• Journal of the Korean Society of Fashion and Beauty
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• v.4 no.1 s.7
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• pp.28-34
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• 2006

The purpose of this study is to historically examine the thoughts and ideas of geometry and to analyze the expression style of design applied to the mass communication such as magazines and world wide webs, by giving definitions on the ideas of geometry and the pattern of cognition. Geometry was evolved to Descartes's analytical geometry, projective geometry, non-Euclidean geometry and Topology at the end of 19th century. When geometry applies to design styles, it is devided into two field, plane geometry and solid geometry. The development of geometry was completed from the Pythagoras symbolic theory of number to Platonic spiritual geometry and Euclidean geometry. It can be studied that those have what kind of symbolic meanings and transformations on each hair design plan. It can also analized how those symbolic forms are appeared on the design form. This tendency means that there is always a try for the use of geometry as reasonable device for hair design. If the hair design and geometry have logical and artistical relation, we can make buildings which have a order, balance and harmony.

• Journal for History of Mathematics
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• v.32 no.5
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• pp.241-255
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• 2019

The proposition that the parallel axiom and the Pythagorean theorem are equivalent in the Hilbert geometry is true when the Archimedean axiom is assumed. In this article, we examine some specific plane geometries to see the existence of the non-archimidean Hilbert geometry in which the Pythagorean theorem holds but the parallel axiom does not. Furthermore we observe that the Pythagorean theorem is equivalent to the fact that the Hilbert geometry is actually a semi-Euclidean geometry.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.849-866
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• 2024

In this paper, we study the rectifying curves in multiplicative Euclidean space of dimension 3, i.e., those curves for which the position vector always lies in its rectifying plane. Since the definition of rectifying curve is affine and not metric, we are directly able to perform multiplicative differential-geometric concepts to investigate such curves. By several characterizations, we completely classify the multiplicative rectifying curves by means of the multiplicative spherical curves.

• Korean Institute of Interior Design Journal
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• v.19 no.5
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• pp.83-94
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• 2010

Conventionally, ornament has developed around linear thinking based on Euclidean geometry, and been explained as simple and lucid natural Euclidean geometrical phenomena. The modular arrangement with vertical, horizontal and diagonal grids has been an organizing principle of classical ornament, but in digital era ornament is found not to be explained only with the principle of traditional arrangement due to the seemingly irregular complex forms. In that sense, this study presents the concept of digital ornament and examined the backgrounds of ornament in digital age, that are complex system and non-Euclidean geometry. Accordingly, the present study takes an approach by dividing new formal types of ornament into algorithmic form, hybrid form and dynamic form to find out a principle of pattern organization. Lately, architects who actively use computer for their architectural designs take the algorithmic strategies in nature and create various and complex patterns by simple rules. The patterns are not the repetition of the same, but the production of singularities. In addition, hybrid form by morphing shows a topologically flexible evolutionary transformation, and is used to create in-between transitional shapes from the source to target. Finally, the patterns by the interaction between the system components which are corresponded to the embedded forces emerge from dynamic simulation of the natural environment. Rather than objects itself, focus is given to the process of generating forms, and the ornamental patterns as the revelation of such implicit order provide not just the formal beauty but also spatial pathways for lights and air, maximizing the effects of lights.

• Journal of Communications and Networks
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• v.12 no.5
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• pp.406-410
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• 2010

This paper presents an approach to the construction of non-binary quasi-cyclic (QC) low-density parity-check (LDPC) codes based on multiplicative groups over one Galois field GF(q) and Euclidean geometries over another Galois field GF($2^S$). Codes of this class are shown to be regular with girth $6{\leq}g{\leq}18$ and have low densities. Finally, simulation results show that the proposed codes perform very wel with the iterative decoding.

• Transactions on Control, Automation and Systems Engineering
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• v.4 no.4
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• pp.330-340
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• 2002

In this Paper. a non-linear approach to a design of model reference adaptive control is presented. The approach is demonstrated by a case study of a simple single-pole and no zero, linear, discrete-time plant. The essence of the idea is to generate a full non-linear model of the plant dynamics and the parameter adaptation dynamics as a gradient descent algorithm with respect to a Riemannian metric. It is shown how a Riemannian metric can be chosen so that the modelled plant dynamics do in fact match the true plant dynamics. The performance of the proposed scheme is compared to a traditional model reference adaptive control scheme using the classical sensitivity derivatives (Euclidean gradients) for the descent algorithm.

• Journal for History of Mathematics
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• v.15 no.3
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• pp.35-48
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• 2002

In this paper, we explore development of mathematical knowledge, especially calculus, non-Euclidean geometry, Euler's theorem, and the comparison of the number of elements in two infinite sets. And we analyze kinds of errors and the roles for errors with respect to increasing knowledge in mathematics.

• School Mathematics
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• v.15 no.4
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• pp.957-973
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• 2013

This study analyzed Secondary Mathematically Gifted' mathematical thinking processes demonstrated from the activities. They configured regular triangle tessellations in the Non-Euclidean hyperbolic disk model. The students constructed the figure and transformation to construct the tessellation in the poincare disk. gsp file which is the dynamic geometric environmen, The students were to explore the characteristics of the hyperbolic segments, construct an equilateral triangle and inversion. In this process, a variety of strategic thinking process appeared and they recognized to the Non-Euclidean geometric system.

• Journal of Educational Research in Mathematics
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• v.23 no.1
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• pp.53-74
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• 2013

The purpose of this paper is to analyze how mathematically gifted middle school students find out the necessary and sufficient condition for a certain hyperbolic line to be parallel to a given hyperbolic line in Non-Euclidean disc model (Poincar$\acute{e}$ disc model) using the Geometer's Sketchpad. We also investigated their characteristic of mathematical thinking and analyze how they express what they had observed while they did mental experiments in the Poincar$\acute{e}$ disc using computer-aided construction tools, measurement tools and inductive reasoning.

• Communications of Mathematical Education
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• v.24 no.3
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• pp.659-689
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• 2010

Taxicab distance function is a practical distance notion which gives us information of real world pathway distance that really taxi can go through. As one of the non-Euclidean geometry, this study of an ideal city with all roads running horizontal or vertical, was introduced by the Russian Mathematician H. Minkowski and synthetically reported by the E. F. Kraus in 1986. After that, there were many reports and papers on this topic and still being researched. At this point of view, our research about taxicab geometry provides its differences from Euclidean plane geometry, and considers about several theorems on plane geometry using the taxicab distance function.