초등수학 영재교육원 학생들의 프랙탈 구성 방법 분석

김상미 대한수학교육학회 2009 수학교육학연구 Vol.19 No.2

The purpose of this study is to show the Fractals activities for mathematically gifted students, and to analyze the constructions on Fractals of the mathematically gifted. The subjects of this study were 5 mathematically gifted students in the Gifted Education Institut and also 6th graders at elementary schools. These activities on Fractals focused on constructing Fractals with the students' rules and were performed three ways; Fractal cards, colouring rules, Fractal curves. Analysis of collected data revealed in as follows: First, the constructions on Fractals transformed the ratios of lines and were changed using oblique lines or curves. Second, to make colouring rules on Fractals, students presented the sensitivities of initial and fractal dimensions on Fractals. In conclusion, this study suggested the importance of communication and mathe- matical approaches in the mathematics classrooms for the mathematically gifted. 본 논문은 영재교육원 초등수학분야 6학년 수업 시간에 진행되었던 프랙탈 활동을 중심으로 그에 따른 학생들이 보여준 프랙탈 구성 방법에 대한 사례연구이다. 첫째로, 프랙탈 카드, 색칠 규칙, 점종이를 활용하여 프랙탈 구성 규칙 만들기, 자연물을 프랙탈 구성 규칙으로 표현하기의 구성 활동 과정을 밝히고, 둘째로 초등수학영재원의 5명 학생이 보여준 변형 과정을 분석하였다. 학생들은 프랙탈 구성 과정에서 기본 규칙의 높은 단계로 반복하기보다는 다른 비율, 사선과 곡선을 도입하여 변형 규칙으로 새로운 프랙탈 상을 얻으려고 하였다. 또한 프랙탈 상을 구현하는 것만이 아니라, 프랙탈의 특성인 ‘초기값 민감성’과 ‘소수 차원’을 제기하여 수학적으로 밝히고자 하였다. 끝으로 영재 수업과 그에 따른 학생들의 학습 과정에서 제기된 프랙탈 구성 방법을 논의하고, 더불어 영재 수업에서 수학적 의사소통의 중요성과 학생들의 수학적인 접근에 대하여 제언하였다.

New Elements Concentrated Planar Fractal Antenna Arrays for Celestial Surveillance and Wireless Communications

Ahmed Najah Jabbar 한국전자통신연구원 2011 ETRI Journal Vol.33 No.6

This research introduces three new fractal array configurations that have superior performance over the well-known Sierpinski fractal array. These arrays are based on the fractal shapes Dragon, Twig, and a new shape which will be called Flap fractal. Their superiority comes from the low side lobe level and/or the wide angle between the main lobe and the side lobes, which improves the signal-to-intersymbol interference and signal-to-noise ratio. Their performance is compared to the known array configurations: uniform, random, and Sierpinski fractal arrays.

B.I.G 건축에 나타난 프랙털 기하학의 표현 특성연구

김효원 ( Kim Hyowon ),윤재은 ( Yoon Jaeeun ) 한국공간디자인학회 2020 한국공간디자인학회논문집 Vol.15 No.8

(Background and Purpose) In the early 1900s, architecture and design were both functional and rational, unlike before that time, and activities were different as well. The influence of learning and the idea of the times have changed as the external forms within them have been shared, and after modernism and post-modernism, they reached modern deconstruction. Increasingly complex social phenomena and architectural forms have started resembling fractals. As architecture denotes the space of the main active environment utilized by human beings, it can describe our past, present, and future because it simultaneously reflects the user behavior, and the current architecture includes our appearance or the direction we are currently pursuing. Among many modern architectural offices, the Bjark Ingels Group (B.I.G.) has emerged as a group that reveals its philosophy in uniquely shaped architectural works. Bjarke Ingels created his own office in 2005 and stood out by winning various architectural awards at a young age and in a short period of time. Moreover, he is drawing attention not only because of his bold design, but also because it is interesting in form and in harmony with the environment . The group aims for an architecture acceptable to both nature and human beings, always striving to reach that goal. Ingels’ architectural form goes beyond simple Euclidean geometry and captivates us with a unique look. Therefore, fractal geometry is easier to understand his architecture. This study examines the expressive characteristics of fractal geometry utilized in the architecture of B.I.G., one of the architects representing the present, to learn about the future direction of architecture should pursue. (Method) Through a literary review, the B.I.G Group and their philosophy, fractal geometry, and fractal geometry characteristics were explored, and the shaping characteristics of fractal geometry were analyzed to derive keywords. We selected and analyzed five B.I.G architectural works using derived keywords. (Results) In this study, eight keywords were extracted through morphological relationships, circulation, non-predictive, and de-scenching, which are architectural molding characteristics of the fractals. It was found that they could be constructed in a form-excellent way, while embracing both humans and nature. (Conclusions) B.I.G.'s architecture includes related, adaptive, evolutionary, and de-authoritative expressions, and such architecture that shows or uses fractal geometry derived from nature could be one of the ways in which we could get closer to it. Currently, a lot of research is underway to make three-dimensional models possible, as well as fractal geometry, and we hope to use them to build new and diverse architectural works that would merge with the environment.

영원철학(The Perennial Philosophy)으로 본 대순사상의 궁극적 실재

허훈 대진대학교 대순사상학술원 2019 대순사상논총 Vol.32 No.-

현대과학자들은 우주라는 복잡계(複雜界)에서 질서의 기본 단위 즉 프랙털(fractal)의 원리를 찾으려고 애쓰고 있다. 프랙털은 수학이나 물리학에서 주로 사용하는 용어이지만, 어떤 궁극적 실재가 다면적 양상을 나타내는 이유를 설명하는 원리로서 적합하다. 프랙털은 이미 과학계에서는 상용화된 원리로서 컴퓨터 그래픽 분야에 널리 응용된다. 본고에서는 프랙털의 원리를 활용하여 대순사상에서 궁극적 실재가 구현되는 양상을 밝힌다. 대순사상에는 도, 상제, 신(신명), 무극, 태극, 천지 등 다양한 궁극적 실재들이 등장하는데, 이들 개념은 서로 회통한다. 즉 궁극적 실재가 프랙털 원리에 의해 구현된다는 사실을 밝힘으로써 궁극적 실재들의 일치ㆍ회통은 현대과학에 의해 뒷받침 되고 있음을 밝힌다. 그러나 전(全)세계의 주류 종교들을 인격신교와 비(非)인격신교로 나누었을 때, 대부분의 종교들은 궁극적 실재를 초월적이며 인격적인 존재로 상정하고 있으며, 이들은 신과 인간의 관계를 프랙털[음양 프랙털, 홀론]의 관계로 상정할 수 없다. 또한 궁극적 실재를 내재적이며 비인격적인 존재로 상정하는 종교들도 홀론의 실현 정도-모든 부분과 전체의 되먹힘-에는 다소 차이를 보이고 있는데, 대순사상은 가장 직접적으로 신(신명)과 인간이 음양 프랙털의 관계임을 명시하고 있다. 즉 “신(신명)은 음(陰), 인간은 양(陽)”, “인간이 곧 신적(神的) 존재”라는 것이다. 나아가 대순사상에서는 이 궁극적 실재를 다양한 관점에서 여러 가지 개념으로 제시하고 있으며, 이들이 회통할 수 있음을 밝히고 있다. 이렇듯, 우주를 홀론(홀라키)으로 파악하는 관점은 영원철학의 핵심 요지(要旨)이기도 하다. 세계의 위대한 영적 스승들, 사상가들, 철학자들, 과학자들이 채택한 보편적인 종교관 즉 영원철학에 따르면 궁극적 실재는 서로 일치하며, 인간과 신은 서로 다르지 않다. 바꿔 말해 대순사상에 나타난 궁극적 실재론의 진리성은 현대 과학과 영원철학에 의해 뒷받침 된다. Modern scientists are trying to find the basic unit of order, fractal geometry, in the complex systems of the universe. Fractal is a term often used in mathematics or physics, it is appropriate as a principle to explain why some models of ultimate reality are represented as multifaceted. Fractals are already widely used in the field of computer graphics and as a commercial principle in the world of science. In this paper, using observations from fractal geometry, I present the embodiment of ultimate reality as understood in Daesoon Thought. There are various models of ultimate reality such as Dao (道, the way), Sangje (上帝, supreme god), Sinmyeong (神明, Gods), Mugeuk (無極, limitlessness), Taegeuk (太極, the Great Ultimate), and Cheonji (天地, heaven and earth) all of which exist in Daesoon Thought, and these concepts are mutually interrelated. In other words, by revealing the fact that ultimate reality is embodied within fractal geometry, it can be shown that concordance and transformation of various models of ultimate reality are supported by modern science. But when the major religions of the world were divided along lines of personality (personal gods) and non-personality (impersonal deities), most religions came to assume that ultimate reality was either transcendental or personal, and they could not postulate a relationship between God and humanity as Yin Yang (陰陽) fractals (Holon). In addition, religions, which assume ultimate reality as an intrinsic and impersonal being, are somewhat different in terms of their degree of Holon realization - all parts and whole restitution. Daesoon Thought most directly states that gods (deities) and human beings are in a relationship of Yin Yang fractals. In essence, “deities are Yin, and humanity is Yang” and furthermore, “human beings are divine beings.” Additionally, in the Daesoon Thought, these models of ultimate reality are presented through various concepts from various viewpoints, and they are revealed as mutually interrelated concepts. As such, point of view regarding the universe wherein Holarchy becomes a models in a key idea within perennial philosophy. According to a universalized view of religious phenomena, perennial philosophy was adopted by the world’s great spiritual teachers, thinkers, philosophers, and scientists. From this viewpoint, when ultimate reality coincides, human beings and God are no longer different. In other words, the veracity of the theory of ultimate reality that has appeared in Daesoon Thought can find support in both modern science and perennial philosophy.

Fractal and laboratory analyses of the crushing and abrasion of granular materials

Vallejo, Luis E.,Chik, Zamri Techno-Press 2009 Geomechanics & engineering Vol.1 No.4

Gravels forming part of the base of flexible pavements experience abrasion and crushing as a result of static and dynamic loads. Abrasion takes place when the sharp corners of the particles of gravel are removed as a result of compressive and shear loads. As a result of abrasion, the particles change in shape. Crushing is caused by the fragmentation of the particles into a mixture of many small particles of varying sizes. In this study, the abrasion and crushing of gravels are evaluated experimentally and analytically. The laboratory component of this study involves gravels that were subjected to abrasion and dynamic compression tests. The evaluation of the abrasion and crushing experienced by the gravel was carried out using fractals. In this study, the fractal dimension concept from fractal theory is used to evaluate: (a) the changes in shape, and (b) the crushing (fragmentation) of the original particles of gravel. It was determined that the fractal dimension of the profile of the particles decreased as a result of abrasion. With respect to crushing, the fragmentation fractal dimension was found to increase with the degree of breakage of the gravel. To understand the influence of crushing on the permeability of the gravels, the hydraulic conductivity of the gravels was measured before and after crushing. The hydraulic conductivity of the gravels was found to decrease with an increase in their level of crushing. Also, changes in the angle of friction of the granular materials as a result of abrasion was calculated using the Krumbein's roundness chart. The angle of friction of the granular materials was found to decrease as a result of abrasion.

Presentation-Oriented Key-Frames Coding Based on Fractals

Luigi Atzori,Daniele D. Giusto,Maurizio Murroni 한국전자통신연구원 2005 ETRI Journal Vol.27 No.6

This paper focuses on the problem of key-frames coding and proposes a new promising approach based on the use of fractals. The summary, made of a set of key-frames selected from a full-length video sequence, is coded by using a 3D fractal scheme. This allows the video presentation tool to expand the video sequence in a “natural” way by using the property of the fractals to reproduce the signal at several resolutions. This feature represents an important novelty of this work with respect to the alternative approaches, which mainly focus on the compression ratio without taking into account the presentation aspect of the video summary. In devising the coding scheme, we have taken care of the computational complexity inherent in fractal coding. Accordingly, the key-frames are first wavelet transformed, and the fractal coding is then applied to each subband to reduce the search range. Experimental results show the effectiveness of the proposed approach.

토성 고리의 프랙탈 차원

강은규,박성일,김상훈 한국물리학회 2008 새물리 Vol.57 No.6

Saturn's rings are composed of numerous inter-stellar particles, including ices, and have been considered to be examples of an astronomic fractal. Analysing a picture of Saturn's rings, we introduced a method to calculate the fractal dimensions of the rings. We showed that a cross-section of Saturn's rings could be a Cantor set with fractal dimension 0.935. Also, we calculated the fractal dimensions of the whole ring pattern which has the form of a Cantor target, as a function of the radius. As the radius was increased, the dimension increased sharply from 0 and oscillating abruptly, to 2 and finally stable. 토성의 고리는 얼음조각을 비롯하여 수많은 성간물질로 구성되어 있어 천체 프랙탈의 한 예로 볼 수 있다. 토성 고리의 사진을 분석하여 고리의 단면과 고리 전체의 프랙탈 차원을 구하고 그 결과를 분석하였다. 토성 고리는 단면의 차원이 0.935인 캔토어 집합으로 볼 수 있다. 반면에 토성 고리 전체는 캔토어 과녁으로 볼 수 있는데 반지름의 변화에 따른 프랙탈 차원의 변화를 구하였다. 고리 중심에서는 0차원이나, 고리가 나타날 무렵부터 차원이 급격히 요동치며 증가하다, 고리 중간을 넘어서면 거의 2차원에 가까운 값으로 안정한다.

공간디자인에 적용된 프랙탈 특성의 인지생태론적 효과

김주미(Kim, Joo-Mi) 한국실내디자인학회 2011 한국실내디자인학회논문집 Vol.20 No.2

The purpose of this study is to propose cognitive ecological effects of fractal patterns in space design. This study investigated the perception and cognition problems regarding landscape patterns showing fractal properties from the cognitive perspective instead of the traditional speculative approach. In particular, the researcher has verified that fractal geometry theory and fractal pattern concept provide insight in space aesthetic values and cognitive effects. Research results are as follows. First, most environmentally-friendly fractal urban forms provide cognitive connectivity. In particular, this space provides a positive emotional response and preference to humans and displays self-organized complexity. This study found that such complexity of space form has characteristics corresponding to parallel cognitive structures of the human brain. Simultaneously, the researcher suggests that the fractal landscape pattern is an alternative for stiff and homogenized modern space. Second, fractal patterns provide hierarchical connectivity within the brain through continuous difference and repetition. In particular, self-similarities of fractal patterns administer significant visual grouping and coherence in human perception. It can be determined whether scaling coherence facilitates easier organization in cognitive organization. Third, fractal patterns in space design provide the basic method for achieving the connection between concept, construction, and urban factors. As a result, the researcher has suggested that scale distribution of geometrical factors, such as fractal patterns, an be a design method to connect various space typologies.

치주질환 진단시 프랙탈 분석의 유용성에 관한 연구

차상윤,한원정,김은경 대한구강악안면방사선학회 2001 Imaging Science in Dentistry Vol.31 No.1

Purpose : To evaluate the usefulness of fractal analysis for diagnosis of periodontitis. Materials and Methods : Each 30 cases of periapical films of male mandibular molar were selected in normal group and patient group which had complete furcation involvement. They were digitized at 300 dpi,256 gray levels and saved with gif format. Rectangular ROIs (10×20 pixel) were selected at furcation, interdental crest, and interdental middle 1/3 area. Fractal dimensions were calculated three times at each area by mass radius method and were determined using a mean of three measurements. We compared fractal dimensions at furcation and interdental crest area of normal group with those of patient group. And then we compared ratio of fractal dimensions at furcation area, interdental crest area to interdental middle 1/3 area. Results : Fractal dimension at interdental crest area of normal group was 1.979±0.018 and that of patient group 1.971±0.012 (p〉0.05). Fractal dimension at furcation area of normal group was 1.986±0.024 and that of patient group 1.974±0.015 (p〈 0.05). The ratio of fractal dimension at interdental crest area to interdental middle 1/3 of normal group was 1.003±0.015 and that of patient group 0.993±0.018 (p〈0.05), The ratio of fractal dimension at furcation area to interdental middle 1/3 of normal group was 1.006±0.018 and that of patient group 0.994±0.018 (p〈 0.05). Conclusion : The ratio of fractal dimension at interdental crest and furcation area to interdental middle 1/3 area showed a statistically significant difference between normal and patient group. In conclusion, it is thought that fractal analysis might be useful for the diagnosis of periodontitis.(Korean J Oral Maxillofac Radiol 2001;31:35-42)

Comparison of Diagnostic Performance of Two-Dimensional and Three-Dimensional Fractal Dimension and Lacunarity Analyses for Predicting the Meningioma Grade

Soopil Kim,박예원,박상현,안성수,장종희,김세훈,이승구 대한뇌종양학회 2020 Brain Tumor Research and Treatment Vol.8 No.1

Background: To compare the diagnostic performance of two-dimensional (2D) and three-dimensional (3D) fractal dimension (FD) and lacunarity features from MRI for predicting the meningioma grade. Methods: This retrospective study included 123 meningioma patients [90 World Health Organization (WHO) grade I, 33 WHO grade II/III] with preoperative MRI including post-contrast T1-weighted imaging. The 2D and 3D FD and lacunarity parameters from the contrast-enhancing portion of the tumor were calculated. Reproducibility was assessed with the intraclass correlation coefficient. Multivariable logistic regression analysis using 2D or 3D fractal features was performed to predict the meningioma grade. The diagnostic ability of the 2D and 3D fractal models were compared. Results: The reproducibility between observers was excellent, with intraclass correlation coefficients of 0.97, 0.95, 0.98, and 0.96 for 2D FD, 2D lacunarity, 3D FD, and 3D lacunarity, respectively. WHO grade II/III meningiomas had a higher 2D and 3D FD (p=0.003 and p<0.001, respectively) and higher 2D and 3D lacunarity (p=0.002 and p=0.006, respectively) than WHO grade I meningiomas. The 2D fractal model showed an area under the curve (AUC), accuracy, sensitivity, and specificity of 0.690 [95% confidence interval (CI) 0.581-0.799], 72.4%, 75.8%, and 64.4%, respectively. The 3D fractal model showed an AUC, accuracy, sensitivity, and specificity of 0.813 (95% CI 0.733-0.878), 82.9%, 81.8%, and 70.0%, respectively. The 3D fractal model exhibited significantly better diagnostic performance than the 2D fractal model (p<0.001). Conclusion: The 3D fractal analysis proved superiority in diagnostic performance to 2D fractal analysis in grading meningioma.