수학이란? 비 과학자에게 과학자 또는 수학 애호가에게 셈은 수학의 작은 영역인 산수에 속할 뿐 숫자를 가지고 셈하는 학문 수학은 질서에 관한 학문 패턴과 구조에 관한 학문 논리적 관계에 대한 학문 셈은 수학의 작은 영역인 산수에 속할 뿐

현대 순수수학의 종류 논리학(Logic)과 기초수학 수, 대칭성을 연구하는 대수학(Algebra) 함수를 연구하는 해석학 (Analysis) 모양을 연구하는 기하학 (Geometry) 공간의 연결구조를 연구하는 위상수학(Topology)

논리학, 기초수학 소크라테스, 라이프니츠, 부울 대수 논리 연산자 암호론 및 컴퓨터의 탄생 ∧∨ (and, or,  …) 암호론 및 컴퓨터의 탄생 튜링, 폰노이만 벤다이어그램, 진리표, 논리 게이트, 투링기계와 처치 연산 힐버트, 러셀, 화이트해드 괴델 정리(incompleteness theorems)

대수학 정수론, 실수, 군론 (대칭성의 연구) 추상적 군, 환, 체, 리군, 리대수 등의 연구 (수 개념의 확장 개념) 다른 수학분야에 응용 암호론에 응용

대칭인 건물

에셔(Escher)의 그림 M. C. Escher (1898 –1972) 네델란드 그래픽 예술가 . In his graphic art, he portrayed mathematical relationships among shapes, figures and space. Additionally, he explored interlocking figures using black and white to enhance different dimensions. Integrated into his prints were mirror images of cones, spheres, cubes, rings and spirals. In addition to sketching landscape and nature in his early years, he also sketched insects, which frequently appeared in his later work. His first artistic work was completed in 1922, which featured eight human heads divided in different planes. Later in about 1924, he lost interest in "regular division" of planes, and turned to sketching landscapes in Italy with irregular perspectives that are impossible in natural form. Relativity, 1953. Although Escher did not have a mathematical training—his understanding of mathematics was largely visual and intuitive—Escher's work has a strong mathematical component, and more than a few of the worlds which he drew are built around impossible objects such as the Necker cube and the Penrose triangle. Many of Escher's works employed repeated tilings called tessellations. Escher's artwork is especially well-liked by mathematicians and scientists, who enjoy his use of polyhedra and geometric distortions. For example, in Gravity, multi-colored turtles poke their heads out of a stellated dodecahedron. The mathematical influence in his work emerged in about 1936, when he was journeying the Mediterranean with the Adria Shipping Company. Specifically, he became interested in order and symmetry. Escher described his journey through the Mediterranean as "the richest source of inspiration I have ever tapped." After his journey to the Alhambra, Escher tried to improve upon the art works of the Moors using geometric grids as the basis for his sketches, which he then overlaid with additional designs, mainly animals such as birds and lions. His first study of mathematics, which would later lead to its incorporation into his art works, began with George Pólya’s academic paper on plane symmetry groups sent to him by his brother Berend. This paper inspired him to learn the concept of the 17 wallpaper groups (plane symmetry groups). Utilizing this mathematical concept, Escher created periodic tilings with 43 colored drawings of different types of symmetry. From this point on he developed a mathematical approach to expressions of symmetry in his art works. Starting in 1937, he created woodcuts using the concept of the 17 plane symmetry groups. In 1941, Escher wrote his first paper, now publicly recognized, called Regular Division of the Plane with Asymmetric Congruent Polygons, which detailed his mathematical approach to artwork creation. His intention in writing this was to aid himself in integrating mathematics into art. Escher is considered a research mathematician of his time because of his documentation with this paper. In it, he studied color based division, and developed a system of categorizing combinations of shape, color and symmetrical properties. By studying these areas, he explored an area that later mathematicians labeled crystallography. Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher’s interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher’s works Circle Limit I–IV demonstrate this concept. In 1995, Coxeter verified that Escher had achieved mathematical perfection in his etchings in a published paper. Coxeter wrote, "[Escher] got it absolutely right to the millimeter." His works brought him fame: he was awarded the Knighthood of the Order of Orange Nassau in 1955. Subsequently he regularly designed art for dignitaries around the world. In 1958, he published a paper called Regular Division of the Plane, in which he described the systematic buildup of mathematical designs in his artworks. He emphasized, "[Mathematicians] have opened the gate leading to an extensive domain." .

해석학 함수 미분. 적분 양자역학의 근본 함수의 연속성, 급수 (코시) 함수의 적분가능성 (리만) 측정 가능 (Measurable) 함수 푸리에, 라플라스, 웨이브렛 변환 해석 미분. 적분 미분방정식 양자역학의 근본

기하학 모양 연구

위상수학 수학에서의 기초적인 역할 공간의 이해, 연결구조의 이해

수학적 사고의 연산 셈, 계산, 열거 반복 a, a 2 , a 3 … 관계 : 순서, 대응, 동치, 역 기수 1, 2, 3, ...., 서수 1st, 2nd, ... 반복 a, a 2 , a 3 … 큰 효과 (Godel, Escher and Bach, Hofstadter, 1979) 관계 : 순서, 대응, 동치, 역 변환: transformation: 조합, 대입 등

수학적 사고의 과정 연역, 귀납 induction, deduction 특수화 Specializing 가설 Conjecturing 일반화 Generalizing 확인 Convincing

수학적 사고의 흐름 능숙하게 다룸 manipulating 유형을 감지 sense of pattern 분명한 표현 articulation 복잡도 complexity by increasing generality or refinement 연결 connectedness . 관계

https://www.youtube.com/watch?v=hgZwSRpfouQ