Integer triangle cosine equals 0, ±1/2, or ±1. The only ones of these giving an angle strictly between 0° and 180° are the cosine value 1/2 with the angle 60°, the cosine value –1/2 with the angle 120°, and the cosine value 0 with the angle 90°. The only combination of three of these, allowing multiple use of any of them and summing to 180°, is three 60° angles. Conditions are known in terms of elliptic curves for an integer triangle to have an integer ratio "N" of the circumradius to the inradius. The smallest case, that of the equilateral triangle, has
Many-angled ones a result, when they manifest in our universe they appear as disconnected floating body parts of some larger beast that is complete in the higher dimension (similar to how a three dimensional being would appear in flatland as its parts pass through the plane of that two-dimensional world). More recently, they have appeared in books by British authors or comics and video games from British comics writers who have worked for "2000 AD". The many-angled ones were mentioned in Charles Stross's "The Atrocity Archives". This work features the usual appearances by "nameless horrors of the abyss," which may or may