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In this section tessellations of the hyperbolic upper half plane by Schwarz triangles will be discussed using elementary methods. For triangles with ideal vertices, i.e. where all three angles are strictly positive, the elementary approach of {{harvtxt|Caratheodory|1954}} will be followed. For triangles with one or two ideal vertices, i.e. one or two angles equal to zero, elementary arguments of {{harvtxt|Evans|1973}}, simplifying the approach of {{harvtxt|Hecke|1935}}, will be used: in the case of a Schwarz triangle with one angle zero and another a right angle, the orientation-preserving subgroup of the reflection group of the triangle is a [[Hecke group]]. For an ideal triangle in which all angles are zero, the existence of the tessellation will be established by relating it to the [[Farey series]] described in {{harvtxt|Hardy|Wright|1979}} and {{harvtxt|Series|2015}}. In this case the tessellation can be considered as that associated with three touching circles on the [[Riemann sphere]], a limiting case of configurations associated with three disjoint non-nested circles and their reflection groups, the so-called "[[Schottky group]]s", described in detail in {{harvtxt|Mumford|Series|Wright|2015}}. Alternatively—by dividing the ideal triangle into six triangles with angles 0, {{pi}}/2 and {{pi}}/3—the tessellation by ideal triangles can be understood in terms of tessellations by triangles with one or two ideal vertices.
===Triangles with no ideal vertices===
Suppose that the [[hyperbolic triangle]] Δ has angles {{pi}}/''a'', {{pi}}/''b'' and {{pi}}/''c'' with ''a'', ''b'', ''c'' integers greater than 1. The hyperbolic area of Δ equals {{pi}} – {{pi}}/''a'' – {{pi}}/''b'' – {{pi}}/''c'', so that
:<math>{1\over a} + {1\over b} + {1\over c} < 1.</math>
The construction of a tessallation will first be carried out for the case when ''a'', ''b'' and ''c'' are greater than 2.<ref> {{harvnb|Caratheodory|1954|pages=177–181}}</ref>
The equality above implies that if one of the angles is a right angle, say ''a'' = 2, then both ''b'' and ''c'' are greater than 2 and one of them, ''b'' say, must be greater than 3. In this case, reflecting the triangle across the side AB gives an isosceles hyperbolic triangle with angles {{pi}}/''c'', {{pi}}/''c'' and 2{{pi}}/''b''. The construction of the tessallation that is given below applies equally well in this case because the doubled triangle is isosceles.<ref>{{harvnb|Caratheodory|1954|pages=181–182}}</ref>
===Triangles with one or two ideal vertices===
===Ideal triangles===
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