• Communications of Mathematical Education
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• v.36 no.1
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• pp.23-38
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• 2022

The study provided a perspective on which readers can see Newton's proof heuristically in order to overcome the difficulty of proof showing 'QT²/QR converges to the latus rectum of ellipse' in the proof of the inverse square law of Newton's Principia. The heuristic perspective is as follows: The starting point of the proof is the belief that if we transform the denominators and numerators of QT²/QR into expression with respect to segments related to diameter and conjugate diameter, we may obtain some constant, the desired value, by their relationship PV × VG/QV² = PC²/CD² in Apollonius' Conic sections. The heuristic perspective proposed in this study is meaningful because it can help readers understand Newton's proof more easily by presenting the direction of transformation of QT²/QR.

• School Mathematics
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• v.18 no.3
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• pp.589-609
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• 2016

In this study, researchers have modernly reinterpreted geometric solving of cubic equations presented by an arabic mathematician, Omar Khayyam in medieval age, and have considered the pedagogical significance of geometric solving of the cubic equations using two conic sections in terms of analytic geometry. These efforts allow to analyze educational application of mathematics instruction and provide useful pedagogical implications in school mathematics such as 'connecting algebra-geometry', 'induction-generalization' and 'connecting analogous problems via analogy' for the geometric approaches of cubic equations: $x^3+4x=32$, $x^3+ax=b$, $x^3=4x+32$ and $x^3=ax+b$. It could be possible to reciprocally convert between algebraic representations of cubic equations and geometric representations of conic sections, while geometrically approaching the cubic equations from a perspective of connecting algebra and geometry. Also, it could be treated how to generalize solution of cubic equation containing variables from geometric solution in which coefficients and constant terms are given under a perspective of induction-generalization. Finally, it could enable to provide students with some opportunities to adapt similar solving procedures or methods into the newly-given cubic equation with a perspective of connecting analogous problems via analogy.

• Journal of Educational Research in Mathematics
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• v.26 no.4
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• pp.715-730
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• 2016

The purpose of this study is to analyze Descartes's point of view on the mathematical connection of algebra and geometry which help comprehend the traditional frame with a new perspective in order to access to unsolved problems and provide useful pedagogical implications in school mathematics. To achieve the goal, researchers have historically reviewed the fundamental principle and development method's feature of analytic geometry, which stands on the basis of mathematical connection between algebra and geometry. In addition we have considered the significance of geometric solving of equations in terms of analytic geometry by analyzing related preceding researches and modern trends of mathematics education curriculum. These efforts could allow us to have discussed on some opportunities to get insight about mathematical connection of algebra and geometry via geometric approaches for solving equations using the intersection of curves represented on coordinates plane. Furthermore, we could finally provide the method and its pedagogical implications for interpreting geometric approaches to cubic equations utilizing intersection of conic sections in the process of inquiring, solving and reflecting stages.

• The Journal of Korean Academy of Prosthodontics
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• v.28 no.1
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• pp.95-122
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• 1990

This study was done to analyze the occlusal curve as one of the factors to be considered for maintenance of occlusal stability in the orthodontic and prosthodontic treatments. Sixty gnathological casts we.e obtained from 43 subjects with normal occlusion and 17 subjects with some of temporomandibular disorders. The occlusal surfaces of gnathologic casts were duplicated by using a Color kit SK-700 and tile reference points of X, Y coordinates were digitized by using the Summagraphic digitizer and 18AT computer system. The Z coordinates of cusp height were measured by 0.01mm measurable caliper. The mathematical computer program of least square method was used to analyze the occlusal curve arranged by three dimensional coordinates of X, Y, Z. The following results were obtained : 1. The occlusal curve of buccal and lingual cusp tips was fitted to the ellipse, and the occlusal curve of anterior teeth was fitted to a part of the circle in the analysis of conic sections. 2. The radius of Spee's curve showed individual differences, but was average 98.7mm in male subjects and 93.7mm in female subjects. 3. The radius fo Spee's curve according to the half of canine width showed the least coefficient of variation. 4. The radius of Spee's curve was not significantly relative to the lateral occlusal contacts on laterotrusion and the absence or presence of temporomandibular disorders. 5. The radius of Wilson's curve showed individual difference and the size of radius was followed by the order of 1st premolar, 2nd premolar, 2nd molar and 1st molar.

• 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.