In this paper we first introduce a fractional form formula among a number of Euler’s formulas. We then extend the formula and with mathematical induction prove the case when the number of terms increases and the exponent is integer. Afterwards, we study the connection between Euler’s formula and Lagrange interpolating polynomial and use the latter to prove part of the extended formula. We then obtain a new formula from this connection. At last, we derive a set of new equations from the extended formula.

No abstract uploaded!

We proved in this paper that 14 triangles are necessary to triangulate a square with every angle no more than 72◦, answering an unsolved problem proposed by Professor D.Eppstein. At first, computer is used to search for topologically feasible solutions of triangulations of a rectangle. Then the geometric feasibility is analyzed case by case by plane geometry. We also proved that any rectangle can be 72◦ triangulated if the number of triangles is not limited.

No abstract uploaded!

No abstract uploaded!