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HyperRogue, by Zeno Rogue, is a roguelike game, about an adventurer in a non Euclidean world. Originally released under the name Hyperbolic Rogue in 2011, it was updated to Hyperbolic Rogue II, for the 2012 Seven Day Roguelike competition. Hyperbolic Rogue III was based on that code, and eventually re-named to HyperRogue for the Steam release.

You explore an infinite world with 66 different lands, each with its own enemies, treasures, and power-ups. The goal of the game is to find and unlock one of the Orbs of Yendor, but you're free to ignore the main quest and go for a high score instead.

The game's main gimmick is that it is set on a hyperbolic plane, which has a lot of strange geometric properties. A regular grid of hexagons and heptagons, triangles whose angles add up to less than 180 degrees, straight lines that seem to curve away from you, infinitely large circles. Returning to the same place twice is very unusual. The creator was inspired by M. C. Escher's hyperbolic tilings for some of the graphics.

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    Hyperbolic Geometry Primer 
Hyperbolic geometry is a type of non-Euclidean geometry, meaning that it does away with some of Euclid's postulates. In particular, it does away with the parallel postulate, stating that given a line and a point, there is exactly one line parallel to the line and going through the point; in hyperbolic geometry, there is an infinite amount of them (see Alien Geometries below for the non-Techno Babble explanation).

Because hyperbolic geometry, much like the surface of a sphere, cannot be represented on a flat screen while preserving both angles and areas, a projection must be used. HyperRogue's default is the Poincaré disk model, which preserves angles, but does away with areas; even though tiles closer to the edge of the disk appear smaller, in hyperbolic geometry, they are exactly the same size. In fact, the entire hyperbolic plane fits in the disk, including points which are infinitely far away (mathematically known as ideal points).

While hyperbolic geometry features many of the same constructs as Euclidean geometry, such as points, lines and polygons, it also features a few constructs with no Euclidean counterparts, many of which are exhibited in HyperRogue. These include:

  • Horocycles, "circles" with infinite area and radius. On the disk model, they look like circles touching the edge of the disk.
  • Hypercycles, "circles" with more than infinite area and radius. On the disk model, they look like circles intersecting the disk at non-right angles (an example can be seen in the photo above with the row of seven-pointed orange stars); if they intersected at right angles, they would be straight lines. They can be generated by including all points that are a given distance from a straight line, and as such, are sometimes also known as equidistants.
  • Limiting parallel lines, which intersect at a single ideal point.
  • Ultraparallel lines, which can be said to have a point where they're "closest" to each other, and beyond it, diverge, all despite both lines being perfectly straight.
  • Ideal polygons: every vertex of an ideal polygon is on an ideal point.
  • Apeirogons: polygons with an infinite amount of sides (to construct one you start with one side of X length, then add two sides connecting it that are, say, half the length of the first side, then two more sides half again in length, and so on). If the sides are short enough, a regular apeirogon approximates a horocycle.

All enemies can kill you in one hit, but most of them also die in one hit, so combat becomes a series of puzzles, not unlike Deadly Rooms of Death. You can only be killed if a monster backs you into a corner, or if several monsters surround you.

The game is free software, with the music released under CC BY-SA 3.0 and everything else released under GPL2, and can be downloaded for free or compiled from source. However, a purchase on Steam, the Play Store or the App Store will give access to the platform-specific leaderboards and achievements, may include features not yet freely released, and of course, supports the developer. The official website explains this in more detail.

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HyperRogue contains examples of:

  • Alien Geometries: The Deconstruction of this trope is the main point of the game; actual non-Euclidean geometries don't suck out your sanity and summon Eldritch Abominations, they're just a Mind Screw. Specifically it uses Hyperbolic Geometry, which is a critical concept for understanding this game. To wit: Compared to Euclidian space (the kind we experience in the real world), in a hyperbolic universe, space itself has more actual space "stuffed in" so to speak, such that if you draw two parallel lines on the ground and extend them indefinitely, they gradually diverge, despite each remaining straight relative to itself. In a euclidian universe like ours, you can only fit four squares to a corner. However a hyperbolic universe could (depending on the size of the squares and the degree of hyperbolic-ness) fit five or more. In this world, such squares would need to have a side length of 1.25374 absolute units (explanation: By analogy, if one were to represent a cube on a sphere as accurately as possible, the cube's edges would need to have a certain ratio to the sphere's radius. With regular tilings in the hyperbolic plane, this applies, too, except the "radius" is completely non-physical and hard to visualize.).
    • The Great Wall between two lands is always a straight line that doesn't cross any other Great Walls. Every land borders infinitely many other lands. Most obvious in the Crossroads, where six or seven of these may be visible at the same time.
    • The Vineyard has a regular arrangement of ultraparallel lines.
    • There are several different regular patterns that serve as a basis for lands. You can also go to Random Pattern Mode where lands and patterns can be combined in weird ways.
    • Hyperbolic space has completely unintuitive (for us) relations between linear size and area. For example: if you consider that an edge of the game's tiles is around 1 meter long, then a circle whose radius is merely 91.6 meters will have greater area than Earth. And yet, any two points will be less than 200 m apart. If it had a radius of 152 m, it would surpass the surface area of a sphere with the radius of Earth's orbit! A circle with radius 1 km would then have an absolutely insane area of 1.65 x 10^140 km2. This is showcased by the Round Table, a circle of a radius of 28 tiles. In Euclidean space, it would have an area of a few thousand tiles; here, it takes up over 30 million.
      • One of the random things Knights say when talked to is the exact capacity of their round table. However, a back-of-the-envelope calculation shows that the number of Knights guarding the outer walls is significantly higher.
    • While retracing your steps exactly is possible, the slightest deviation will be amplified enormously, taking you to unexplored areas.
    • Less obviously, the Running Dogs follow you on a path always one step to your left or your right (since they can't walk on tiles you've walked on). If you run in a straight line, you'll slowly outpace them, due to the above mentioned factor of parallel lines diverging by necessity in a hyperbolic world (for them to catch up with you they would need to be following on the exact tiles you've been walking in, but they can't, so they can only run in a line equidistant to yours).
    • And of course, R'Lyeh, with its infinitely nested Temples of C'thulhu.
  • And Your Reward Is Clothes: The reward for collecting the Holy Grail is becoming one of the Knights of the Round Table, getting you a spiffy new cape.
  • Anti-Frustration Features: The game will prevent you from walking into an enemy's attack radius, or otherwise doing anything that would instantly get you killed. You only lose the game when all your options would instantly get you killed.
  • Commonplace Rare: Your hoard of treasures includes precious gems, alchemical ingredients, powerful magic items... and wine bottles. Apparently "wine grown under a hyperbolic sun would be extremely prized in your home location".
  • Clingy Jealous Girl: If you manage to save two Princesses, they will murder each other until only one is left.
  • Crazy Jealous Guy: If you manage to save two Princes, they will also murder each other until only one is left. Apparently the political ramifications of heads of state assassinating one another isn't a concern for anyone.
  • Damn You, Muscle Memory!: Clicking on an adjacent tile will move you there. Clicking on a distant tile will move you in that direction... unless you have an orb with a ranged power, in which case it will use that power instead.
  • Damsel in Distress: A Princess is held prisoner in the Palace and there is a sidequest to find and rescue her. (Changing the settings lets you rescue a Prince instead, which is the default for female PCs.)
  • Developers' Foresight: There are lots of unexpected reactions from luring monsters into other Lands.
    • The Dead Cave appears to be an Evil Counterpart of the Living Cave. Bringing a Rock Troll from the Living Cave and killing it there will cause life to spread out from its body, turning the entire Dead Cave into a Living Cave (and quite possibly trapping you as the walls rearrange themselves).
    • The description of Demon Sharks says that they are demons from Hell that fell into the water. Sure enough, if a demon follows you into the water, they transform.
    • Most monsters trigger explosions in the Minefield when they step on a mine. Birds, however, don't, as they fly. But if you kill a bird and it happens to be on a mine, the fall of its dead body will set it off. On the other hand, Ghosts also won't trigger mines, but if you kill them, there's no corporeal body to fall, so the mine won't be set off.
  • Dynamic Difficulty: The more treasures you collect, the more monsters will spawn. Only in that land, though, so you can alleviate this by going somewhere else.
  • Easy-Mode Mockery: Playing on casual mode disables achievements.
  • 11th-Hour Superpower: The Land Of Power. Orbs that were rare finds until this point now litter the ground, and although the enemies are more numerous and powerful than anywhere else, you'll easily defeat them once you accumulate four or five powers at once. Unfortunately, you lose all your powers when you go somewhere else.
  • Endless Game: Getting an Orb of Yendor is considered "beating the game" and awards you 50 points and almost all the orb powers, but you can keep playing and try to find the Holy Grail, the Hyperstones, or just get on the high score table.
  • Equipment-Based Progression: The only way to get stronger is to collect various magical Orbs. And their effects are temporary, too, so it's more like No Progression.
  • Evil Chancellor: The Viziers in the Palace.
  • Gay Option: Standardly, the gender of the Prince/Princess you can rescue from the Palace is opposite of your character's gender. But there is a configuration option that lets you make it the same gender.
  • Fire and Brimstone Hell
  • Invincible Minor Minion:
    • Sandworms, Cthulu's Tentacles, and Rock Snakes can't be attacked, but they'll die if they're lured into a corner with nowhere to go.
    • Greater Demons can't be attacked, but when you Level Up, they become easily-slain Lesser Demons.
    • Running Dogs aren't inherently any more invulnerable than any other standard enemy. But in the Land of Eternal Motion, you can't stop running for long enough to swing a sword at them. You can still kill them with magic, by luring them out of their Land, or by tricking them into a corner with no escape.
    • The Shadow appears occasionally in the Graveyard, and simply can't be killed. However, it doesn't move directly towards you — instead it retraces your steps.
  • Jungle Japes: The Jungle. It's filled with Ivies that constantly extend, possibly attacking you, but destroying an Ivy's core will destroy all vines attached to it as well.
  • Gameplay and Story Segregation: No matter how many of the life-extending Elixirs of Life you collect, you'll still die in one hit. The description lampshades this.
  • Giver of Lame Names: The Knights of the Round Table have an infinite number of castles, and they're all called Camelot.
  • Golden Snitch: Originally the Orbs of Yendor gave 1000 points each. Every other treasure gives 1 point, and you'll usually have just over 100 by the time you unlock the Orbs. Eventually, the Orbs were changed to give 50 points, so that they weren't such a game-breaker on high score runs.
  • Hub Level: The Crossroads. Unusually, it doesn't teleport you to distant lands — hyperbolic geometry lets it get away with just having a lot of borders, which works because all of them are ultraparallel lines.
  • Humans Are the Real Monsters: If you look at the player's description, it says "This monster has come from another world, presumably to steal our treasures." This is fairly accurate, considering that monster are docile unless they see the player.
  • Infinite: The mutant ivy in the clearing. The mutant ivy is an infinitely large monster that occupies a horocycle, and thus, has infinite area. You aren't supposed to actually defeat it, just try and take some of its apples.
    • The world itself is infinite, too, and has infinitely many infinitely large regions (duh!).
  • Lethal Joke Item: The Dead Orbs. Other orbs let you throw fire and lightning, teleport, summon golems, or make you completely invulnerable. The Dead Orb doesn't do anything, and the description comments that you might as well drop it.
    It's actually the only item you can drop, so you can use it to mark areas you plan to return to. This is easily the most practical way to complete the Yendor and Grail quests.
  • Level Grinding: Averted. Unusually for a roguelike, your character doesn't get any stronger as the game continues. Orbs give you new powers, but all of their effects are temporary.
    • The enemies in Hell allow this in a strange way. Greater Demons are invulnerable, but the description explains that killing 10 Lesser Demons will make you strong enough to fight them, at least until you encounter even stronger demons. In fact, this simply turns all Greater Demons on screen into Lesser Demons, and does nothing to affect future Greater Demon spawns.
  • Mirror Self: Your main asset in Mirror Land. Mirages make the same moves as you; Mirror Images are left-right reversed. Because of the non-Euclidean geometry, they tend to wander off after a few moves.
  • No Fair Cheating: Cheating blocks you from the highscore list, and the menu shows how many times you cheated in big red letters.
  • Notice This: There's an option to give enemies and items glowing white outlines. Pressing Alt will activate it temporarily.
  • One-Hit Point Wonder: The player, and most of the monsters that aren't invincible.
  • Puzzle Boss: Every enemy in the game, since you have to figure out how to get to them, But in particular the Hedgehog Warriors, Pikemen, and Flail Guards, which cannot be attacked directly and are killed by moving across adjacent tiles.
  • Save-Game Limits: You can only save immediately after collecting an Orb of Safety. This doesn't come up very often; the game is a roguelike and you're expected to win or die in one sitting.
    The reason for the limit is that you'll explore a huge area in the course of the game, which would normally make the saved game unwieldy. The Orb of Safety teleports you to a completely different area, clearing the computer's memory of all the already-explored regions.
  • Set a Mook to Kill a Mook: Hyperbugs will fight off anyone who intrudes into their hive. You can use this to lure one swarm into another's hive, whereupon they will ignore you to focus on each other. You can then mop up the survivors.
  • Slippy-Slidey Ice World: The Icy Land and Cocytus.
  • Shout-Out: Many of the Lands are shout-outs to some other franchise.
  • Stalked by the Bell: If you wait for long enough in one place — usually because you're surrounded on all sides by walls, or enemies that can't move enough to actually kill you — the game will spawn armies of ghosts to force a Game Over.
  • Temporary Platform: The Land of Eternal Motion has nothing but this. The only creatures that live here are the ever-moving Running Dogs.
  • Tilesweeper: The Minefield, a land based on Minesweeper where the player is told how many mines are adjacent to any given tile and must reach treasures without stepping on mines.
  • 20 Bear Asses: Collecting ten of any treasure will allow magic orbs to spawn in that area. Some lands also require you to collect specific treasures before they'll unlock — to get to Hell, you need to collect ten treasures in each of nine different lands.
  • Unexpected Gameplay Change:
    • Euclidean Mode shows how the game would play on a regular hexagonal grid, with none of the hyperbolic weirdness. This changes several things, making various parts of the game easier or harder and some areas completely unplayable.
      • Running away becomes more difficult, especially from multiple enemies at a time. You have to use walls to bottleneck them, you can't rely on walking across heptagons any more. And some Lands don't have walls.
      • Obstacles become more substantial. In many lands, solid lines of impassable terrain will block off entire directions from you. This is still true in hyperbolic mode, but as the creator puts it, "there are many more directions".
      • Several Lands flat-out don't work because they require hyperparallel lines that simply don't exist in Euclidean geometry. This includes the Great Walls and the Crossroads, so travelling from Land to Land is usually impossible.
      • Mirages get much more useful, since they'll stay in the same position relative to you, allowing you to accumulate truly massive armies.
      • The Yendor Quest gets much easier.
    • In later versions, this was expanded to a full-on "experiment with geometry" menu, which gives you a lot more options for geometry, including spherical, Euclidean, hyperbolic, toroidal, triangles, squares, pentagons, octagons, whatever you fancy. You aren't limited to two dimensions, either: want to play in a grid of dodecahedra? You're the king. It even includes geometries that work in three dimensions but not in two, such as Solv (explanation: XY and XZ planes are hyperbolic, but oriented orthogonally, so receding objects will appear to stretch. As another example, taking ten steps up, one step forward and ten steps down can result in net movement of as far as a thousand steps forward.) and Nil (explanation: moving forward, then left, then backward, then right results in net movement up, allowing "impossible" Penrose triangles and staircases to exist.).
    • Version 7.0 adds Shoot-'em-up Mode (sometimes lovingly nicknamed "shmup"), where you have a ranged attack, your movement (and movement of monsters) no longer needs to follow the grid, and the game plays in real-time, with most mechanics kept unchanged.
    • In Orb Strategy Mode, orbs are presented to the player for collecting the land's treasure, and are saved up, allowing the player to escape what otherwise would be a checkmate. However, these are limited supply, and in addition, the requirements for unlocking lands are higher.
    • Version 11.0 adds Racing Mode, which is Exactly What It Says on the Tin. However, hyperbolic geometry presents yet another twist here: whereas in an Euclidean racing game, veering off to the left or right a bit doesn't matter that much, here, such veering off may result in massive detours which will significantly slow you down.
  • Where It All Began: The Icy Land, where you start the game, shares a lot with the post-endgame land Cocytus. The main difference is that, in the Icy Land, your body heat will melt the walls and create passages for you. In Cocytus, your body heat melts the floor, revealing impassable water.
  • Yet Another Stupid Death: No matter how well the game is going, you're still a One-Hit Point Wonder, and it's all too easy to get killed if you stop paying attention.

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