• East Asian mathematical journal
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• v.30 no.3
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• pp.303-310
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• 2014

We study the existence of regular polygons and regular solids whose vertices have integer coordinates in the three dimensional space and study side lengths of such squares, cubes and tetrahedra. We show that except for equilateral triangles, squares and regular hexagons there is no regular polygon whose vertices have integer coordinates. By using this, we show that there is no regular icosahedron and no regular dodecahedron whose vertices have integer coordinates. We characterize side lengths of such squares and cubes. In addition to these results, we prove Ionascu's result [4, Theorem2.2] that every equilateral triangle of side length $\sqrt{2}m$ for a positive integer m whose vertices have integer coordinate can be a face of a regular tetrahedron with vertices having integer coordinates in a different way.

• Journal of the Korean Mathematical Society
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• v.38 no.5
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• pp.1069-1105
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• 2001

Regular maps and hypermaps are cellular decompositions of closed surfaces exhibiting the highest possible number of symmetries. The five Platonic solids present the most familar examples of regular maps. The gret dodecahedron, a 5-valent pentagonal regular map on the surface of genus 5 discovered by Kepler, is probably the first known non-spherical regular map. Modern history of regular maps goes back at least to Klein (1878) who described in [59] a regular map of type (3, 7) on the orientable surface of genus 3. In its early times, the study of regular maps was closely connected with group theory as one can see in Burnside’s famous monograph [19], and more recently in Coxeter’s and Moser’s book [25] (Chapter 8). The present-time interest in regular maps extends to their connection to Dyck\`s triangle groups, Riemann surfaces, algebraic curves, Galois groups and other areas, Many of these links are nicely surveyed in the recent papers of Jones [55] and Jones and Singerman [54]. The presented survey paper is based on the talk given by the author at the conference “Mathematics in the New Millenium”held in Seoul, October 2000. The idea was, on one hand side, to show the relationship of (regular) maps and hypermaps to the above mentioned fields of mathematics. On the other hand, we wanted to stress some ideas and results that are important for understanding of the nature of these interesting mathematical objects.

• Bulletin of the Korean Chemical Society
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• v.33 no.11
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• pp.3777-3787
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• 2012

Three kinds of 3D isomorphous and isostructural coordination polymers, namely, $\{[Ln_2(PDA)_3(H_2O)_3]{\cdot}0.25H_2O\}_{\infty}$ (Ln = La(1), Sm(2), and Gd(3)) ($PDA^{2-}$ = pyridine-2,6-dicarboxylate anion) have been synthesized under hydrothermal conditions and characterized by elemental analyses, IR spectroscopy, thermal analyses and single crystal X-ray diffraction. In these MOFs, Ln(III) centers adopt eight-coordinated and nine-coordinated with the $N_1O_7$ and $N_2O_7$ donor sets to construct distorted trianglar dodecahedron geometry and tricapped trigonal prism configurations, respectively. Based on the building block of tetranuclear homometallic $Ln_4C_4O_8$ unit (16-membered ring), 1-3 are connected into highly ordered 2D sheets via O-C-O linkers and further constructed into 3D architectures through hydrogen bonds. Crystallographic parameters suggest that the lanthanide contraction effect exist in these coordination polymers. Luminescent properties of the lanthanide-based MOFs (metal-organic frameworks) have been measured at room temperature, which reveal that they presenting ionselective characters toward certain metals, such as $Mg^{2+}$, $Cd^{2+}$ and $Pb^{2+}$ ions.