• 한국수학교육학회지시리즈A:수학교육
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• 제44권2호
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• pp.297-306
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• 2005

In this paper, we propose a computer environment class for student's mental representations about non-Euclidean geometry. Through the software Cinderella, students construct knowledge about non-Euclidean geometry and recognize differentness between Euclidean and non-Euclidean geometry. Also they recognize an existence of non-Euclidean geometry newly and its mental representations with images represented in Cinderella. In geometry class, we make students can use many representations systematically and can figure a visual internal image by emphasizing a transform process. And then students can reason about non-Euclidean geometry.

• 과학교육연구지
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• 제47권3호
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• pp.263-272
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• 2023

예비교사가 비유클리드 기하학에서 수학적 정의를 이용한 이차곡선의 학습으로 유클리드 기하학의 다양한 개념을 어떻게 이해하고 활용할 수 있는지를 살펴본다. 본 연구에서는 D 대학교 수학교육과 3학년 수업에서 수학적 정의를 이용하여 택시기하, 민코프스키 거리공간과 같은 비유클리드 공간의 이차곡선 학습이 예비교사들에게 새로운 기하학적 개념을 습득하고 수용하는 능력 향상에 도움을 줄 수 있음을 보였다. 이러한 결과로부터 택시기하와 민코프스키 거리공간에서의 정의를 활용한 이차곡선 학습이 창의적이고 유연한 사고를 유도하여, 예비교사들의 유클리드 기하학 교육 전문성 향상에 기여할 것으로 기대된다.

• 한국수학사학회지
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• 제24권4호
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• pp.21-31
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• 2011

이 글의 목적은 비유클리기하의 제안자의 한사람인 로바체프스키의 수학철학이 현대수학철학의 일종의 저수지였음을 보이고, 그의 수학철학이 비유클리드기하의 탄생에 기여했음을 밝히는 것이다.

• 복식
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• 제60권5호
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• pp.117-127
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• 2010

In this study, hyperspace is a result of imagination created by means of facts and fiction, represents a transfer to determination and indetermination, and means an extension to an open form. In other words, hyperspace is a high dimensional space expanded to imagination through the combination of the viewpoint on facts in this dimension and fiction. When the 2D plane surface or 3D symmetry is destroyed, or when the frame is twisted or entangled, the non-Euclidean geometry is created eventually. And when the twisting leads to transmutation and the destruction of the form reaches the extreme; this in turn became the twisting like Mbius band. Likewise, the non-Euclidean geometry is co-related to the asymmetry of the Higgs mechanism. When the 'destruction of symmetry' is considered, symmetric theory and asymmetric world can be connected. The asymmetry in turn can maintain balance by arranging the uneven weights at different distances from the shaft. Moreover, at this the concept of the upper, lower, left and right, which was included in the original form, may be crumbled down. The destruction of the symmetry is essential in order to present forecast that coincides with the phenomenon of the real world. Non-Euclidean geometry characteristic is expressed by asymmetry, twists, and deconstruction and its representative characteristic is ambiguity. The boundary between the front, back, upper, lower, inner and outer is unclear, and it is difficult and vague to pinpoint specific location. The design that does not clearly define or determine the direction of wearing costume is indeed the non-oriented design that can be worn without getting restricted by specific direction such as front and back. Non-Euclidean geometry characteristic of hyperspace have been applied to create new shapes through the modification of the substance from traditional clothing of the eastern world to modern fashion. The way of thinking in the 'hyperspace' that used to be expressed in the costumes of the east and the west in the past became the forum for unlimited creation.

• 한국수학교육학회지시리즈A:수학교육
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• 제4권1호
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• pp.1-15
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• 1966

In accordance with the tendency of Modern Mathematics laying emphasis on Mathematical structure, that is, on axioms, it is necessary for students to be interested in structure of Geometry on Mathematics Education. In fact, it is of importance not only to obtain new ideas but also to forget old ones in the development of Mathematics. Most students do not understand the Mathematical significance of axioms, and do not know what Mathemetical truth is. Now Non-Euclidean Geometry offers opportunity to understand the essence of Mathematics better, and is no less effective than Euclidean Geometry in training student in logical inference. This thesis is a study with regard to what should be taught and how student should be guided at High school Mathematics. Chiefly Hyperbolic Geometry is discussed in connection with Abosolute Geometry. As Non-Euclidean Geometry has not appeared in our curriculum, some experiments are required before putting it into actual curriculum to find out how much students understand and how much pedagogically useful it can be. This is only a. presentation of a tentative plan, which needs to be criticized by many teachers.

• East Asian mathematical journal
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• 제39권4호
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• pp.479-492
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• 2023

Although Euclidean plane geometry is implemented in the middle school course, there are three major problems in high school space geometry that can be intuitively taken for granted or misinterpreted as circular arguments. In order to solve this problem, this study proved three major problems using sets, Euclidean plane geometry, and parallel line postulates. This corresponds to a logical sequence and has mathematical and mathematical educational values. Furthermore, it will be possible to configure spatial geometry using sets, and by giving legitimacy to non-Euclidean spatial geometry, it will open the possibility of future research.

• 한국수학사학회지
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• 제33권3호
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• pp.155-165
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• 2020

Pythagorean theorem is a proposition on the relationship between the lengths of three sides of a right triangle. It is well known that Pythagorean theorem for Euclidean geometry deforms into an interesting form in non-Euclidean geometry. In this paper, we investigate a new perspective that replaces right triangles with 'proper triangles' so that Pythagorean theorem extends to non-Euclidean geometries without any modification. This is seen from the perspective that a rectangle is an equiangular quadrilateral, and a right triangle is a half of a rectangle. Surprisingly, a proper triangle (defined by Paolo Maraner), which is a half of an equiangular quadrilateral, satisfies Pythagorean theorem in many geometries, including hyperbolic geometry and spherical geometry.

• 호남수학학술지
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• 제34권1호
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• pp.103-111
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• 2012

Wetzel[5] proved if ${\Gamma}$ is a closed curve of length L in $E^n$, then ${\Gamma}$ lies in some ball of radius [L/4]. In this paper, we generalize Wetzel's result to the non-Euclidean plane with much stronger version. That is to develop a lower bound of length of a triangle inscribed in a circle in non-Euclidean plane in terms of a chord of the circle.

• Journal of applied mathematics & informatics
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• 제31권1_2호
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• pp.79-86
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• 2013

Taxicab plane geometry and Cinese-Checker plane geometry are non-Euclidean and more practical notion than Euclidean geometry in the real world. The ${\alpha}$-distance is a generalization of the Taxicab distance and Chinese-Checker distance. It was first introduced by Songlin Tian in 2005, and generalized to n-dimensional space by Ozcan Gelisgen in 2006. In this paper, we studied the isoperimetric inequality in ${\alpha}$-plane.

• 한국실내디자인학회논문집
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• 제19권2호
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• pp.90-98
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• 2010

The concepts of laws like regularity or persistence or recurrence those are discovered in nature, became the essential elements in speculative philosophy, study and scientific technology. Western civilization was spread out by these natural laws. As this background, this study is aimed to research the theories of natural laws and the development of geometry as the descriptive tools and the development aspects of the concepts of space. According to Whitehead's four theories on the natural laws, the result of this study that aimed like that as follows. First, the theories on the immanence and imposition of the natural laws were the predominant ideas from ancient Greek to before the scientific revolution, the theory on the simple description like the positivism made the Newton-Cartesian mechanism and an absolutist world view. The theory on the conventional interpretation made the organicism and relativism world view according to non-Euclidean geometry. Second, the geometrical composition of ancient Greek architecture was an aesthetics that represented the immanence of natural laws. Third, in the basic symbol of medieval times, the numeral symbol was the frame of thought and was an important principal of architecture. Fourth, during the Renaissance, architecture was regarded as mathematics that made the order of universe to visible things and the geometry was regarded as an important architectural principal. Fifth, according to the non-Euclidean geometry, it was possible to present the natural phenomena and the universe. Sixth, topology made to lapse the division of traditional floor, wall and ceiling in contemporary architecture and made to build the continuous space. Seventy, the new nature was explained by fractal concepts not by Euclidean shapes, fractal presented that the essence of nature had not mechanical and linear characteristic but organic and non-linear characteristic.