• The Mathematical Education
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• v.53 no.4
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• pp.493-507
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• 2014

The purpose of this study is to analyze how high school students form and interpret the mental image in the process of solving regular polyhedron problems. For this purpose, a set of problems about the regular polyhedron's vertex is developed on the base of the regular polyhedron's duality and circulation. and applied to 2 students of the 12th graders in D high school. After 2 hours of teaching and learning and another 2 hours of mental image-analysis process, the following research findings are obtained. Fisrt, a student who recorded medium high-level grade in the national scholastic test can build the dynamic image or the patten image in the process of solving regular polyhedron's vertex problems by utilizing the 3D geometry program. However, the other student who recorded low-level grade can build the concrete-pictorial image. Second, pattern image or dynamic image can help students solve the regular polyhedron's vertex problems by proper transformation of informations and the mental images while the concrete-pictorial image does not help. Hence, it is recommended that the mathematics teachers should develop teaching and learning materials about the regular polyhedron's duality and circulation and also give students suitable questions to build the various mental images.

• The Mathematical Education
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• v.45 no.2 s.113
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• pp.191-204
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• 2006

Computer technology has a potential to change the contents of school mathematics and the way of teaching mathematics. But in our country, the problem whether computer technology should be introduced into mathematics classroom or not was not resolved yet. As a tool, computer technology can be used by teachers who are confident of the effectiveness and who can use it skillfully and can help students to understand mathematics. The purpose of this study was to investigate the effective way to introduce and utilize computer technology based on the status quo of mathematics classroom setting. One possible way to utilize computer technology in mathematics classroom in spite of the lack of computer and the inaccessibility of useful software is using domain specific simulation software like 'Polyhedron'. 'Polyhedron', as we can guess from the name, can be used to explore regular and semi regular polyhedra and the relationship between them. Its functions are limited but it can visualize regular polyhedra, transform regular polyhedra into other polyhedra. So it is easier to operate than other dynamic geometry software like GSP. To investigate the effect of using this software in mathematics class, three classes(one in 6th grade from science education institute for the gifted, two in 7th grade) were chosen. Activities focused on the relationship between regular and semi regular polyhedra. After the class, several conclusions were drawn from the observation. First, 'Polyhedron' can be used effectively to explore the relationship between regular and semi regular polyhedra. Second, 'Polyhedron' can motivate students. Third, Students can understand the duality of polyhedra. Fourth, Students can visualize various polyhedra by reasoning. To help teachers in using technology, web sites like NCTM's illuminations and NLVM of Utah university need to be developed.

• Bulletin of the Korean Mathematical Society
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• v.33 no.3
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• pp.397-409
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• 1996

Let X be a connected compact polyhedron and $f : X \to X$ a selfmap of X. The Reidemeister number of f is denoted by R(f).

• Journal of Computational Design and Engineering
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• v.2 no.1
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• pp.47-54
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• 2015

In recent years, 3D printers have become popular as a means of outputting geometries designed on CAD or 3D graphics systems. However, the complex user interfaces of standard 3D software can make it difficult for ordinary consumers to design their own objects. Furthermore, models designed on 3D graphics software often have geometrical problems that make them impossible to output on a 3D printer. We propose a novel AR (augmented reality) 3D modeling system with an air-spray like interface. We also propose a new data structure (octet voxel) for representing designed models in such a way that the model is guaranteed to be a complete solid. The target shape is based on a regular polyhedron, and the octet voxel representation is suitable for designing geometrical objects having the same symmetries as the base regular polyhedron. Finally, we conducted a user test and confirmed that users can intuitively design their own ornaments in a short time with a simple user interface.

• Journal for History of Mathematics
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• v.16 no.4
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• pp.91-106
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• 2003

We seek the possibility of application of ICT(Information ＆ Communication Technology) to the experience learning in mathematics. Especially we focused on regular polyhedron in the middle school mathematics and on geodesic dome. We investigate related historical tales about the subjects which seem quite helpful of motivating to understand them. Then by using them and by ICT we develop an experience learning program for geodesic dome as an application of regular polyhedron. Through the program we expect that students show positive attitude in mathematical tendency and curiosity about mathematics and that the application of ICT to the experience learning in mathematics is very effective to improve their learning skill in mathematics.

• Journal of Korea Multimedia Society
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• v.3 no.3
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• pp.298-308
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• 2000

We present an efficient algorithm for finding the sequence of extreme vortices of a moving regular convex polyhedron of with respect to a fixed plane H.. The algorithm utilizes the spherical Voronoi diagram that results from the outward unit normal vectors nF$_{i}$ 's of faces of P. It is well-known that the Voronoi diagram of n sites in the plane can be computed in 0(nlogn) time, and this bound is optimal. However. exploiting the convexity of P, we are able to construct the spherical Voronoi diagram of nF$_{i}$ ,'s in O(n) time. Using the spherical Voronoi diagram, we show that an extreme vertex problem can be transformed to a spherical point location problem. The extreme vertex problem can be solved in O(logn) time after O(n) time and space preprocessing. Moreover, the sequence of extreme vertices of a moving regular convex polyhedron with respect to H can be found in (equation omitted) time, where m$^{j}$ $_{k}$ (1$\leq$j$\leq$s) is the number of edges of a spherical Voronoi region sreg(equation omitted) such that (equation omitted) gives one or more extreme vertices.

• Korean journal of applied entomology
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• v.32 no.4
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• pp.407-413
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• 1993

Twelve recombinant viruses with wider host range were plaque purified after coinfectian of Autographa cahjornica and Bombyx mOT! NPVs into Sf9 ar BmN-4 cells. Restriction endonucleases analysis of the recombinant's DNAs showed that the recombinatIOn between AcNPV and BmNPV genomes had occurred more than once. When the recombinam RecB-8, derived from BrnN-4 cells, was observed by electron rntcroscopy, the shape of the polyhedron was a regular tetrahedron, and few virions were occluded into a polyhedron.

• Journal of Gifted/Talented Education
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• v.18 no.1
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• pp.53-77
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• 2008

This study was conducted with the focus on the process of constructing 3 definition and produced definitions as well as gifted students' conceptions of a mathematical definition In the study, the students made five types of regular polyhedrons (section1), observed them and stated their characteristics (section2) and then constructed a definition of regular polyhedrons based on their observations (section3). We divided students into two groups by analyzing students' definitions. One group made definitions that were consist with a mathematical definition of regular polyhedrons, the other one made definitions that were not. We checked if they fulfilled requirements for a mathematical definition. Researchers sought to gain various suggestions through the analysis of the observations and definition laid down by the students and through the characteristics shown by the students in the process of defining the concept.

• Journal of Elementary Mathematics Education in Korea
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• v.17 no.2
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• pp.351-370
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• 2013

The purpose of this study is to analyze the modes of visualization which appears in the process of thinking that mathematically gifted 6th grade students get to understand components of the three-dimensional shapes on the duality of regular polyhedrons, find the duality relation between the relations of such components, and further explore on whether such duality relation comes into existence in other regular polyhedrons. The results identified in this study are as follows: First, as components required for the process of exploring the duality relation of polyhedrons, there exist primary elements such as the number of faces, the number of vertexes, and the number of edges, and secondary elements such as the number of vertexes gathered at the same face and the number of faces gathered at the same vertex. Second, when exploring the duality relation of regular polyhedrons, mathematically gifted students solved the problems by using various modes of spatial visualization. They tried mainly to use visual distinction, dimension conversion, figure-background perception, position perception, ability to create a new thing, pattern transformation, and rearrangement. In this study, by investigating students' reactions which can appear in the process of exploring geometry problems and analyzing such reactions in conjunction with modes of visualization, modes of spatial visualization which are frequently used by a majority of students have been investigated and reactions relating to spatial visualization that a few students creatively used have been examined. Through such various reactions, the students' thinking in exploring three dimensional shapes could be understood.

• Korean Journal of Microbiology
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• v.23 no.2
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• pp.129-137
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• 1985

The RK-temperate phage which infected with Bacillus cereus was isolated and the characters were investigated. The induction of RK-temperate phage from host bacterium attained by ultraviolet light irradiation (15W, 30cm, 30-120sec) and mitomycin C treatment (0.2-2 ug/ml). The host range of RK-temperate phage was not revealed with lysogenic and related strains of B. cereus. But B. cereus(PS) 352 which obtained by N-nitrosoguanidine treatment (1,000{$\mu}g/ml)$ to phage infected with host bacteria was sensitive bacteria of RK-temperate phage. RK-temperate phage was stabilized at the condition of nutrient broth (pH 7-8), Tris-buffer (pH 7-8) and ammonium buffer (pH 8-9) and Sorensen's phosphate buffer (pH 6-7), but unstabilized at other salt solutions and pH range. Also, thermostability was to $45^{\circ}C$ but unstabilized at above $50^{\circ}C$. At RK-temperate phage, the measurment values of head, neck, mid tail and end tail were 59nm, $9{\times}16nm,\;10{\times}189nm,\;and\;10{\times}14nm$ respectively. The morphology of head was regular polyhedron, and the end tail was coneate form. On the one hand, the number of capsid protein layer of tail were consist of 4, 35, and 1 at neck, mid tail, and end tail, respectively. RK-temperate phage was identified with DNA phage and G+C contents were 38.63. The latent time of RK-temperate phage was 30 minutes and the burst size was 70-80. And the host bacteria was lysed in case of multi-infection, above moi 1.