By Francisco Botana, Tomas Recio
This ebook constitutes the completely refereed post-proceedings of the sixth foreign Workshop on automatic Deduction in Geometry, ADG 2006, held at Pontevedra, Spain, in August/September 2006 as a satellite tv for pc occasion of the foreign Congress of Mathematicians, ICM 2006.
The thirteen revised complete papers awarded have been rigorously chosen from the submissions made as a result of a decision for papers - in the scope of ADG - almost immediately after the assembly. The papers convey the energetic number of subject matters and strategies and the present applicability of computerized deduction in geometry to diverse branches of arithmetic and to different sciences and technologies.
Read Online or Download Automated Deduction in Geometry: 6th International Workshop, ADG 2006, Pontevedra, Spain, August 31-September 2, 2006, Revised Papers PDF
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Additional resources for Automated Deduction in Geometry: 6th International Workshop, ADG 2006, Pontevedra, Spain, August 31-September 2, 2006, Revised Papers
Each surface is planar, spherical, or conical. Two surfaces intersect along a segment of a curve. Since the surfaces are quadrics, each curve has degree at most four. If the degree of the curve is one, it is a line. If the degree of the curve is two, then it is a conic section. In particular it is a planar curve. When the intersection of a cone and a sphere is reducible, the irreducible components are planar. When the intersection of two cones is reducible, the intersection consists either of planar curves or of a line and an irreducible nonplanar cubic.
Let f ∈ F. Let C1 , . . , Ck be the boundary curves of f that are congruent to C. Let T be the set of all congruences T : Ci → C, as i ranges from 1 to k. The group of motions of R3 acts on F by (T∗ f )(x) = f (T −1 x). With this action, T∗ χ(A) = χ(T A). We define DC (f ) = T ∈T T∗ f ∈ F. This is well-defined in the sense that it does not depend on the expression for f as a sum of characteristic functions. Note that DC depends on f through the list C1 , . . , Ck of boundary curves, so it is not a linear operator on F .
For example, a regular tetrahedron is not equidecomposable with a cube of the same volume. The proof of this result is that the cube and the tetrahedron have different Dehn invariants, but all equidecomposable polyhedra have the same Dehn invariant. In two dimensions, the corresponding result is true: any two polygons of the same area are equidecomposable. That is, if they have the same area, the first polygon can be cut into finitely many triangles in such a way that they can be reassembled into the second polygon.