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→Tessellation by Schwarz triangles: start of construction of tessellation |
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The construction of a tessellation 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 original triangle Δ gives a convex polygon ''P''<sub>1</sub> with 3 vertices. At each of the three vertices the triangle can be successively reflected through edges emanating from the vertices to produce 2''m'' copies of the triangle where the angle at the vertex is {{pi}}/''m''. The triangles do not overlap except at the edges, half of them have their orientation reversed and they fit together to tile a neighborhood of the point. The union of these new triangles together with the original triangle form a connected shape ''P''<sub>2</sub>. It is made up of triangles which only intersect in edges or vertices, forms a convex polygon with all angles less than {{pi}} and each side being the edge of a reflected triangle.
The equality above for ''a'', ''b'' and ''c'' 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 tessellation above through increasing convex polygons adapts word for word to this case except that around the vertex with angle 2{{pi}}/''b'', only ''b''—and not 2''b''—copies of the triangle are required to tile a neighborhood of the vertex. This is possible because the doubled triangle is isosceles.<ref>{{harvnb|Caratheodory|1954|pages=181–182}}</ref>
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