Escher http www mcescher com Regular tiling Edge

Escher http: //www. mcescher. com /

Regular tiling Edge to edge tilings: Regular polygons 3, 4, 6 Semi-regular tilings: two or more regular polygons so that the vertex figures are same size and shape.

Miscellaneous tiling 3 -gons 4 -gons (quadilateral)

pentagon tiling (최소 14개)

hexagons Irregular tilings of three types

tilings 3 -gons all are possible 4 -gons all are possible 5 Unknown 14 or more 6 -gons 3 7 - 8 -… gons None

연산표 * I V W X Y Z I I V W X Y Z V V W I Z X Y W W I V Y Z X X X Y Z I V W Y Y Z X W I V Z Z X Y V W I

Strip patterns and symmetry Translation symmetry reflection symmetry, glide reflection All symmetries are of this form. 이것들은 군을 이룬다.

Rigid Motions 각과 거리를 보존하는 움직임 2 -차원에는 translation rotation reflection (about a line) glide reflection 어떻게 이들을 분류하는가? 2차원에는 17개의 벽지 (wallpaper) 군이 존 재함 (html파일들) http: //www. clarku. edu/%7 Edjoyce/wallpap er/seventeen. html

Aperiodic tiling (Penrose tiling)

quasicrystals http: //www. lassp. cornell. edu/lifshitz/ quasicrystals. html

http: //mathworld. wolfram. com/Kepler. Conjecture. html http: //mathworld. wolfram. com/Cubic. Close. Packing. html http: //mathworld. wolfram. com/Hexagonal. Close. Packing. html

오류 정정 부호 통신(휴대폰, 인터넷), CD, DVD 플레이 어등등 많은 응용--- 소음 등을 극복하 는 방법 Hamming code만 설명 http: //www. eccpage. com/ http: //mathworld. wolfram. com/Error. Correcting. Code. html