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
Semi-invariant of a quiver the coefficients of the characteristic polynomial In case Q has neither loops nor cycles the variety "k"[Rep(Q,d)] has a unique closed orbit corresponding to the unique d-dimensional semi-simple representation, therefore any invariant function is constant. Elements which are invariants with respect to the subgroup SL(d) := ∏ SL(d("i")) form a ring, SI(Q,d), with a richer structure called ring of semi-invariants. It decomposes as where A function belonging to SI(Q,d) is called semi-invariant of weight "σ". Consider the quiver Q: Fix d = ("n","n"). In this case "k"[Rep("Q",("n","n"))] is congruent to the set of square matrices of size "n": "M"("n"). The