Orb of the Sword
The Orb of the Sword is a special weapon found in the Kraken Depths after the player has collected 10 Sunken Treasures. While it is active, it generates a sword of pure energy that floats next to the player and moves along with the player. It is a very powerful weapon, capable of killing many monsters that usually cannot be attacked directly, as well as destroying barrow walls in the Burial Grounds along with a few other wall objects, like the trees in the Caribbean. Monsters also recognize its power, and will avoid moving towards the tile the sword occupies.
However, the player cannot directly rotate the sword to point it at enemies; once the Orb is picked up, the sword always points in the same direction relative to the player as the player moves around. In order to use the sword effectively, the player must exploit the hyperbolic geometry of the game in order to indirectly cause the sword's orientation to shift relative to the orientation of the surrounding world. The key is the fact that the interior angles of a triangle in the hyperbolic plane do not add up to 180°; this fact can be exploited to cause the sword to become rotated relative to the rest of the world in spite of its orientation remaining completely unchanged relative to the player.
The Orb of the Sword is required for retrieving treasure from the Burial Grounds. If the player successfully retrieves 10 ancient treasures, a second orb, the Orb of the Sword II, will be unlocked, which behaves like the Orb of the Sword but generates a sword pointing in the opposite direction. Using both Orbs together will give the player two swords on either side. Successfully retrieving 25 ancient treasures will give a further power-up of both Orbs, causing the swords to become capable of destroying even more objects.