The narrator recounts how a friend of his, Halpin Chalmers, took to experimenting with consciousness-expanding drugs and Mental Time Travel. After taking an exotic drug, Chalmers claims to have pushed beyond curved space-time into angular time, where he saw horrifying and incomprehensible entities — and was in turn seen by them. Fearing that the entities, which he refers to as "the Hounds of Tindalos", mean to pursue him back into the mundane world, Chalmers attempts to foil them by clearing out a room and plastering over the corners and edges to make a space containing only curves, no angles through which the Hounds can manifest. However, an earthquake shatters the plaster, and Chalmers is found dead the following morning.
This story contains examples of:
- Agent Scully: The narrator spends most of the story disbelieving what Chalmers says, thinking he's just had a bizarre drug-induced hallucination.
- Alien Geometries: The Hounds are ravenous creatures of weird geometry who travel through time and space, and the only way to avoid them once they're on your trail is to completely avoid sharp angles (such as in a completely circular room).
- Apocalyptic Log: A police officer investigating Chalmers' sudden death finds a few scraps of paper he had written on, the last of which was apparently scribbled as he was attacked and killed by the titular monsters. Amusingly, the doomed writer literally transcribes his dying scream:Chalmers' notes: They are breaking through! Smoke is pouring from the corners of the wall. Their tongues—ahhhhh—
- Bizarre Alien Biology: The Hounds themselves are never described, but the strange blue secretion they left behind on Chalmers' corpse consists of some sort of living matter that, weirdly, contains no trace of recognizable enzymes.
- Bizarre Alien Locomotion: The Hounds are strange, angular creatures who existed long before single-celled organisms first evolved. They are normally invisible as they inhabit the "angles of time" as opposed to the "curves of time" that humans and other life-forms do. Thus, they can freely travel through time as well as materialize through any corner of a wall or object if it's sharp enough. Thus, the only way to avoid getting hunted and eaten by one is to stay in a room with no angles.
- Call a Smeerp a "Rabbit": The Hounds of Tindalos are named mainly for their persistence in tracking down anyone whose "scent" they have picked up. Otherwise they're vaguely-described abominations that apparently hunt their victims through time, can materialize from any nearby corner they find, and presumably don't bear much if any family resemblance to canines as we know them at all.
- Clock Roaches: The hounds are chasing Chalmers because he traveled back in time too much.
- Cryptic Background Reference: In his Apocalyptic Log, Chalmers mentions a bunch of unexplained stuff that's presumably connected to his researches.Chalmers' notes: I do not believe they can reach me, but I must beware of the Doels. Perhaps they can help them break through. The satyrs will help, and they can advance through the scarlet circles.
- First-Person Peripheral Narrator: Chalmers' friend who witnesses his fateful drug trip and subsequent deterioration. Chalmers addresses him as "Frank", suggesting that it is Long himself.
- Hell Hound: The Hounds of Tindalos are Hellhounds mixed with Eldritch Abomination. Their existence predates multicellular life on Earth, they are immortal, and they can travel freely through time and space. Since they can exist in the "angles" of time (everything else lives in the "curves"), they can manifest through any corner anywhere. The Hounds aren't named as such for their appearance, but for their relentlessness. The Hounds hunger for something other living creatures possess that they lack; once they become aware of something or someone they will never stop hunting them. An easy way to gain their attention is to travel through time.
- Higher Understanding Through Drugs: The story centers on a man who tries to use an obscure drug as a form of Mental Time Travel. At first it works brilliantly, until he goes a little too far into the past...
- Hyperspace Is a Scary Place: While not hyperspace per se, the dimension dwelt in by the Hounds of Tindalos is a pretty nasty place to be, as if you travel through it, you set the Hounds on you. And as they can enter the world through any angle, and will never stop; this is bad to say the least.
- Poke in the Third Eye: The story depicts an experiment in mental time travel that goes horribly wrong when the traveler is spotted by the eponymous beings, spectral creatures from another plane of existence who follow him back through time and eventually kill him.
- Time Abyss: The Hounds of Tindalos are from an age "before space and time". They can move casually through the angles of time, to hunt those humans who have travelled in time and devour them.