• East Asian mathematical journal
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• v.17 no.1
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• pp.115-128
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• 2001

We present old unsolved problems on BCK-sequences connected with convex congruences on BCK-algebras. We posed also some new problems on subalgebras.

• Communications of Mathematical Education
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• v.38 no.2
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• pp.215-230
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• 2024

Congruences and symmetry are familiar concepts that can be encountered in everyday life. In order to effectively understand and acquire these concepts, the role of appropriate visual examples is important. This analysis examined various visual examples used in the process of learning the concepts of congruence and symmetry in elementary mathematics textbooks and focused on the use of convex polygons and concave polygons. As a result of the analysis, various types of polygons were used as visual examples for teaching and learning of congruence and symmetry in textbooks. The frequency of use of concave polygons was higher in the order of congruence, line symmetry, and point symmetry, and it was confirmed that it was used more frequently in the process of exploring properties than in the introduction of the concept. Based on these results, a plan to utilize concave polygons in teaching and learning of congruence and symmetry was sought.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.823-833
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• 2009

It is well-known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of the modular group, then its translates by the group elements form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such hyperbolically convex polygons can be obtained by using Dirichlet's and Ford's polygon constructions. Another method of obtaining a fundamental domain for subgroups of the modular group is through the use of a right coset decomposition and we call such domains standard fundamental domains. In this paper we give subgroups of the modular group which do not have hyperbolically convex standard fundamental domain containing only inequivalent cusps.

• Journal for History of Mathematics
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• v.20 no.4
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• pp.49-60
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• 2007

In this article, the concept of rigidity of smooth surfaces in the three dimensional Euclidean space which naturally arises in elementary geometry is introduced, and the natural process of the development of rigidity theory for compact surfaces and its generalizations are investigated.