• 한국수학교육학회지시리즈A:수학교육
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• 제50권2호
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• pp.233-246
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• 2011

The geometry field in the current high school curriculum deals mainly with analytic geometry and the reference to logic geometry leaves much to be desired. This study investigated the construction on a heptagon by using conic sections as one of measures for achieving harmony between analytic geometry and logic geometry in the high school curriculum with the Geometer's Sketchpad(GSP), which is a specialized software prevalent in mathematics education field and is intended to draw an educational suggestion on it.

• 한국수학사학회지
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• 제22권2호
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• pp.99-116
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• 2009

본 연구에서는 18세기 출판된 '자와 컴퍼스의 방법'에 제시된 정다각형의 작도방법들 중에서 오류를 포함하는 작도를 분석하였다. 정7각형과 정9각형은 자와 컴퍼스를 이용하여 작도불가능 하다는 것이 알려져 있지만, '자와 컴퍼스의 방법'에서는 이들 정다각형을 작도하는 두 가지 방법이 제시되어 있다. 본 연구에서는 이들 작도가 오류를 포함하고 있음을 보였고, 이에 관련된 몇몇 정다각형 작도 방법도 오류를 포함하고 있음을 보였다. 이를 통해 정다각형 작도문제의 해결을 위한 노력에서 성공적이지 못한 시도에 관련된 새로운 자료를 제공할 것으로 기대된다.

• 한국진공학회:학술대회논문집
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• 한국진공학회 2009년도 제38회 동계학술대회 초록집
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• pp.340-340
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• 2010

Various structures of vacancy defects in graphene layers and carbon nanotubes have been reported by high resolution transmission electron microscope (HR-TEM) and those arouse an interest of reconstruction processes of vacancy defects. In this talk, we present reconstruction processes of vacancy defects in a graphene and a carbon nanotube by tight-binding molecular dynamics (TBMD) simulations and by first principles total energy calculations. We found that a structure of a dislocation defect with two pentagon-heptagon (5-7) pairs in graphene becomes more stable than other structures when the number of vacancy units is ten and over. The simulation study of scanning tunneling microscopy reveals that the pentagon-heptagon pair defects perturb the wavefunction of electrons near Fermi level to produce the $\sqrt{3}\;{\times}\;\sqrt{3}$ superlattice pattern, which is in excellent agreement with experiment. It is also observed in our tight-binding molecular dynamics simulation that 5-7 pair defects play a very important role in vacancy reconstruction in a graphene layer and carbon nanotubes.

• 대한수학회지
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• 제48권2호
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• pp.367-382
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• 2011

We establish a necessary and sufficient condition for a heptagonal knot to be figure-8 knot. The condition is described by a set of Radon partitions formed by vertices of the heptagon. In addition we relate this result to the number of nontrivial heptagonal knots in linear embeddings of the complete graph $K_7$ into $\mathbb{R}^3$.

• 한국진공학회:학술대회논문집
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• 한국진공학회 2011년도 제40회 동계학술대회 초록집
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• pp.428-429
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• 2011

Recently, we performed ab initio total energy calculation and tight-binding molecular dynamics (TBMD) simulation to study structures and the reconstruction of native defects in graphene. In the previous study, we predicted by TBMD simulation that a double vacancy in graphene is reconstructed into a 555-777 composed of triple pentagons and triple heptagons [1]. The structural change from pentagon-octagon-pentagon (5-8-5) to 555-777 has been confirmed by recent experiments [2,3] and the detail of the reconstruction process is carefully studied by ab initio calculation. Pentagon-heptagon (5-7) pairs are also found to play an important role in the reconstruction of vacancy in graphene and single wall carbon nanotube [4]. In the TBMD simulation of graphene nanoribbon (GNR), we found the evaporation of carbon atoms from both the zigzag and armchair edges is preceded by the formation of heptagon rings, which serve as a gateway for carbon atoms to escape. In the simulation for a GNR armchair-zigzag-armchair junction, carbon atoms are evaporated row-by-row from the outermost row of the zigzag edge [5], which is in excellent agreement with recent experiments [2, 6]. We also present the recent results on the formation and development of dislocation in graphene. It is found that the coalescence of 5-7 pairs with vacancy defects develops dislocation in graphene and induces the separation of two 5-7 pairs. Our TBMD simulations also show that adatoms are ejected and evaporated from graphene surface due to large strain around 5-7 pairs. It is observed that an adatom wanders on the graphene surface and helps non-hexagonal rings change into stable hexagonal rings before its evaporation.

• 반도체디스플레이기술학회지
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• 제3권3호
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• pp.23-25
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• 2004

The defect formation energy of boron nitride (BN) nanotubes is investigated using molecular-dynamics simulation. Although the defect with tetragon-octagon pairs (4-88-4) is favored in the flat cap of BN nanotubes, BN clusters, and the growth of BN nanotubes, the formation energy of the 4-88-4 defect is significantly higher than that of the pentagon-heptagon pairs (5-77-5) defect in BN nanotubes. The 5-77-5 defect reduces the effect of the structural distortion caused by the 4-88-4 defect, in spite of homoelemental bonds.

• 한국반도체및디스플레이장비학회:학술대회논문집
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• 한국반도체및디스플레이장비학회 2004년도 춘계학술대회 발표 논문집
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• pp.59-62
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• 2004

The defect formation energy of boron nitride (BN) nanotubes is investigated using molecular-dynamics simulation. Although the defect with tetragon-octagon pairs (4-88-4) is favored in the flat cap of BN nanotubes, BN clusters, and the growth of BN nanotubes, the formation energy of the 4-88-4 defect is significantly higher than that of the pentagon-heptagon pairs (5-77-5) defect in BN nanotubes. The 5-77-5 defect reduces the effect of the structural distortion caused by the 4-88-4 defect, in spite of homoelemental bonds. The instability of the 4-88-4 defect generates the structural transformation into BNNTs with no defect at about 1500 K.

• 한국진공학회:학술대회논문집
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• 한국진공학회 2010년도 제39회 하계학술대회 초록집
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• pp.54-55
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• 2010

Vacancy defects in graphene can be created by electron or ion irradiation and those induce ripples which can change the electronic properties of graphene. Recently, the formation of defect structures such as vacancy defects and non-hexagonal rings has been reported in the high resolution transmission electron microscope (HR-TEM) of reduced graphene oxide [1]. In those HR-TEM images, it is noticed that the dislocations with pentagon-heptagon (5-7) pairs are formed and diffuses. Interestingly, it is also observed that two 5-7 pairs are separated and diffuse far away from each other. The separation of 5-7 pairs has been known to be due to their self-diffusion. However, from our tight-binding molecular dynamics simulation, it is found that the separation of 5-7 pairs is due to the diffusion of single vacancy defects and coalescence with 5-7 pairs. The diffusion and coalescence of single vacancy defects is too fast to be observed even in HR-TEM. We also implemented Van der Waals interaction in our tight-binding carbon model to describe correctly bi-layer and multi-layer graphene. The compressibility of graphite along c-axis in our tight-binding calculation is found to be in excellent agreement with experiment. We also discuss the difference between single layer and bi-layer graphene about vacancy diffusion and reconstruction.