The Perpahedron is a 3-D robotic version of the Polygon Disciple Perpendiculus. Boss idea by Diepmon. Original concept by Fillygroove.


Despite being a very annoying machine that has no purpose than to annoy the heck out of anyone who meets it, it has no relation to the Hesperian League. In fact, the Perpahedron was the inspiration for Lord Sassafras to create the Sassafras Supreme way back then. As for the Perpahedron itself, its origins are largely conjectural. Some say that Perpendiculus created this laughing machine himself to annoy The Lich, while others suggest it might be a creation of Octrax as a way to humiliate the square disciple. Another outrageous theory suggests that it was the creation of Dr. Lacus back in his days as Dr. Tanker, while yet another puts forth that it was a creation of the Hesperians (see above for why that isn't possible.) The most likely out of these four theories is that of Perpendiculus being its creator, although he evidently had some help. For all his rightened square-ness, Perpendiculus can't read a lick of Runic.


The Perpahedron is really just a static 3-D rendition of Perpendiculus, except with eyes and an animated, HAWing mouth with buck teeth. It otherwise is colored the same and looks very much like its namesake.


The Perpahedron appears randomly in any area of the Realm, in any location except Imperium and the Tritellmennian Hills. However, it will always appear within 300 tiles of a player if there is a Square Nest nearby. When it spawns, it comes crashing down from the sky in the Square Nest, scattering squares everywhere and leaving a crater on the ground, where it immediately activates and HAWs its head off continuously. Since it is permanently stuck in the ground, the Perpahedron is completely immobile and cannot move from its position.

Touching the Perpahedron is highly advised against, as its immense body damage, even on its hands, will deal a good amount of damage to players. The Perpahedron is passive unless attacked, at which it will begin spawning a ton of squares and fire upon the offending entity.

In terms of square spawning rates, the Perpahedron can spawn up to a maximum of 500 squares (including normal, Summoner Squares, Sentry Squares, and Splitter Squares) in the vicinity, with an additional limit of 500 squares that it can spawn through its attacks. When inactive, the Perpahedron spawns one normal square every 2 seconds that moves into the general area around it. This spawn rate will triple if provoked, and will also allow it to spawn hostile squares.

The Perpahedron can use any of the below attacks, cycling through them at random. Regardless of the attack, its mouth and eyes will always be animated, saying HAW continuously at varying volume and its eyes jiggling all over the place. Its entire body will also remain static, regardless of attack type (bullets may appear to come from nothing, same for squares.) The Perpahedron has only one phase, and destroying all the hands will cause the body to become vulnerable, like Perpendiculus.


  • Fist Armor: The Perpahedron gains 8 Yellow Squares and Summoner Squares (alternating 4 and 4) circling each of its hands, 32 in total. It will start "punching" tanks that get too close, dealing very high damage with each punch, stacked on top of the additional Squares' body damage. Each square that is a part of the fist armor will have 100 HP and will deal 12 HP of body damage. Summoner Squares that are destroyed will break off and chase players.
  • Pew Pew: The Perpahedron starts firing off bullets that are targeted at players at long range for medium damage. Lasts for 12 seconds.
  • Squarenova: Every square within 25 tiles is brought together and explodes in a massive explosion of squares that are sent out at high speed.
  • Square-off: The Perpahedron spawns four Square Nest Defenders and sends them to attack players There can be duplicates. Usually used at least 45 seconds after the last of the previous four have been destroyed.
  • Square In The Head: The Perpahedron summons five Beta Squares right on top of five players. Said players are immobile until their allies destroy the squares on top of them. They can still fire, but not hit the square directly unless they have drones or a top-mounted Auto Turret.
  • Rightened Lightning: The Perpahedron fires four lightning streams from each hand, doing high damage and inflicting Paralyzed and Stunned for 10 seconds. The lightning moves rather slowly in a sine-wave pattern, and has a range of 40 tiles.
  • Quadrilateral Bombardment: The Perpahedron will release a series of louder "HAW HAW HAW"s as it fires a bunch of squares quickly upward off-screen. Yellow square-shaped targets will begin to dot the map, slowly becoming red. There will be tens of these quickly appearing over the course of about six seconds. When a target becomes fully red, it will flash, vanish, and a Yellow Square will land there, firing four bullets evenly spaced outward. Players standing in the target when the Square hits will take minor damage + the Square's body damage. The bullets themselves do much more damage. The Perpahedron's cannon arms are Armored as this ability happens.
  • Square Wannabes: The Perpahedron summons an Alpha Square right on top of a Necromancer, Factory, or Hellfire, dealing the Alpha Square's body damage.
  • Square Swarm: The Perpahedron spawns 75-110 Summoner Squares or 160-220 Swarm Squares in front of it. It then directs it in one group in a fat line formation. Existing squares are not affected.
  • Squares Squares Squares!: The Perpahedron spawns 225 squares that orbit around it, blocking attacks. No further action is taken to attempt to focus defense with this square ring, due to the Perpahedron having the intelligence of a brick.
  • HAW HAW HAW: The Perpahedron begins HAWing faster and faster as it begins to suck up all the squares in the vicinity into its bottom spawner. It then spits out a gigantic clump of mixed squares that is launched out at high speed in a random direction. Upon contact with anything, even the tiniest bullet, all the squares are released out like Squarenova.
  • HAWt and Cold: The Perpahedon fires two streams of bullets from its barrels, one fiery red-orange and one icy light blue. Both bullet streams have particles behind them of the same color. The fire bullets inflict Burning for 3 seconds and medium damage, while the ice bullets inflict medium-low damage and Slowed for 4 seconds.
  • Haw: The Perpahedron gains Frenzy and begins to spontaneously generate hawthorn fruit that appears and disappears at random intervals in the vicinity, dealing medium damage to players hit by the fruit.


  • AW HAW HAW HAW!: Only used if it spawned near Perpendiculus's fortress. If Perpendiculus is currently in battle and the Perpahedron drops from the sky, a cutscene will play for the players inside Perpendiculus's fortress where Perpendiculus and his minions will stop attacking after a gigantic crash can be heard outside. The awkward silence is broken by the HAW HAW HAW of the Perpahedron bursting through the doors of Perpendiculu's inner sanctum, carried by a group of Square Nest Defenders. It then is tossed on the floor of the room, acting as an annoying setpiece that can't be destroyed (but nor can attack) for the rest of Perpendiculus's battle, which resumes after the cutscene ends.



The original Perpahedron, created by the cat who goes by many names.

  • The Perpahedron started out as a joke image on Discord created by Fillygroove (Tacocat247) sometime back in mid-2017 with pictures of a 3-D Perpendiculus that seems to be falling based on its downwards-facing angle. It was created specifically for Text Wall's comic series Aeroventures, issue #7 (In the Shadow of the Perpetrator.) In 2018, it became an attack of Lord Sassafras, and now this in 2019.
  • The Perpahedron is trademarked, and is correctly stylized as "The Perpahedron™".
  • Its name comes from "Perpendiculus" and "-hedron," a stem applied to geometric solids to describe them having a specific number of faces or type of face. In this case, "Perpahedron" can be literally taken to mean "Solid with faces of that of a Perpendiculus."
  • Its theme is the appropriately named "Hawtiness," by Graviatar (GComet.)
  • Most of its attacks are taken from Perpendiculus.
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