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? How many angels can dance on the head of a pin? The question "How many angels can dance on the head of a pin?" (alternatively ""How many angels can stand on the "point" of a pin?"") is a reductio ad absurdum of medieval scholasticism in general, and its angelology in particular, as represented by figures such as Duns Scotus and Thomas Aquinas. It is first recorded in the 17th century, in the context of Protestant apologetics. The question has also been linked to the fall of Constantinople, with the imagery of scholars debating about minutiae while the Turks besieged the