Ideal simplices and double-simplices, their non-orientable hyperbolic manifolds, cone manifolds and orbifolds with Dehn type surgeries and graphic analysis

In connection with our works in Molnár (On isometries of space forms. Colloquia Math Soc János Bolyai 56 (1989). Differential geometry and its applications, Eger (Hungary), North-Holland Co., Amsterdam, 1992), Molnár (Acta Math Hung 59(1–2):175–216, 1992), Molnár (Beiträge zur Algebra und Geometrie 38/2:261–288, 1997) and Molnár et al. (in: Prékopa, Molnár (eds) Non-Euclidean geometries, János Bolyai memorial volume mathematics and its applications, Springer, Berlin, 2006), Molnár et al. (Symmetry Cult Sci 22(3–4):435–459, 2011) our computer program (Prok in Period Polytech Ser Mech Eng 36(3–4):299–316, 1992) found 5079 equivariance classes for combinatorial face pairings of the double-simplex. From this list we have chosen those 7 classes which can form charts for hyperbolic manifolds by double-simplices with ideal vertices. In such a way we have obtained the orientable manifold of Thurston (The geometry and topology of 3-manifolds (Lecture notes), Princeton University, Princeton, 1978), that of Fomenko–Matveev–Weeks (Fomenko and Matveev in Uspehi Mat Nauk 43:5–22, 1988; Weeks in Hyperbolic structures on three-manifolds. Ph.D. dissertation, Princeton, 1985) and a nonorientable manifold $$M_{c^2}$$ M c 2 with double simplex $${\widetilde{{\mathcal {D}}}}_1$$ D ~ 1 , seemingly known by Adams (J Lond Math Soc (2) 38:555–565, 1988), Adams and Sherman (Discret Comput Geom 6:135–153, 1991), Francis (Three-manifolds obtainable from two and three tetrahedra. Master Thesis, William College, 1987) as a 2-cusped one. This last one is represented for us in 5 non-equivariant double-simplex pairings. In this paper we are going to determine the possible Dehn type surgeries of $$M_{c^2}={\widetilde{{\mathcal {D}}}}_1$$ M c 2 = D ~ 1 , leading to compact hyperbolic cone manifolds and multiple tilings, especially orbifolds (simple tilings) with new fundamental domain to $${\widetilde{{\mathcal {D}}}}_1$$ D ~ 1 . Except the starting regular ideal double simplex, we do not get further surgery manifold. We compute volumes for starting examples and limit cases by Lobachevsky method. Our procedure will be illustrated by surgeries of the simpler analogue, the Gieseking manifold (1912) on the base of our previous work (Molnár et al. in Publ Math Debr, 2020), leading to new compact cone manifolds and orbifolds as well. Our new graphic analysis and tables inform you about more details. This paper is partly a survey discussing as new results on Gieseking manifold and on $$M_{c^2}$$ M c 2 as well, their cone manifolds and orbifolds which were partly published in Molnár et al. (Novi Sad J Math 29(3):187–197, 1999) and Molnár et al. (in: Karáné, Sachs, Schipp (eds) Proceedings of “Internationale Tagung über geometrie, algebra und analysis”, Strommer Gyula Nemzeti Emlékkonferencia, Balatonfüred-Budapest, Hungary, 1999), updated now to Memory of Professor Gyula Strommer. Our intention is to illustrate interactions of Algebra, Analysis and Geometry via algorithmic and computational methods in a classical field of Geometry and of Mathematics, in general.

József Engel de Szepeslőcse – Linguist and Physician

József Engel (1807–1870) originates from an intellectual family from the Northern part of Hungary, he got to Marosvásárhely (Târgu Mureş) due to family relations. There he was assistant in the Golden Deer Pharmacy. Later he graduated the theoretical course of Chemistry at the Pest University. His thesis was printed. Then he studied medicine at the same university between 1830 and 1836. He wrote his thesis about the measles (De Morbilis). Meanwhile studying at Pest he got interested in Hungarian linguistics. He elaborated a study on the stem words of the Hungarian language which won a competition of the Hungarian Academy of Science. Engel was a much appreciated general practitioner at Marosvásárhely. The famous mathematician, János Bolyai was also his patient. He subscribed to German medical journals, collected plants and minerals. But his major interest was linguistics. In the middle of the 1850-s there was a movement at Kolozsvár (Cluj-Napoca) to establish a Museum Society. Then Engel’s linguistic research work was rediscovered. Some articles were published about him and even a fragment of his work in progress got printed. In 1857 Engel moves to Kolozsvár to help the founding of the Transylvanian Museum Society and to finish his treatise. As a general practitioner he could hardly make his living. In 1859 the Hungarian Academy of Science elected him corresponding member. He finished his thesis in linguistics and sent it to the Academy as an inaugural address. It was presented, but not published. His conception was considered obsolete. He died quite forgotten at Kolozsvár. At the Academy Henrik Finály held a memorial speech upon his life and activity. His two sons and two grandsons continued the medical traditions.

The Ramsey theory of the universal homogeneous triangle-free graph

The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math. 38(1) (1971) 69–83] and denoted [Formula: see text], is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.[Formula: see text] , eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and culminating in work of Sauer [Coloring subgraphs of the Rado graph, Combinatorica 26(2) (2006) 231–253] and Laflamme–Sauer–Vuksanovic [Canonical partitions of universal structures, Combinatorica 26(2) (2006) 183–205], the Ramsey theory of [Formula: see text] had only progressed to bounds for vertex colorings [P. Komjáth and V. Rödl, Coloring of universal graphs, Graphs Combin. 2(1) (1986) 55–60] and edge colorings [N. Sauer, Edge partitions of the countable triangle free homogenous graph, Discrete Math. 185(1–3) (1998) 137–181]. This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph [Formula: see text], there is a finite number [Formula: see text] such that for any coloring of all copies of [Formula: see text] in [Formula: see text] into finitely many colors, there is a subgraph of [Formula: see text] which is again universal homogeneous triangle-free in which the coloring takes no more than [Formula: see text] colors. This is the first such result for a homogeneous structure omitting copies of some nontrivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code [Formula: see text] and the development of their Ramsey theory.