• Journal of the Korean School Mathematics Society
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• v.12 no.4
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• pp.365-388
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• 2009

In this paper we solve various triangle construction problems given by three points(a midpoint of side and other two points). We investigate relation between these construction problems, draw out a base problem, and make hierarchy of solved construction problems. In detail we describe analysis for searching solving method, and construction procedure of required triangle.

• The Mathematical Education
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• v.48 no.3
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• pp.287-302
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• 2009

This study investigated how 5th grade students found and understood triangle congruence conditions (SSS, SAS, ASA). In particular, this study focused on children's processes of discovering triangle congruence conditions and the obstacles which they encountered in the process of making sense of these conditions. Our data indicates that inquiring the cases in which less than three factors of triangle are given is helpful for children to guess triangle congruence conditions and understand the minimal characteristic of these conditions. And the degree of difficulty of discovering each congruence condition is different. Children discovered SAS condition and ASA condition easily, but it was hard for them to discover and understand that SSS was also a triangle congruence condition because they connected the length of a given side with the use of a scaled ruler not a compass.

• Journal of the Korean Society of Clothing and Textiles
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• v.47 no.2
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• pp.392-408
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• 2023

This study concerns the flower-shaped silver crown ornament and inverted triangle-shaped crown ornament of Baekje, which were worn frequently during the Sabi period. The purpose of this study is to present a new estimated shape of the crown and ornaments. Individual excavation cases and archaeological data were analyzed. The flower-shaped silver crown ornament appears as a thin silver plate with buds on the center and side branches and is symmetrically bent from the center to form a ∧ shape. The inverted triangle-shaped crown ornament resembles two right-angle triangles that are back-to-back. The crown to which the two ornaments were added appears to be a triangular crown that was made by covering birch bark of with fabric. Both ornaments were believed to have been located on the front of crown, but that is incorrect. The flower-shaped silver crown ornament was inserted on the front of the crown, and the inverted triangle-shaped crown ornament was fixed with a tip at the top of the crown and then obliquely on the crown's side. The inferred design was confirmed with real reproductions. This study is significant in that it identifies the crown of Baekje during the Sabi period.

• School Mathematics
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• v.18 no.2
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• pp.425-440
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• 2016

Concept images play an important role in the acquisition of mathematical concept. However, sometimes concept images are derivatives of student's misconceptions. In addition, students always learn concept images despite teachers' efforts to teach concept definitions. Therefore, teachers need to know about all the concept images of a particular concept. This study aimed to analyze the concept image that students have about the triangle when they have already learned about the triangle in school. It was found that some students have different concept images about the triangle between Semo. Moreover, many students have misconceptions about vertices, sides, and angles. In particular, students think Gak denotes a side, although it means angle. Based on these results, I suggest that the curriculum and textbook require improvement.

• Journal of the Korean Society of Costume
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• v.63 no.8
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• pp.76-89
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• 2013

This study is about 'asymmetry triangle-Mu' Jeoksam and Hansam in the early days of Joseon Dynasty. A study was done regarding the records of Jeoksam and Hansam in literature, the present state of the excavated 'asymmetry triangle-Mu' clothing Jeoksam and Hansam, and finally a deduction of the reason for the appearance of the 'asymmetry triangle-Mu' clothing Jeoksam and Hansam. The width of front length of 'asymmetry triangle-Mu' clothing in the early days of Joseon Dynasty is 29.5~35 cm and the width of one breath of the sleeve is 29.5~35 cm. The width of 'asymmetry triangle-Mu' is 9.5~16 cm and it is relatively big. Comparing to the width of one breath of the sleeve, it is almost 1:2.2~3.6 ratio. Therefore, when the sleeve was cut, the Mu was linked in order to save fabric the gusset of sleeve had to be folded and turned, and finally it became asymmetric. As a result of the above consideration, since the width of upper garments of $16{\sim}17^{th}$ century was big, the wearing of short tops of Jeoksam or Hansam without side vent as a small 'triangle-Mu' was uncomfortable. Because of this reason, the size had no option but to become bigger. So, during the $16^{th}$ and $17^{th}$ century, a period where mass production of fabric was difficult, the 'asymmetry triangle-Mu' type was considered to be a reasonable cutting method. After the middle of $17^{th}$ century, it can be estimated that 'asymmetry triangle-Mu' clothing disappeared according to the narrow aspect of clothing type.

• Journal of Fashion Business
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• v.20 no.6
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• pp.79-93
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• 2016

Since the development of patterns using tessellation is applied to a wide range of fields such as clothing, architecture, environment, and products, etc. and its expression principle is also found in various fields such as mathematics and science, etc. However, this pattern is mostly used as a math material with little studies on fashion and culture. In addition, it is thought that Korean traditional culture products need more various and modern design development methods and pattern through preliminary investigation which is simple copy of traditional items, simple copy of Korean Alphabet, Chinese character, and folk paintings. Therefore, it will present the method to make more design cases using Tessellation. Tessellation that combines mathematics and art will be the infinite form of designing of designers as well as creative training way to understand the composition principles of old culture and to raise sense of modern design. Tessellation of regular triangle, regular square, and regular hexagon was performed on the patterns which have meaning of wealth and prosperity of Korean traditional patterns. As the concrete method, first, each side of the regular triangle is developed symmetrically with patterns of fish, turtle, and cicadas. Second, rotational movement after symmetry movement about middle point of one side ${\times}$ 1 symmetry movement about middle point ${\times}$ 1 using crane and cloud, of the regular triangle was performed. Third, the regular square was tessellated parallel movement ${\times}$ 2 with "Da(multi)" and dragon pattern as the source image. Fourth, the sitting tiger was tessellated with symmetry movement about middle point ${\times}$ 2 and parallel movement ${\times}$ 1. Fifth, three bat patterns are tessellated by again rotational movement of two sides after rotational movement of one side and rotational movement of the other side. In addition, It developed traditional culture product design of the scarf, umbrella, aprons, neckties.

• Journal for History of Mathematics
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• v.32 no.4
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• pp.159-173
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• 2019

In this paper, we examine the brief history of the ring of four almonds regarding Mesopotamian mathematics, and present reasons why the Omar Khayyam's triangle, a special right triangle in a ring of four almonds, was essential for artisans due to its unique pattern. We presume that the ring of four almonds originated from a point symmetry figure given two concentric squares used in the proto-Sumerian Jemdet Nasr period (approximately 3000 B.C.) and a square halfway between two given concentric squares used during the time of the Old Akkadian period (2340-2200 B.C.) and the Old Babylonian age (2000-1600 B.C.). Artisans tried to create a new intricate pattern as almonds and 6-pointed stars by subdividing right triangles in the pattern of the popular altered Old Akkadian square band at the time. Therefore, artisans needed the Omar Khayyam's triangle, whose hypotenuse equals the sum of the short side and the perpendicular to the hypotenuse. We presume that artisans asked mathematicians how to construct the Omar Khayyam's triangle at a meeting between artisans and mathematicians in Isfahan. The construction of Omar Khayyam's triangle requires solving an irreducible cubic polynomial. Omar Khayyam was the first to classify equations of integer polynomials of degree up to three and then proceeded to solve all types of cubic equations by means of intersections of conic sections. Omar Khayyam's triangle gave practical meaning to the type of cubic equation $x^3+bx=cx^2+a$. The work of Omar Khayyam was completed by Descartes in the 17th century.

• The Mathematical Education
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• v.41 no.2
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• pp.227-231
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• 2002

The Pythagorean theorem and Pythagorean triple are well known. We know some Pythagorean triples, however we don't Cow well that every natural number can belong to some Pythagorean triple. In this paper, we show that every natural number, which is not less than 2, can be a length of a leg(a side opposite the acute angle in a right triangle) in some right triangle, and list some Pythagorean triples.

• East Asian mathematical journal
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• v.31 no.4
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• pp.459-481
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• 2015

In this study, we investigated whether the theorem is established even if we replace a 'square' element in the Euclidean proof of the Pythagorean theorem with different figures. At this time, we used different figures as equilateral, isosceles triangle, (mutant) a right triangle, a rectangle, a parallelogram, and any similar figures. Pythagorean theorem implies a relationship between the three sides of a right triangle. However, the procedure of Euclidean proof is discussed in relation between the areas of the square, which each edge is the length of each side of a right triangle. In this study, according to the attached figures, we found that the Pythagorean theorem appears in the following three cases, that is, the relationship between the sides, the relationship between the areas, and one case that do not appear in the previous two cases directly. In addition, we recognized the efficiency of Euclidean proof attached the square. This proving activity requires a mathematical process, and a generalization of this process is a good material that can experience the diversity and rigor at the same time.

• Journal of the Korean Society of Costume
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• v.59 no.3
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• pp.131-144
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• 2009

This study is about single-layered $Chog{\breve{a}}ri$ excavated from SongHyosang(宋效商, 1430-1490, SHS hereafter), SongHeeJong(宋喜從, the late 1500s, SHJ hereafter) tombs. There are 7 single-layered $Chog{\breve{a}}ris$ for men. We focus on comparison of their design and sowing method. 1. Design: Investigating collar, $Chog{\breve{a}}ris$ from SHS have MokpanGit and $Chog{\breve{a}}ris$ from SHJ have KalGit. KalGit has been seen from SHJ to 17th, 18th and 19th centuries. After the late 1500s, there is no MokpanGit single-laTered $Chog{\breve{a}}ri$ for men. Side panel under arm has various shapes(triangle, trapezoid, triangle+trapezoid) in 15th century. After the late 1500s, It changes into no side panel. Two $Chog{\breve{a}}ris$ with no side panel from SHJ reveals that the late 1500s is a period of transition. 2. Sewing method: First, researching lengthwise grainline of the fly, the left fly has lengthwise grainline outside In four, inside in three. The right fly has lengthwise grainline outside in just one, the others have lengthwise grainline inside. Compared with today's way, there is a great difference, but in those times there isn't an established rule. This is true of side panel under arm. The sewing method are backstitch, running stitch, and hemming. Researching the construction method of seam, in putting two selvages together, open seam and plain seam are used. In putting selvage and bias, bias and bias together, flat felled seam and french seam are used. This study shows that single-layered $Chog{\breve{a}}ri$ far men from 15C. to 16C. has changes of design such as collar(Git) and side panel undo. arm. But there is little change in sewing method.