Presentation on theme: "Area – Scissors style. Puzzle! 3 RULES: 1. You can not talk, point, nudge, indicate 2. You can’t take pieces from others, you can only give them! 3. You."— Presentation transcript:
Converse If polygon P can be decomposed into pieces that are rearranged to make Q, then P and Q have the same area. Is the opposite true? If P and Q are polygons of equal area, can it be decomposed into pieces that can be put together to make Q?
Bolya-Gerwein Theorem The answer is yes! Proved independently by Bolyai and Gerwein in the 1830’s
Equi Two polygons are “equi” if you can cut one into pieces and rearrange those pieces to get the second polygon If P and Q are polygons that are equi, we say P~Q
Properties P~P If P~Q then Q~P If P~Q and Q~R then P~R.
Steps 1.Every Triangle is equi to a Rectangle 2.Parallelograms with a common base and the same height are Equi 3.Two rectangles with the same area are Equi 4.Every polygon can be dissected into triangles 5.Every polygon is Equi to a rectangle 6.Two polygons with the same area are Equi
Step 4: Dissect a polygon into triangles Induct: Base Case: n =3, a triangle, we are done Induction Step: The base case is three N is the number of sides Find a diagonal
Step 5: Polygons are equi to rectangles Use Step 4 freely Make rectangles with the same base Squish them together
Step 6: Finish it! So any two polygons with the same area can be made equi to some rectangles. Since these rectangles have the same area, they are equi and we are done!!!! Bolyai and Gerwein Rectangles of the same area We are almost done
Extensions We can ask, what are the minimum number of cuts necessary? What kind of motions are allowed? – Parallel translation – Central Symmetries In general, you need both, the Hadwiger-Glur result classified what polygons you need just parallel translations for
Hadwiger-Glur l + c - a b - J l = a-b-c Two polygons can are equi through parallel translations alone if they have the same J l for every line l. Furthermore, the only shapes that are equi to a square by using parallel translations alone are centrally symmetric polygons.
Classical Dissections Aha! Solutions, Martin Erickson Wikipedia, Henry Dudeney
Third Dimension Hilbert’s Third Problem Max Dehn showed that the regular tetrahedron and the Cube of the same volume were not Equi in 1902. Still an open problem in Non-Euclidean geometries
References Dissections: plane & fancy by Greg N. Frederickson Aha! Solutions by Martin Erickson Equivalent and Equidecomposable Figures by V.G. Boltyanskii http://mathworld.wolfram.com/Dissection.html http://www.cut-the-knot.org/Curriculum/Geometry/CarpetWithHole.shtml The “Two Basics” mathematics teaching approachand open ended problem solving in China by Zhang, Dianzhou1and Dai, Zaiping