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he scissor group 𝕊poly<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>𝕊</mi><mo class="qopname">poly</mo><!--nolimits--></math> of polygons is defined as the free abelian group subject to these two families of relations. In some sense, this entire article is an exploration of scissor and congruence relations in diverse contexts. By and by, we will construct several closely related scissor groups 𝕊poly<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>𝕊</mi><mo class="qopname">poly</mo><!--nolimits--></math>, 𝕊count<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>𝕊</mi><mo class="qopname">count</mo><!--nolimits--></math>, 𝕊ring<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>𝕊</mi><mo class="qopname">ring</mo><!--nolimits--></math>, 𝕊cover<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>𝕊</mi><mo class="qopname">cover</mo><!--nolimits--></math>, and 𝕊mot<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>𝕊</mi><mo class="qopname">mot</mo><!--nolimits--></math>, each constructed as a free abelian group modulo scissor and congruence relations

Any polygon in the plane can be cut into finitely many triangles that can be reassembled into a rectangle of unit width. Figure 1 illustrates three steps (2<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn></math>, 3<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>, and 4<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn></math>) of the general algorithm. The algorithm consists of 5<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>5</mn></math> elementary transformations. (1) Triangulate the polygon. (2) Transform triangles into rectangles. (3) Fold long rectangles in half. (4) Rescale each rectangle to give it an edge of unit width. (5) Stack all the unit width rectangles end to end. The length of the unit width rectangle is the area.