Rolling simplexes and their commensurability. I: The axiom and criterion of incompressibility and the momentum lemma (Q1947655)

The author of this paper intended to provide a characterization of Pappian affine planes in terms of oriented area equality. Unfortunately the characterization is false. She first defines the notion of \textit{rolling}: a triangle \(ABC\) can be rolled into a triangle \(A'B'C'\), if the latter can be obtained from the former through a finite sequence of \textit{simple moves}, a triangle \(ABY\) being obtained from a triangle \(ABX\) through a simple move if and only if \(XY\parallel AB\). She then states, without proof, Proposition 1, which states that, given two triangles \(ABC\) and \(A'B'C'\), the latter can be rolled into \(ABD\), where \(D\) is a point on the line \(BC\). A proof of this proposition, albeit in the presence of an additional requirement that ``rolling'' includes the translation of a triangle \(ABC\) along any segment \(AX\) with \(X\) on \(AB\) (axiom A10), was provided by the reviewer in pages 93--94 of [Aequationes Math. 66, No. 1--2, 90--99 (2003; Zbl 1085.51003)]. Then, three axioms are presented, of which the first, called \(R0\), is claimed to play a rather miraculous role of proving the other two, and of implying Desargues and Pappus. \(R0\) states that, if \(D\) is a point on the line \(BC\), and is different from \(C\), then the triangle \(ABD\) cannot be rolled into triangle \(ABC\) (this is called the incompressibility axiom). The problems in the purported proof occur already in the \textit{Momentum lemma}, which is a key ingredient in the author's attempted proof of Desargues. Its proof claims a use of \(R0\), but in fact, the conclusion drawn, namely that \(Q_1SP_3\) can be rolled into \(Q_1SP_1\), follows from the list of triangles that can be ``sequentially rolled into each other''. In other words, the proof of Desargues theorem (in Corollary 1) is out of thin air, i.e. out of the axioms for affine planes! That the paper's result is false, i.\ e., that not only is the proof flawed, but that there cannot be a proof of it, follows from the results by \textit{L. Lesieur} [Math. Z. 93, 334--344 (1966; Zbl 0151.26001)] and \textit{J.-C. Petit} [Math. Z. 94, 271--306 (1966; Zbl 0156.19303)], where it is shown that a notion of directed area satisfying axioms implying \(R0\), can be introduced in affine planes more general than Moufang planes. The same results appear in [\textit{Yu. P. Razmyslov}, Mosc. Univ. Math. Bull. 66. No. 5, 223--226 (2011)] with similar proofs, and without proof in [\textit{O. V. Gerasimova} and \textit{Yu. P. Razmyslov}, J. Math. Sci. 177, No. 6, 860--861 (2011; Zbl 1308.51002)].