• 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.

• 제어로봇시스템학회:학술대회논문집
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• 1991.10b
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• pp.1591-1596
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• 1991

The present paper describes a method of recognizing a polyhedron employing the notion of network constraint analysis. Typical difficulties in three-dimensional object recognition, other than shading, reflection, and hidden line problems, include the case where appearances of an object vary according to observation points and the case where an object to be recognized is occluded by other objects placed in its front, resulting in incomplete information on the object shape. These difficulties can, however, be solved to a large extent, by taking account of certain local constraints defined on a polyhedral shape. The present paper assumes a model-based vision employing an appearance-oriented model of a polyhedron which is provided by placing it at the origin of a large sphere and observing it from various positions on the surface of the sphere. The model is actually represented by the sets of adjacent faces pairs of the polyhedron observed from those positions. Since the shape of a projected face gives constraint to that of its adjacent face, this results in a local constraint relation between these faces. Each projected face of an unknown polyhedron on an acquired image is examined its match with those faces in the model, producing network constraint relations between faces in the image and faces in the model. Taking adjacency of faces into consideration, these network constraint relations are analyzed. And if the analysis finally provides a solution telling existence of one to one match of the faces between the unknown polyhedron and the model, the unknown polyhedron is understood to be one of those memorized models placed in a certain posture. In the performed experiment, a polyhedron was observed from 320 regularly arranged points on a sphere to provide its appearance model and a polyhedron with arbitrarily postured, occluded, or imposed another difficulty was successfully recognized.

• Journal of Educational Research in Mathematics
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• v.14 no.1
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• pp.19-37
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• 2004

In school mathematics, polygon and polyhedron are defined by vague terms such as "surrounded" or "formed" Moreover, the inclusion of boundary and interior in the definition of polygon and polyhedron is varied according to the context. Polygon and polyhedron are considered as "context-dependent concept" in school mathematics. Elementary school mathematics introduces a surface only in the context of solid, yet secondary school mathematics explains a surface as the trace of the line movement. From the perspective of fallabilism, it is possible and desirable to lead students to revise and improve their conceptions on polygon, polyhedron, and surface. It is more appropriate to name a face, an edge, and a vertex rather than to express a face of polyhedron, an edge of polyhedron, and a vertex of polyhedron in textbooks. The term "surface as a polygon" in secondary mathematics textbooks shows a conflict between intuitive approach in elementary school and logical approach in secondary school.

• 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.

• Journal of Sericultural and Entomological Science
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• v.16 no.2
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• pp.55-75
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• 1974

Transovarial transmission of cytoplamic polyhedrosis virus in the silkworm was studied by observing the phenomena of induction, interference and virulence enhancement in the larvae from moths inoculated with hexagonal Polyhedra of cytoplasmic polythedrosis virus. The experimental results obtained are as followings. 1. The effect of inoculation with tetragonal polyhedra of cytoplasmic polyhedrosis virus on the rate of infective induction with hexagonal polyhedron virus and with hexagonal plus tetragonal polyhedron viruses in the larvae from moths infected with hexagonal polyhedron virus was studied. Infection rate was higher by 40 to 60 percent in the larvae from infected female group than in tile larvae from noninfected female group. 2. In the studies of the effect of formalin-feeding on the induction of infection with hexagonal polyhedron virus, infection rate was higher by 40 percent in the larvae from infected female group than in the larvae from noninfected female group. However, there was no significant difference in the infection rates between the two formalin-concentration groups. 3. The effect of cold treatment on the induction of infection with hexagonal polyhedron virus was studied. Infection rate was higher by 50 percent in the larvae from infected female group than ill the larvae from noninfected female group. No difference was found in the infection rates of the two treatment groups of 12 and 48 hours. 4. The phenomena of induction and interference were studied by observing rate of infection with hexagonal polyhedron virus induced by the inoculation with tetragonal polyhedron virus. The degree of interference of primary hexagonal polyhedron virus by secondary tetragonal Ployhedron virus was increased as the dosage of secondary virus was increased. At the concentration of 1${\times}$10$\^$8/m1 of the secondary virus, the degree of interference was similar to. that of control group. On the other hand, infection with tetragonal polyhedron virus at low concentration was interfered by the primary virus. At the concentration of 1N10f/m1 of tetragonal polyhedron virus, however, the rate of infection with tetragonal polyhedron virus was increased sharply, which is still lower by 30 percent than that of control group. 5. In the studies of induction and virulence enhancement, based on the 1ate of mixed infection with hexagonal and tetragonal polyhedron viruses, the highest difference of infection rate between experimental group and control group exceeded 40 percent when the concentration of tetragonal polyhedron virus was 1${\times}$10$\^$7/m1. However, the infection rate of control group was not affected by concentrations of tetragonal polyhedron virus.

• Journal for History of Mathematics
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• v.23 no.2
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• pp.101-121
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• 2010

This article includes an introduction, a history of Pick's theorem on lattice polyhedron and its proof, Reeve's theorem on 3-dimensional lattice polyhedrons extended from the Pick's theorem and Ehrhart polynomial generalized as an n-dimensional lattice polyhedron, and then shows the relationship between the volume of 3-dimensional polyhedron and the number of its lattice points by means of Reeve's theorem. It is aimed to apply the relationship to the visualization of sums in sequences.

• 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 Sericultural and Entomological Science
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• v.17 no.2
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• pp.101-118
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• 1975

The pathogenicity of hexagonal and tetragonal cytoplasmic polyhedron virus to one of present leading silkworm varieties, Jam 103${\times}$Jam 104, and its parents, Jam 103 and Jam 104, was investigated. The activation of occult virus by oral inoculation of foreign virus as well as the interference phenomena between the activated and inoculated cytoplasmic polyhedron viruses was observed and the results obtained are as follows. 1. The pathogenicity of cytoplasmic polyhedrosis viruses. 1) There was a high significant difference in the pathogenicity of hexagonal polyhedron virus between the hybrid and its parents showing the lowest infection rate in the hybrid (Jam 103${\times}$Jam 104), medium infection rate in the Japanese line (Jam 103) and the highest infection rate in the Chinese line (Jam 104). In the pathogenicityof tetragonal polyhedron virus, a significant difference was observed only between the hybrid (Jam 103${\times}$Jam 104) and the Chinese line (Jam 104) by showing a higher infection rate in the Chinese line than in the hybrid. 2) The pathogenicity of both hexagonal and tetragonal polyhedron viruses showed a high significant difference in different silkworm instars inoculated by showing a higher infection rate at the second instar than at the fourth instar. 3) The pathogenicity of both hexagonal and tetragonal polyhedron viruses was increased as the concentration of viruses inoculated increased. 2. The phenomena of induction and interference by oral inoculation of foreign virus. 1) The induction rate of cytoplasmic polyhedrosis virus was higher in the parents than in the hybrid. In the parents. a higher rate in the Chinese line than in the Japanese line was observed. 2) The effect of inoculation at different instars on the induction was studied and the induction rate was higher at the second instar inoculated than at the fourth instar inoculated. 3) The degree of activation of hexagonal polyhedron virus with inoculation of tetragonal polyhedron virus was very high when a lower concentration of virus was inoculated and it was very low when a higher concentration of virus was inoculated 4) The degree of activation of tetragonal polyhedron virus with inoculation of hexagonal polyhedron virus was very low when a lower concentration of virus was inoculated and it was not observed when a higher concentration of virus was inoculated. 5) The mixed infection rate with hexagonal and tetragonal polyhedron viruses was higher at the second instar inoculated than at the fourth instar inoculated. 6) It was observed that the secondary hexagonal pc]yhedron virus activated interfered with the primary tetragonal polyhedron virus inoculated when the inoculated concentration of the primary virus is low and the primary virus inoculated interfered with the secondary virus activated when the inoculated concentration of the primary virus is high.

• Proceedings of the KSME Conference
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• 2005.11a
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• pp.11.2-11.2
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• 2005

• Journal of the Korean Mathematical Society
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• v.36 no.5
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• pp.911-921
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• 1999

We define the coincidence index of pairs of maps p, f : $\widetilde{X}$ $\rightarrow$ X where p is a covering of a polyhedron X. We use a polyhedral transversality Theorem due to T. Plavchak. When p=identity we get the classical fixed point index of self map of polyhedra without using homology.