Apéry, François; Yoshida, Masaaki Pentagonal structure of the configuration space of five points in the real projective line. (English) Zbl 0919.52013 Kyushu J. Math. 52, No. 1, 1-14 (1998). The configuration space of five points in the real projective line \(P^1\) is defined as the orbit space \(X^\circ\) of the diagonal action of PGL\(_2{\mathbb R}\) on the set \(\{ (x_1,\ldots,x_5)\in(P^1)^5\mid x_i \not= x_j\) for \(i\not=j \}\). The orbits of all \(5\)-tuples \((x_1,\ldots,x_5)\) such that no three of the \(x_i\) coincide form a smooth compactification \(X\) of \(X^\circ\). This space is the union of \(12\) pentagons which are glued along their edges and their vertices. These pentagons form a \(\{5,4\}\)-tesselation of \(X\). The two-fold unbranched covering \(\widetilde{X}\) of \(X\) equals Poncet’s great dodecahedron which is a orientable surface of genus \(4\). Identifying antipodal points of \(\widetilde{X}\) one recovers \(X\) as a non-orientable closed surface of genus \(5\). Moreover, lifting the tesselation of \(X\) to \(\widetilde{X}\) yields a \(\{5,4\}\)-tesselation of the great dodecahedron by \(24\) pentagons. Finally, the authors give a modular interpretation of \(X\) as a quotient of the real \(2\)-ball. Reviewer: H.Löwe (Braunschweig) Cited in 5 Documents MSC: 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 51M15 Geometric constructions in real or complex geometry 14P99 Real algebraic and real-analytic geometry Keywords:configuration space; great dodecahedron; closed surface; tesselation PDFBibTeX XMLCite \textit{F. Apéry} and \textit{M. Yoshida}, Kyushu J. Math. 52, No. 1, 1--14 (1998; Zbl 0919.52013) Full Text: DOI