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
Circular chess advance only opens one square for the bishop. Some players advance the central pawns first anyway, while others prefer to advance the rooks' pawns in order to open lines of attack for the more powerful pieces; it is not known which move, if any, is objectively best. The different geometry of the square and circular boards creates considerable differences in endgame theory: three of the four "basic checkmates" on a square board (those with king and rook, king and two bishops, or king and bishop and knight against a lone king) rely on forcing the defending king into the corner