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.

프랙털(Fractal) 특성을 적용한 형상 문법 생성 연구 - 프랙털 기하학 특성이 적용된 형상 문법 건축 조형의 미의식 중심으로 -

김가영,조택연 한국공간디자인학회 2024 한국공간디자인학회논문집 Vol.19 No.8

(연구 배경 및 목적) 형상 문법은 건축 설계와 조형물 제작에 있어 복잡한 형태와 구조를 생성하는 규칙 기반의 시스템으로, 건축가와 디자이너들의 작업에서 다양한 형태와 패턴을 구체화하는 데 도움을 준다. 또한 형상 문법은 복잡한 건축 디자인 문제의 해결 시, 규칙에 기반한 접근 방식으로 문제를 구조화하여 체계적인 해결책 탐색을 가능하게 함으로써 효율적인 도구 역할을 한다. 그럼에도 형상 문법에 의해 생성된 각각의 건축물과 조형물은 미적 매력이 일관되게 드러나지 않는다. 본 연구는 형상 문법을 통해 생성된 조형물의 미적 매력이 일관되지 않은 원인을 탐구하고, 조형의 프랙털 구조와 미의식 사이 특성을 연구하여 이러한 조형물의 미적 호감도를 증가시키는 방법을 제시하는 데 그 목적을 둔다. (연구 방법) 본 연구는 일반적인 형상 문법 조형물과 프랙털의 특성을 적용한 형상 문법 조형물의 미적 매력을 비교하는 행동학적 연구로 진행된다. 설문에 참여한 총 273명의 피험자들은 형상 문법만으로 생성된 조형물과 프랙털 특성이 적용된 조형물을 보고 미적 호감도를 평가하게 되며, 이때 조형물의 디자인은 2차원 형상 문법과 3차원 형상 문법(그룹 A), 프랙털 특성이 적용된 3차원 형상 문법 1단계와 2단계(그룹 B)의 네 가지로 제시된다. 설문 종료 후에는 수집된 데이터가 가진 프랙털 호감 구조를 통계 분석을 통해 추출하고, 이렇게 추출된 프랙털 특성이 조형물의 심미적 매력에 미치는 영향을 검토한다. 그리고 결과를 바탕으로 형상 문법을 통한 건축 조형물의 다양성과 잠재적 가치에 대해 논한다. (결과) 연구 결과, 일반적인 형상 문법 조형물보다 프랙털 특성을 적용한 형상 문법 조형물에서 미적 호감도가 증가한 것으로 나타남을 발견하였다. 추가적으로 연령별, 성별, 디자인 관련 교육 및 경험에 따른 미적 호감도의 차이가 있는지 분석하였고 이 결과, 연령과 성별에 따른 미적 호감도의 차이가 존재함을 발견하였다. 그러나 디자인 관련 교육 및 경험에 따른 호감도의 유의미한 차이는 발견되지 않았다. 이러한 결과는 연령별 및 성별에 따른 호감도의 차이는 있지만, 형상 문법을 통해 조형물을 생성하는 과정에서 프랙털의 특성 요소를 더하면, 보편적으로 프랙털 특성이 적용된 조형물의 미적 매력이 높게 보임을 시사한다. 이는 형상 문법을 통해 생성된 조형물의 다양성과 잠재적 가치가 프랙털 특성의 적용을 통해 크게 증진될 수 있음을 입증한다. (결론) 프랙털 특성을 적용한 형상 문법 디자인 연구는 건축가와 디자이너들에게 자연의 복잡한 미를 조형물에 반영하게 함으로써 형상 문법 디자인의 범위 확장 및 활용도 향상에 기여할 수 있을 것으로 보인다. 형상 문법의 잠재적 가치 증진을 통해 앞으로 이러한 디자인 방식이 건축 조형 작업에 다양하게 적용될 수 있기를 기대한다. (Background and Purpose) Shape grammar is a rule-based system that aids in generating complex forms and structures in architectural design and sculptural creation, enabling architects and designers to concretize diverse shapes and patterns. Additionally, shape grammar serves as an effective tool by structuring complex architectural design problems in a rule-based approach, facilitating a systematic exploration of solutions. However, the aesthetic appeal of individual architectural and sculptural works generated by shape grammar does not consistently emerge. This study aims to investigate the reasons for the inconsistent aesthetic appeal of sculptures generated through shape grammar and proposes methods to enhance the aesthetic appeal of these sculptures by examining the characteristics between the fractal structure of shapes and aesthetic consciousness. (Method) This study is conducted as a behavioral research project comparing the aesthetic appeal of general shape grammar sculptures and shape grammar sculptures incorporating fractal characteristics. A total of 273 participants evaluate the aesthetic appeal of sculptures created solely through shape grammar and those incorporating fractal characteristics. The sculptures are presented in four categories: two-dimensional shape grammar and three-dimensional shape grammar (Group A) and three-dimensional shape grammar with first- and second-level fractal characteristics (Group B). After the survey, the fractal preference structure within the collected data is extracted through statistical analysis, and the impact of these extracted fractal characteristics on the aesthetic appeal of the sculptures is examined. Based on the findings, this study discusses the diversity and potential value of architectural sculptures created through shape grammar. (Results) The study found that the aesthetic appeal of shape grammar sculptures incorporating fractal characteristics was higher than that of general shape grammar sculptures. Additionally, an analysis of differences in aesthetic appeal based on age, gender, and education & experience in design was conducted. The results indicate significant differences in aesthetic appeal according to age and gender, while no significant difference was found based on design-related education and experience. These findings suggest that while there are differences in appeal by age and gender, incorporating fractal characteristics into the process of creating sculptures through shape grammar generally enhances the aesthetic appeal of fractal-based sculptures. This supports the notion that the diversity and potential value of shape grammar-generated sculptures can be greatly enhanced by applying fractal characteristics. (Conclusions) Research on shape grammar design with applied fractal characteristics may contribute to extending the scope and utility of shape grammar design by enabling architects and designers to reflect the intricate beauty of nature in sculptures. Through the enhancement of the potential value of shape grammar, it is hoped that this design approach can be widely applied in architectural sculptural works in the future.

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.

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.

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.