: 0. Abstract : 4. Related facts.

5. An open problem.

Example 8. When we thought that our investigation was over, one student made an another beautiful figure. He calculated $ (10^{12n+2} - 10^{6n + 2} + 7 \times 10^{3n - 1} - 1)^{y} $ for $ n = 26$ and $ y = 11 $, and gave it colours.

Then he got the following figure. It is not a sector. It is the shape of a convex lens that reminds us of our own universe. We have not understood the mathematical structure of this shape.

\includegraphics[height=6cm,clip]{graph9-18.eps}

References.
[1] R. Miyadera and Y. Kotera . Una Bella Curva Che Troviamo in Connessione con lo Sviluppo di Archimede 2/2005.
[2] T.Takahashi, R.Miyadera, D.Minematsu and S.Hashiba. How high school and undergraduate students could present original research of mathematics using computer algebra system, Symbolic and Algebraic Maripulation Japan Society for Symbolic and Algebraic Computation to appear.
[3] R. Miyadera, K. Miyabe, D. Kitajima, N. Fujii and K. Fujii . How high school students could present original math research using Mathematica Challenging the Boundaries of Symbolic Computation: Proceeding of the 5th International Mathematical Symposium London University@ Imperial College Press 2003.