타일링과 테셀레이션

개요

• 테셀레이션 - 동일한 모양을 이용해 평면이나 공간을 빈틈이나 겹쳐지는 부분 없이 채우는 것
• 2차원 평면의 테셀레이션

메모

• Gerard Westendorp Platonic tilings of Riemann surfaces
• 타일링 http://www.flickr.com/photos/syngola/sets/72157603528308920/
• 이슬람의 문양에서 많이 발견됨.
• 예술가 에셔의 작품에는 이러한 것을 주제로 한 작품이 많음
• 에셔는 스페인의 알함브라 궁전에서 이러한 것을 보고 영감을 받았는데, 알함브라는 무어인(북아프리카의 무슬림)들이 스페인 남긴 것. http://www.youtube.com/watch?v=YPoNYrZDraI
• Tilings Encyclopedia
• http://mathoverflow.net/questions/46502/on-the-number-of-archimedean-solids
• 준결정 (quasicrystal)

하위페이지

• 17 Plane Crystallographic groups
• 7개의 프리즈 패턴
• 아르키메데스 타일링
• 정다면체
• 2차원 쌍곡기하학의 테셀레이션

관련된 항목들

• 반사 변환
• 반전 사상(inversion)
• 유한반사군과 콕세터군(finite reflection groups and Coxeter groups)
• 축구공의 수학

사전 형태의 자료

• http://en.wikipedia.org/wiki/Domino_%28mathematics%29
• http://mathworld.wolfram.com/TruchetTiling.html

에세이, 리뷰, 강의노트

• Dana Mackenzie, A Hyperbolic Plane Coloring and the Simple Group of Order 168, The American Mathematical Monthly, Vol. 102, No. 8 (Oct., 1995), pp. 706-715

관련논문

• Nelson, Roice, and Henry Segerman. “Visualizing Hyperbolic Honeycombs.” arXiv:1511.02851 [math], November 7, 2015. http://arxiv.org/abs/1511.02851.
• Gao, Honghao, Nan Shi, and Min Yan. 2013. “Spherical Tiling by 12 Congruent Pentagons.” Journal of Combinatorial Theory, Series A 120 (4): 744–76. doi:10.1016/j.jcta.2012.12.006.
• Mendelsohn, N. S. 2004. “Tiling with Dominoes”. The College Mathematics Journal 35 (2) (3월 1): 115-120. doi:10.2307/4146865.

관련기사

• [재미있는수학교실빈틈없이 평면 덮기] 정미자 신림고 수학교사 세계일보, 2007-07-09
• [예술속 수학이야기(45)에셔와 테셀레이션] 김정하·인천건지초등학교교사 경향신문, 2007년 12월 04일

블로그

• 에셔의 예술에 공헌한 수학 피타고라스의 창, 2008-1-8
• E8이란 무엇인가 (2) : 8차원에서 내려온 그림자 피타고라스의 창, 2008-8-2

노트

1. A semi-regular tessellation is made of two or more regular polygons.
2. To name a tessellation, go around a vertex and write down how many sides each polygon has, in order ... like "3.12.12".
3. 1953 ROTATION - A Tessellation which the shape repeats by rotating or turning.
4. A Tessellation which the shape repeats by reflecting or flipping.
5. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons.
6. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace.
7. Tessellations are sometimes employed for decorative effect in quilting.
8. In 1619 Johannes Kepler made an early documented study of tessellations.
9. Tessellations are a fun, hands-on way to explore STEAM, whether you are in art class, math class, or in a STEM or STEAM classroom.
10. Tessellations are patterns resulting from arranging, or tiling, shapes without any gaps.
11. This is the type of tessellation you can make easily with a sticky note (as shown below).
12. Rotation tessellations are accomplished by (you guessed it!) rotating the tessellated shape.
13. A tiling of regular polygons (in two dimensions), polyhedra (three dimensions), or polytopes ( dimensions) is called a tessellation.
14. In the plane, there are eight such tessellations, illustrated above (Ghyka 1977, pp. 76-78; Williams 1979, pp.
15. A tessellation of -dimensional polytopes is called a honeycomb.
16. The tessellation process is divided into three stages which form an optional part of Vertex Processing in the rendering pipeline.
17. The amount of tessellation done in this case is taken from default values set into the context.
18. This stage is only executed if a tessellation evaluation shader (TES) is active in the current program or program pipeline.
19. The TES can also force the generation of the tessellation as a series of points rather than triangles or lines by providing the primitive.
20. But tessellations can be formed from multiple shapes.
21. The word tessellation can also refer to the act of tessellating—forming such a pattern.
22. Example: The building was designed to look like a tessellation in the form of a honeycomb pattern.
23. There are three regular shapes that make up regular tessellations: the equilateral triangle, the square and the regular hexagon.
24. Equilateral triangles, squares and regular hexagons make up regular tessellations.
25. Semi-regular tessellations are made of more than one kind of regular polygon.
26. Within the limit of the same shapes surrounding each vertex (the points where the corners meet), there are eight such tessellations.
27. : Create a tessellation by deforming a triangle, rectangle or hexagon to form a polygon that tiles the plane.
28. Tessellation is when shapes fit together in a pattern with no gaps or overlaps.
29. Alain Nicolas, the great French tessellation artist, has posted a gallery of new original tessellations that are quite amazing.
30. Try coloring our new "Angry Birds" tessellation.
31. Brick walls, tiled floors, and the honeycomb in bee hives are all tessellations.
32. Yes, we'd be happy to post your class's tessellations in a "school gallery" on www.Tessellations.org.
33. We can study the way regular polygons interact with each other, and one way they can do so is through tessellations.
34. A regular tessellation is one made using only one regular polygon.
35. A semi-regular tessellation uses two or more regular polygons.
36. A tessellation is a repeated series of geometric shapes that covers a surface with no gaps or overlapping of the shapes.
37. Tessellations are used in works of art, fabric patterns or to teach abstract mathematical concepts, such as symmetry.
38. All regular tessellations must be made of regular polygons.
39. However, not all regular polygons can be used to create a tessellation because their sides do not line up evenly.
40. The words tessellate and tessellation come from a Latin word which means "small stones" and "to pave with small stones".
41. We will now look at different types of tessellations that deal with regular polygons.
42. A tessellation is a pattern of shapes repeated to fill a plane.
43. Tessellations are something we often see in quilts, carpets, floors, and more.
44. For the tessellation above composed of squares to the left, the sum of the angles at a vertex are 90°+90°+90°+90°=360°.
45. A regular tessellation is made up of regular congruent polygons.
46. That function computes triangle edge and inside tessellation factors.
47. This next example shows a surface shader that does some displacement mapping without using tessellation.
48. This is not needed yet, but it’s more efficient for tessellation to use as small structure as possible.
49. Purely distance based tessellation is effective only when triangle sizes are quite similar.
50. As one journey — the classification of all convex polygon tessellations — ends, another is just beginning.
51. The squinting eyes, the jut of the chin, the precise tessellation of the lower lip and upper lip stay the same.
52. Start with creating a tessellation shape using the "translation pattern" ( see the steps below ).
53. There are few patterns that you can follow to construct a tessellation.
54. This tessellation was based on a hexagon.
55. Do you know what is the definition of tessellation and what does it take to create one?
56. A fundamental region is a shape that is repeated in order to form a tessellation.
57. One shape of a tile in a tessellation is called a prototile.
58. Based on the types of polygons, tessellations are classified as regular, semi-regular and non-regular or irregular.
59. Therefore, these complex structures can be interpreted as interwoven tessellations of the α and β phases.
60. 4e–g) reveal the detailed structures around the tessellation vertices.
61. The angular sum of the polygons involved is therefore 360°, resulting in perfect tessellation.
62. These complex tessellations can be regarded as two different phases interweaving in multiple ways.
63. Here you can create your own tessellations using regular polygons.
64. At every vertex in the tessellation, the internal angles of multiple different polygons meet.
65. Tessellations in Art Many artists, architects and designers use tessellations in their work.
66. All the tessellations we saw so far have one thing in common: they are periodic.
67. A tessellation of the plane is an arrangement of polygons which cover the plane without gaps or overlapping.
68. The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.
69. In particular, students, perhaps in groups, should be encouraged to produce their own (non-regular) tessellations of the plane.
70. Tessellations and Fractals are two different art techniques that often get mistaken for one another despite being nothing alike.
71. Regular tessellations are tile coverings made up of only one shape.
72. Semi-regular tessellations: When two or three different polygonal shapes share a common vortex, it is called a semi-regular tessellation.
73. C. Escher used tessellation patterns extensively in his work, often to great effect.
74. Tessellation is a system of shapes which are fitted together to cover a plane, without any gaps or overlapping.
75. The word tessellation itself derives from the Greek tessera, which is associated with four, square and tile.
76. Tessellations are a common feature of decorative art and occur in the natural world all around us.
77. Traditionally, the pattern formed by a tessellation is repetitive.
78. Tessellations run the gamut from basic to boggling.
79. All tessellations, even shapely and complex ones like M.C. Escher's, begin with a shape that repeats without gaps.
80. No tessellation talent outshines Dutch graphic artist M.C. Escher.
81. In the context of quilting, tessellation refers to regular and semiregular of tessellation of either patch shapes or the overall design.
82. Then make your own tessellations inspired by artist M.C. Escher .
83. To our knowledge, this is a genuine molecular-level realization of a 2D superstructure exhibiting this kind of surface tessellation.

메타데이터

위키데이터

• ID : Q3751781

Spacy 패턴 목록

• [{'LEMMA': 'domino'}]