It’s believed that the answers to various problems of basic math consisting of elementary calculus and geometry faced by high-school and college students have already been well-established, and there might not be any room for further study or discussion. But actually, pitfalls are found almost everywhere in this area, especially when it comes to graphing functions to understand their behavior in a higher dimensional space that we can neither experience nor access directly. Although topological abstraction plays an exceptionally important role in modern mathematics, it can greatly reduce the amount of information to what is needed for the problem to be just identified and lead to a lot of misconceptions. Rather than such abstraction processes, we advocate that figurative approaches are often considered as prime solution candidates. The work presented at Cyberworlds had been completely rewritten in Wolfram language. At the conclusion, it was also shown that the fifth* representation of the real projective plane could be visualized.
*The fourth one had already described in the appendix of Anschauliche Geometrie (by Hilbert and Cohn-Vossen, first published in 1952).
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