Seems really interesting

EDIT: OK, I bought the game (Steam version is only 1 Euro) and tried it out. It's actually very good! The author makes ingenious use of the fact that hyperbolic space has huge amount of space available.This allows him to have zones which both extent indefinitely AND border many other zones.

The game is simple, but difficult - you move in the space defined by truncated (3,7) tiling, picking up the treasure and killing monsters. Combat is puzzle-based, you simply kill the monsters if you attack them before they can attack you -- otherwise you die. Monsters move based on deterministic algorithms so you can predict their moves in advance. The game map is theoretically infinite (new parts are generated as you move out of the old ones), you only have limited visibility (which ties nicely with graphic limitations of Poincare projection).

The game has many zones, all with their own monsters and gimmicks. So far I've seen:

Crossroads - relatively safe area whose main feature is that connections to other zones are incredibly common.
Icy Land - frozen waste divided up by ice walls. Each space has a temperature defined and if you stand near a wall, it will evetually melt through your body heat.
Living Cave - walls in here use cellular automaton algorithm to grow and shrink.
Jungle - this is overgrown by plants who actively try to get you.
Desert - inspired by the Dune novels, this desert is divided by impassable dunes, and it's home to huge sandworms who are slow, but grow instead of moving and cannot be killed directly.
Alchemist Lab - place with two colors of tiles, one serving as floors, one as walls, that can be switched under certain circumstances.
Land of Mirrors - full of objects that create your duplicates when you touch them.
Land of Eternal Motion - bleak place where every tile you step on disappears. Enemies can be easily outrun thanks to the fact that there are no parallel lines, and so they have to move along longer curves to move in the same direction as you.
Dry Forest - place where you can cut down trees. Apparently, it is one spark away from a wildfire, though.

And there are still more.

The difficulty is also done interestingly: each zone has its associated type of treasure. As you collect the treasures, the number of monsters in that zone grows, so it's in your interest to only collect a few items and then move to another place.

[ICN5D]

Cool! Just watched some gameplay vids. I might need to check this out. There's way more space in there than I can imagine.

[quickfur]

I've actually written a roguelike game with a 4D world map. The nice thing about roguelikes is that ASCII graphics are acceptable, which makes displaying a 4D slice of the game world as a 2D array of 2D arrays possible (and even easy!). Unfortunately I don't have access to a Windows dev machine so it only runs in Linux... that basically reduced the size of my already non-existent audience by half. So it never actually got to the point where interesting things could happen (e.g., interacting with AI-controlled entities), it remained mainly as an exploration and collect-the-items style game.

[Marek14]

Wendy, do you know of a simple way to embed an apeirogon or horocycle in the (6,6,7) tiling?

[wendy]

The groups with finite-extent symmetries, such as x3x7o, really don't have what it takes to have a horogon, because the f.e. groups represent something like integers, and you just can't keep dividing the integers as often as you want, which is what the horogon symmetry can do.

You need something with a finite-content group (ie one or more vertices on the horizon), or a kind of broken symmetry that is only 'inwards-symmetric', such as the infinite stack of horogonal cupolae support. Here, they support eg 1 -> 2, which looks identical inwards, but not outwards.

The tiling formed by 5 5 5 3 supports horocyclic symmetry, but this is because it has a horogonal-inwards symmetry at some large scale.

[Marek14]

The groups with finite-extent symmetries, such as x3x7o, really don't have what it takes to have a horogon, because the f.e. groups represent something like integers, and you just can't keep dividing the integers as often as you want, which is what the horogon symmetry can do.

You need something with a finite-content group (ie one or more vertices on the horizon), or a kind of broken symmetry that is only 'inwards-symmetric', such as the infinite stack of horogonal cupolae support. Here, they support eg 1 -> 2, which looks identical inwards, but not outwards.

The tiling formed by 5 5 5 3 supports horocyclic symmetry, but this is because it has a horogonal-inwards symmetry at some large scale.

Yeah, I tried yesterday and one of the problem is that a grid like that doesn't really support convergent parallels (since they eventually come closer than grid resolution)... Spent some time debating this with the developer yesterday. But it seems there are interesting changes ahead -- one of the new lands being planned is a place with hyperbolic bug armies (stemming partially from my wondering about the differences between war in Euclidean and hyperbolic space), and the developer also discovered a very interesting periodic covering of hyperbolic plane composed of 40 cells (28 hexagons, 12 heptagons) and he is basing several lands on this.

[Klitzing]

... and the developer also discovered a very interesting periodic covering of hyperbolic plane composed of 40 cells (28 hexagons, 12 heptagons) and he is basing several lands on this.

Come on - details! - which patch?
--- rk

[Marek14]

... and the developer also discovered a very interesting periodic covering of hyperbolic plane composed of 40 cells (28 hexagons, 12 heptagons) and he is basing several lands on this.

Come on - details! - which patch?
--- rk

Well, I don't have a picture on me and he says that the new version will be out soon I'll post a screenshot then.

[Marek14]

OK, game was updated, so this is the pattern:

http://roguetemple.com/z/hyper/period.png

EDIT: OK, after looking at the pattern, I wonder what the symmetry group is. There's 3-fold symmetry at center of every pattern. Looks like it might have something to do with either {6,5} or {12,4}. Each main pattern has 6 long "edges" shared with exactly one other color, but I'm not sure what structure of corners would correspond to what I see there.

The game itself now has a "pattern editor" (accessible through the settings) where you can move along this pattern.

[quickfur]

It looks like x3x7o.

[Marek14]

It looks like x3x7o.

Well, yes, the basic tiling is x3x7o. My question is how to classify tiling with the pattern, i.e. the colored block? I think the first step should be to compute the exact inner angles of the hexagons and heptagons. This would give us their area, and then the area of the whole pattern (28 hexagons, 12 heptagons). If it's isomorphic to another tiling, it must have the same area as the basic polygon of that tiling...

EDIT: Or not. Even if I can't compute the area of individual hexagon or heptagon, it turns out I CAN compute the total area of 7 hexagons + 3 heptagons:

Start with triangle joining centers of 3 cells around a vertex. Its angles are pi/3, pi/3 and 2*pi/7, so the defect is pi - pi/3 - pi/3 - 2*pi/7 = pi/21.

This triangle can be divided into 3 areas, two belonging to a hexagon and one to a heptagon:
2a + b = pi/21.

Now, a complete hexagon has area 6a and complete heptagon has 7b.

If we multiply the equation by 21, we'll get:

42a + 21b = pi.

42a is 7 * 6a - area of 7 hexagons and 21b is 3 * 7b - area of 3 heptagons. Since the pattern is 4 times that much, its area is exactly 4*pi.

This is equivalent of infinite hexagon (inner angle 0), or of:

pi/3 9-gon (3/9) of {9,6}
pi/2 12-gon (6/12) of {12,4}
2*pi/3 18-gon (12/18) of {18,3}

The question is: what is the exact symmetry of the pattern tiling?

[Marek14]

Currently (v6.0) there are some new interesting geometric patterns. For example, seeing the concentric circles in the Hive from afar is very weird. Or the dense straight line pattern in Vineyard.

[Marek14]

The 6.2 version introduces some interesting challenges. One of them is the Round Table.

You see a round table. It's round, but it's also quite big. The Knights tell you to find the Holy Grail, which is in the center.

The challenge is: how to find a center of a big circle in hyperbolic plane? The circle is big enough that you don't see the edge when you're deep inside.

I have my own idea for an algorithm, but I would like to hear your ideas as well Remember that you play on (6,6,7)-grid and there is an item (Dead Orb) which you can carry in vast quantities and drop on cells to mark them.

How to find the Holy Grail?

[ubersketch]

I like the use of horocycles in here. I wasn't expecting them actually.

[Marek14]

I like the use of horocycles in here. I wasn't expecting them actually.

Well, they are something typically hyperbolical.

[ubersketch]

I like the use of horocycles in here. I wasn't expecting them actually.

Well, they are something typically hyperbolical.

Yeah, as a big fan of roguelikes, this is like having an unstoppable force collide with an immovable object. What I learned from this game is this weird uniform tiling the creator discovered on his own and that navigating in hyperbolic space is nearly impossible.

[ubersketch]

Wendy, do you know of a simple way to embed an apeirogon or horocycle in the (6,6,7) tiling?

The game does this, many of the worlds use horocycles in the (6,6,7) tiling.

[ubersketch]

The 6.2 version introduces some interesting challenges. One of them is the Round Table.

You see a round table. It's round, but it's also quite big. The Knights tell you to find the Holy Grail, which is in the center.

The challenge is: how to find a center of a big circle in hyperbolic plane? The circle is big enough that you don't see the edge when you're deep inside.

I have my own idea for an algorithm, but I would like to hear your ideas as well Remember that you play on (6,6,7)-grid and there is an item (Dead Orb) which you can carry in vast quantities and drop on cells to mark them.

How to find the Holy Grail?

Dang, I didn't know about the Dead Orbs. This is an interesting mathematical problem.

[Marek14]

The 6.2 version introduces some interesting challenges. One of them is the Round Table.

You see a round table. It's round, but it's also quite big. The Knights tell you to find the Holy Grail, which is in the center.

The challenge is: how to find a center of a big circle in hyperbolic plane? The circle is big enough that you don't see the edge when you're deep inside.

I have my own idea for an algorithm, but I would like to hear your ideas as well Remember that you play on (6,6,7)-grid and there is an item (Dead Orb) which you can carry in vast quantities and drop on cells to mark them.

How to find the Holy Grail?

Dang, I didn't know about the Dead Orbs. This is an interesting mathematical problem.

Well, you do realize that this is an old topic? Right now, the game is on version 10-something. For example, my question about embedding horocycles is no longer relevant since the solution was found.

Also not sure what you mean by "this weird uniform tiling the creator discovered on his own" -- the (6,6,7) tesselation is well-known. Zeno has a series of posts here: http://zenorogue.blogspot.cz/ where he explains where the various elements come from.

Though there are a few other periodical tilings in the game made from the main grid tiles that might qualify as weird uniform tilings...

[Marek14]

Reviving this topic for a bit to show some graphics. Last year I talked to the developer and showed him some of my work in uniform hyperbolic tilings, and he added Conway's orbifold system to the game. This allows rendering them in a way that is much more visually striking than what the old Hyperbolic Applet could do (and you can even play the game on them, if you choose so).

For example, here are the three distinct variants of (4,4,4,4,3):

34444-3.png

(878.7 KiB) Not downloaded yet

34444-5b.png

(867.93 KiB) Not downloaded yet

34444-7.png

(868.55 KiB) Not downloaded yet

The game can automatically sort the faces into classes of equivalence and color them like so. I've been experimenting a bit with making big, more complicated tilings into a series of images centered on each kind of face.

[quickfur]

Kinda tangential, but this gives me an idea: has anyone ever studied anisotropic space before? I.e., it could be hyperbolic in one direction, spherical in another direction, and Euclidean in a 3rd direction. Is such a thing possible? How would it behave?

[Klitzing]

has anyone ever studied anisotropic space before? I.e., it could be hyperbolic in one direction, spherical in another direction, and Euclidean in a 3rd direction. Is such a thing possible? How would it behave?

Marek brought this very topic up just some month ago. Just look here: http://hi.gher.space/forum/viewtopic.php?f=3&t=2353.
--- rk

[Marek14]

Kinda tangential, but this gives me an idea: has anyone ever studied anisotropic space before? I.e., it could be hyperbolic in one direction, spherical in another direction, and Euclidean in a 3rd direction. Is such a thing possible? How would it behave?

Specifically, it seems that there are several ways to do this. One way is just more or less Cartesian product. The other approach, which I looked into, is based on creating various sets of improper points in projective geometry. This, however, usually leads to SPACETIME geometries, not space geometries; i.e. you will end up with zero or imaginary distances between certain points, which can be better understood using the relativistic concept of interval.

[PatrickPowers]

Kinda tangential, but this gives me an idea: has anyone ever studied anisotropic space before? I.e., it could be hyperbolic in one direction, spherical in another direction, and Euclidean in a 3rd direction. Is such a thing possible? How would it behave?

Seems to me that string theory is like this. The Ads/CFT correspondence certainly is.

The geometry of our Universe is like this. Einstein relativity combines hyperbolic, elliptical, and Euclidean elements. Whether you could do it with a rigid background geometry I don't know, but it seems possible.

[Marek14]

Well, as for anisotropic spaces, check out this:

https://www.youtube.com/watch?v=2LotRqz ... e=youtu.be

HyperRogue now supports the Solv geometry; I'm not sure if there is a better implementation of that one around.

[mr_e_man]

Well, as for anisotropic spaces, check out this:

https://www.youtube.com/watch?v=2LotRqz ... e=youtu.be

HyperRogue now supports the Solv geometry; I'm not sure if there is a better implementation of that one around.

I found that the Solv geometry occurs in (H²)², the product of two hyperbolic planes.

A single plane H² with curvature -1/a² can be given horocyclic coordinates, for which the metric is

ds² = a² (du₁² + exp(2u₁) du₂²).

If a=1, then u₂ is length along the horocycle, and u₁ is radial distance away from the horocycle.

Combining the metrics for two planes,

ds² = a² (du₁² + exp(2u₁) du₂² + du₃² + exp(2u₃) du₄²).

The Solv geometry is on the hypersurface where u₁+u₃=0. The metric is

ds² = a² (2 du₁² + exp(2u₁) du₂² + exp(-2u₁) du₄²).

If we set the scale by a²=1/2, and define v₂=au₂, v₄=au₄, then this becomes

ds² = du₁² + exp(2u₁) dv₂² + exp(-2u₁) dv₄²,

which has the same form as the metric given by ZenoRogue.

[Marek14]

Interesting! Do the remaining two Thurston geometries (Nil and universal cover of (SL,R)) also occur as subspaces of higher-dimensional product spaces?

[lllllllllwith10ls]

Also, hyperrogue has the dimensional crystal which we can use to visualize higher dimensions potentially

[mr_e_man]

Interesting! Do the remaining two Thurston geometries (Nil and universal cover of (SL,R)) also occur as subspaces of higher-dimensional product spaces?

They do occur as subsets of higher-dimensional pseudo-Euclidean spaces. You might call it spacetime (with more than one time dimension). And the metric and the symmetry are maintained.

I could give details (e.g. Nil occurs in ℝ¹² with signature (+)⁶(-)⁶), if you're interested. But I'm not even sure if I'm interested. I worked on this for a while, but it seems like the intrinsic view of the geometry is more useful than the extrinsic, especially when there's such a large difference in dimension.

[Marek14]

Interesting! Do the remaining two Thurston geometries (Nil and universal cover of (SL,R)) also occur as subspaces of higher-dimensional product spaces?

They do occur as subsets of higher-dimensional pseudo-Euclidean spaces. You might call it spacetime (with more than one time dimension). And the metric and the symmetry are maintained.

I could give details (e.g. Nil occurs in ℝ¹² with signature (+)⁶(-)⁶), if you're interested. But I'm not even sure if I'm interested. I worked on this for a while, but it seems like the intrinsic view of the geometry is more useful than the extrinsic, especially when there's such a large difference in dimension.

So, six spatial and six time dimensions?