**Continuation of the Tangram**

**Lemma 1:** If sixteen equal isosceles right triangles are combined into a convex polygon, then a rational side of one triangle does not lie along an irrational side of another.

**Lemma 2:** If sixteen equal isosceles right triangles are combined into a convex polygon, then the sides of the polygon are formed by sides of the same kind (rational or irrational) of the triangles. Moreover, if a side of the polygon which is formed by the rational or the irrational sides of the triangles is said to be a rational or an irrational side (respectively) of the polygon, then in general the rational and the irrational sides of the polygon alternate. In particular, if an angle of the polygon is a right angle, the two adjacent sides are both rational or both irrational.

**Lemma 3:** If sixteen equal isosceles right triangles are combined into a convex polygon, then the number of the sides of the polygon does not exceed eight.

**Lemma 4:** If sixteen equal isosceles right triangles are combined into a convex polygon, then this polygon can be inscribed in a rectangle with all the rational or the irrational sides of the polygon as the sides of the rectangle.

These lemmas will lead us to this equation,

**a ^{2}+b^{2}+c^{2}+d^{2}=2xy-16**

Through 4 cases, of which I will be presenting 1, we will find that there are 20 possible solutions. However, only 13 of these are actual solutions due to the fact that in a real Tangram set, the pieces are not actually all right isosceles triangles.

This talk was not only intriguing, but also brought home a rather popular theorem. Emilie’s presentation of the result (in her second talk) walked us through the proof that there are precisely thirteen convex polygons which may be formed by use of tangrams. By use of four lemmas, the result may be broken into four cases centering around inscribing the shape in a rectangle. Using these cases, it is possible to show that there are precisely twenty possibilities. Of these twenty possible creations using sixteen right isosceles triangles, seven of them turn out to be impossible to create using the tangrams due to constraints resulting from various special shapes in the tangram set. Fantastic!

Emilie’s talk very interesting. I had not known very much about tangrams before listening to Emilie’s talk. I was very impressed by her proofs and her ability to clearly walk the class through them. Her results were interesting and presented very clearly (as Chris explained, seven out of the twenty possible shapes were proven to be impossible to create).