타일 떠붙임 시멘트 모르타르의 배합비 변화에 따른 품질 특성 및 시공성에 대한 관능 평가

황인성 ( Hwang Yin-seong ),기태경 ( Ki Tae-kyoung ),한동엽 ( Han Dong-yeop ),노상균 ( Noh Sang-kyun ) 한국건축시공학회 2021 한국건축시공학회지 Vol.21 No.1

The aim of the research is providing a fundamental data on quality and constructability of direct tile setting method depending on various cement to sand ratio for tiling dry cement mortar. A large number of tile setting failures reported is related with the cement mortar and its construction for tiling. Because of different materials of tiles, the properties of tiling dry cement mortar, an adhesive for tiling, can influence on quality and constructability of tiling differently. Practically, the easiest way of controlling the properties of the tiling dry cement mortar is to control the proportion of cement and sand. Hence, in this research, sand to cement ratio (S/C) was controlled. Since there is no standarized method on evaluating performance of dry cement mortar for tiling, a several sensory evaluation methods were suggested and executed. According to the experiments conducted in this research, the adhesive performance of cement mortar for tiles can be different depending on the sides such as tile and substrate. Additionally, depending on S/C, finishability, initial adhesive performance, and tile shifting resistance can be changed for ceramic tile. Therefore, under the conditions of this research, about 5 of S/C can be recommended for appropriate performace of tiling dry cement mortar.

Girih 타일링을 이용한 초등수학영재 프로그램 개발 및 적용 연구

박혜정,조영미 韓國英才學會 2012 영재교육연구 Vol.22 No.3

girih 타일링은 비주기적이면서도 규칙적인 구조를 가진 타일링으로, 최근 ‘이슬람 사원 장식에 숨어있는 수학의 비밀’로 주목받고 있다. 본 연구에서는 이를 소재로 하여 비주기적인 규칙성 안에 숨어 있는 수학이 만들어내는 유용성과 아름다움을 체험할 수 있는 초등 수학영재 프로그램을 개발·적용하고 그 결과를 분석하는 데 목적을 두었다. 개발한 초등수학영재 프로그램은 ‘이슬람 문양 속 girih의 비밀을 찾아서’이며, Renzulli의 3부 심화학습 형식에 따라 적용하였다. 이 프로그램은 대전광역시 유성구에 소재하고 있는 D 초등학교 5, 6학년 통합영재반 6명에게 적용한 결과를 토대로 수정, 보완 하였으며 개발된 프로그램 및 학습 자료는 초등수학영재 교육을 위한 소재 개발과 방법에 있어 도움이 될 것으로 기대한다. The purpose of this study is to develop a new program for elementary math-gifted students by using 'Girih Tililng' and apply it to the elementary students to improve their math-ability. Girih Tililng is well known for ‘the secrets of mathematics hidden in Mosque decoration’ with lots of recent attention from the world. The process of this study is as follows; (1) Reference research has been done for various tiling theories and the theories have been utilized for making this study applicable. (2) The characteristic features of Mosque tiles and their basic structures have been analyzed. After logical examination of the patterns, their mathematic attributes have been found out. (3) After development of Girih tiling program, the program has been applied to math-gifted students and the program has been modified and complemented. This program which has been developed for math-gifted students is called ‘Exploring the Secrets of Girih Hidden in Mosque Patterns’. The program was based on the Renzulli’s three-part in-depth learning. The first part of the in-depth learning activity, as a research stage, is designed to examine Islamic patterns in various ways and get the gifted students to understand and have them motivated to learn the concept of the tiling, understanding the characteristics of Islamic patterns, investigating Islamic design, and experiencing the Girih tiles. The second part of the in-depth learning activity, as a discovery stage, is focused on investigating the mathematical features of the Girih tile, comparing Girih tiled patterns with non-Girih tiled ones, investigating the mathematical characteristics of the five Girih tiles, and filling out the blank of Islamic patterns. The third part of the in-depth learning activity, as an inquiry or a creative stage, is planned to show the students' mathematical creativity by thinking over different types of Girih tiling, making the students' own tile patterns, presenting artifacts and reflecting over production process. This program was applied to 6 students who were enrolled in an unified(math and science) gifted class of D elementary school in Daejeon. After analyzing the results produced by its application, the program was modified and complemented repeatedly. It is expected that this program and its materials used in this study will guide a direction of how to develop methodical materials for math-gifted education in elementary schools. This program is originally developed for gifted education in elementary schools, but for further study, it is hoped that this study and the program will be also utilized in the field of math-gifted or unified gifted education in secondary schools in connection with 'Penrose Tiling' or material of 'quasi-crystal'.

비주기적 타일링을 이용한 융합교육 프로그램 개발 및 적용 연구

이정엽,허혜자 대한수학교육학회 2024 학교수학 Vol.26 No.1

본 연구는 타일링과 수학을 통해 창의력과 수학에 대한 긍정적 태도를 육성하기 위한 프로그램 개발 및 적용 연구이다. 학교수학에서는 초등학교 4학년에서 평면도형의 밀기, 뒤집기, 돌리기를 통한 규칙적인 무늬 만들기 활동을 하고, 중학교 1학년에서 정다각형의 내각에 대해서 다룬다. 고등학교 선택과목인 실용수학(2015개정 교육과정)과 수학과문화(2022개정 교육과정)는 쪽매맞춤 활동을 포함하고 있다. 타일링은 테셀레이션 또는 쪽매맞춤으로 불리기도 하는데, 알함브라 궁전과 같은 유명 건축물 뿐아니라 가정집의 현관, 욕실, 베란다, 광장 등 일상생활 속에서 흔하게 접할 수 있으며, 이러한 타일링을 예술로 승화시킨 미술가 에셔의 작품은 널리 알려져 있다. 타일링은 미술과 수학을 연결한 융합교과 프로젝트 수업으로 학생들의 호기심을 자극하고창의력을 기르기에 충분한 주제이다. 특히, 최근에 발표된 ‘비주기적인 모노타일(aperiodic monotile)’의 발견을 소재로 “무한히 넓은 욕실 바닥에 타일을 깔려고 한다. 비주기적으로만 깔 수 있는 타일 모양이 있을까?”와 “무한히 넓은 욕실 바닥에 타일을깔려고 한다. 한 가지 모양의 타일을 뒤집거나 회전시키는 것만 허용하여 욕실 바닥을 까는데 깔 때마다 비주기적으로만 깔리는경우가 있을까?”라는 두 개의 문제를 중심으로 융합교육 프로그램을 개발하고, 고등학생을 대상으로 10차시 수업을 진행하였으며, 그 결과 학생들의 창의력 신장과 수학에 대한 긍정적 태도의 향상 가능성을 볼 수 있었다. This study is a program development and application study to foster creativity and positive attitudes toward mathematics through tiling and mathematics. In school math, fourth graders in elementary school engage in regular pattern-making activities by pushing, flipping, and rotating flat shapes, and first graders in middle school cover interior angles of regular polygons. Practical Mathematics (2015 Curriculum), and Mathematics and Culture (2022 Curriculum), which are elective high school subjects, include tessellation activities. Tiling, also called tessellation, is commonly encountered in everyday life, such as in famous buildings like the Alhambra, as well as in home entrances, bathrooms, verandas, and squares. The works of artist Escher, who elevated such tiling into art, are widely known. Tiling is a convergence class subject which connects art and mathematics, and is a topic sufficient to stimulate students' curiosity and foster creativity. Recently there was a discovery of ‘aperiodic monotile’. Based on this discovery we raised following two questions: “We are planning to tile an infinitely wide bathroom floor. Are there tile shapes that can only tile the floor aperiodically?” and “We are planning to tile an infinitely wide bathroom floor. Is there a single tile shape that can only tile the floor aperiodically, when we only allow to use rotated versions and reflected versions of the tile?” Centered on these two questions, we developed a convergence education program and held a 10-session class for high school students. As a result, it was possible to see the possibility of increasing students' creativity and improving their positive attitude toward mathematics.

Tiling of Closed Plane Curves

M.S El-Ghoul,M.E. Basher 충청수학회 2005 충청수학회지 Vol.18 No.2

In this paper, we introduced the tiling, for closed plane curves $\alpha (s)$, and we discussed the properties of tiling. Also if $\alpha (s)$ was arbitrary plane closed curve equipped by tiling $\Im $ then we studied the effect of retraction and tiling retraction on it.

TILING OF CLOSED PLANE CURVES

El-Ghoul, Mabrouk Salem,Basher, Mohamed Esmail 충청수학회 2005 충청수학회지 Vol.18 No.2

In this paper, we introduced the tiling, for closed plane curves ${\alpha}(s)$, and we discussed the properties of tiling. Also if ${\alpha}(s)$ was arbitrary plane closed curve equipped by tiling ${\Im}$ then we studied the effect of retraction and tiling retraction on it.

Girih tiling을 이용한 초등수학영재 프로그램 개발 및 적용 연구

박혜정,조영미 한국초등수학교육학회 2012 한국초등수학교육학회 연구발표대회 논문집 Vol.2012 No.08

본 연구는 최근 ‘이슬람 사원 장식에 숨어있는 수학의 비밀’로 주목받고 있는 비주기적이면서 도 규칙적인 구조를 가진 Girih tiling을 소재로 하여 초등 수학영재 프로그램을 개발, 적용하는데 그 목적을 두었다. 본 연구에서 개발한 초등수학영재 프로그램은 ‘이슬람 문양 속 Girih의 비밀을 찾아서’이며, Renzulli의 3부심화학습모형을 적용하였다. 이 프로그램은 대전광역시 유성구에 소재 하고 있는 D초등학교 5,6학년 통합영재반 6명에게 적용한 결과를 토대로 수정, 보완 하였으며 개 발된 프로그램 및 학습 자료는 초등수학영재 교육을 위한 소재 개발과 방법에 있어 방향을 제시 할 것으로 기대된다. 또한 개발된 영재프로그램은 초등수학에 초점을 두고 있지만, Girih tiling을 Penrose tiling이나 준결정(Quasicrystal) 물질 등과 관련지어 프로그램을 개발한다면 중등수학영 재교육 프로그램이나 융합영재교육 프로그램으로 확장, 심화 시킬 수 있을 것이다.

GENERALIZED DOMINOES TILING'S MARKOV CHAIN MIXES FAST

KAYIBI, K.K.,SAMEE, U.,MERAJUDDIN, MERAJUDDIN,PIRZADA, S. The Korean Society for Computational and Applied M 2019 Journal of applied mathematics & informatics Vol.37 No.5

A generalized tiling is defined as a generalization of the properties of tiling a region of ${\mathbb{Z}}^2$ with dominoes, and comprises tiling with rhombus and any other tilings that admits height functions which can be ordered into a distributive lattice. By using properties of the distributive lattice, we prove that the Markov chain consisting of moving from one height function to the next by a flip is fast mixing and the mixing time ${\tau}({\epsilon})$ is given by ${\tau}({\epsilon}){\leq}(kmn)^3(mn\;{\ln}\;k+{\ln}\;{\epsilon}^{-1})$, where mn is the area of the grid ${\Gamma}$ that is a k-regular polycell. This result generalizes the result of the authors (T-tetromino tiling Markov chain is fast mixing, Theor. Comp. Sci. (2018)) and improves on the mixing time obtained by using coupling arguments by N. Destainville and by M. Luby, D. Randall, A. Sinclair.

GENERALIZED DOMINOES TILING'S MARKOV CHAIN MIXES FAST

K.K. Kayibi,U. Samee,Merajuddin,S. PIRZADA 한국전산응용수학회 2019 Journal of applied mathematics & informatics Vol.37 No.5

A generalized tiling is defined as a generalization of the properties of tiling a region of Z^2 with dominoes, and comprises tiling with rhombus and any other tilings that admits height functions which can be ordered into a distributive lattice. By using properties of the distributive lattice, we prove that the Markov chain consisting of moving from one height function to the next by a flip is fast mixing and the mixing time τ(ε) is given by τ(ε)≤(kmn)^3 (mn ln k + ln ε^-1 ), where mn is the area of the grid Γ that is a k-regular polycell. This result generalizes the result of the authors (T-tetromino tiling Markov chain is fast mixing, Theor. Comp. Sci. (2018)) and improves on the mixing time obtained by using coupling arguments by N. Destainville and by M. Luby, D. Randall, A. Sinclair.

금속비의 확장 가능성에 관한 연구

정제윤,황은준,송상헌 한국과학영재교육학회 2022 과학영재교육 Vol.14 No.1

The metallic means are defined as the only positive real roots of x-1/x=n or x^2-nx=1. One of the most prominent applications of the metallic means is known as aperiodic tilings, such as the Penrose tiling or the Ammann-Beenker tiling. The Penrose tiling is already generalized to 3-dimensions as well. Another application of metallic means is on polyhedra : the golden ratio and silver ratio appear in various regular, Archimedean, and Catalan solids as well. This study categorizes previous studies on metallic means by the method of research, confirms whether the 3D Penrose tiling can be generalized to 4D figures, and verifies whether the golden ratio and silver ratio appear even in n-dimensional figures. As a result, it was revealed that the 4D Penrose tiling does not exist under certain conditions. Also, it was confirmed that the golden ratio and silver ratio could not appear in regular and semiregular polytopes in 5 dimensions or higher.

적응적 타일링 및 블록 매칭을 통한 포토 모자이크 알고리즘

서성진,김기웅,김선명,이해연 한국정보처리학회 2012 정보처리학회 논문지 Vol.(이전)19 No.1

모자이크란 여러 가지 빛깔의 재료를 조각조각 붙여서 무늬나 영상을 만드는 기법을 말하며, 최근에는 디지털 이미징 기술의 발달로 인하여사진을 이용하여 영상을 만드는 포토 모자이크 기술들이 활용되고 있다. 본 논문에서는 적응적 타일링 및 블록 매칭을 통하여 포토 모자이크 영상을 만드는 컴퓨터 알고리즘을 제안한다. 제안한 알고리즘은 사진 데이터베이스의 생성 단계와 포토 모자이크 생성 단계로 구성된다. 사진 데이터베이스란 모자이크에 사용되는 사진(또는 타일)을 의미하며, 타일을 4×4로 분할한 후에 각 영역의 RGB 평균값을 특징값으로 저장한다. 포토모자이크 생성 단계는 입력 영상에 대하여 기 설정된 블록 크기로 분할한 후에 특징을 추출하는 과정, 인접한 블록들 사이의 유사도를 비교하여병합하는 적응적 타일링 과정, 적응적 타일링을 통해 생성된 블록들을 사진 데이터베이스의 타일들과 유클리드 차이로 유사도를 비교하여 유사한 타일을 찾는 블록 매칭 과정 및 매칭된 타일의 명암값을 해당 블록의 명암값으로 교체하여 영상의 유사도를 높이는 밝기값 조정 과정으로 구성된다. 또한 인접 블록간 타일의 중복성을 최소화하는 기법을 적용하여 영상의 품질도 향상하였다. 제안한 알고리즘의 성능을 분석하기 위하여안드레아 모자이크 소프트웨어와 비교하였고, 정량적인 분석 및 정성적인 분석에 있어서 제안한 알고리즘이 우수한 것으로 나타났다. Mosaic is to make a big image by gathering lots of small materials having various colors. With advance of digital imaging techniques,photomosaic techniques using photos are widely used. In this paper, we presents an automatic photomosaic algorithm based on adaptive tiling and block matching. The proposed algorithm is composed of two processes: photo database generation and photomosaic generation. Photo database is a set of photos (or tiles) used for mosaic, where a tile is divided into 4×4 regions and the average RGB value of each region is the feature of the tile. Photomosaic generation is composed of 4 steps: feature extraction, adaptive tiling, block matching, and intensity adjustment. In feature extraction, the feature of each block is calculated after the image is splitted into the preset size of blocks. In adaptive tiling, the blocks having similar similarities are merged. Then, the blocks are compared with tiles in photo database by comparing euclidean distance as a similarity measure in block matching. Finally, in intensity adjustment, the intensity of the matched tile is replaced as that of the block to increase the similarity between the tile and the block. Also, a tile redundancy minimization scheme of adjacent blocks is applied to enhance the quality of mosaic photos. In comparison with Andrea mosaic software, the proposed algorithm outperforms in quantitative and qualitative analysis.