Hyperbolic geometry or hypercycle, then the triangle has no circumscribed circle. As in spherical and elliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent. A special polygon in hyperbolic geometry is the regular apeirogon, a uniform polygon with an infinite number of sides. In Euclidean geometry, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180 degrees and the apeirogon approaches a straight line. However, in hyperbolic geometry, a regular apeirogon has sides of
How many angels can dance on the head of a pin? about how many could perch at the same time on the head of a pin"". Dorothy L. Sayers argued that the question was "simply a debating exercise" and that the answer "usually adjudged correct" was stated as, "Angels are pure intelligences, not material, but limited, so that they have location in space, but not extension." Sayers compares the question to that of how many people's thoughts can be concentrated upon a particular pin at the same time. She concludes that infinitely many angels can be located on the head of a pin, since they do not occupy any space there: