Tiling, tessellations by
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1 Par Marcel Morales Professeur à l IUFM de Lyon Rattaché pour la recherche à Tiling, tessellations by MARCEL MORALES hand Tessellations 2010 [TA P E Z L ' A D R E S S E D E L A S O C I E T E ]
2 Tiling, tessellations by hand 2 Table des matières Tiling p1or R Lattice and fundamental region for p1 (R0) Tiling p2 or R Lattice and fundamental region for p2 (R2) Tiling p3 ou R Lattice and fundamental region for p3 (R3) Tiling p4 ou R Lattice and fundamental region for p4 (R4) Tiling p6 ou R Lattice and fundamental region for p6 (R6) Tiling pg ou M Lattice and fundamental region for pg (M0) Tiling pgg ou M0R Lattice and fundamental region for pgg (M0R2) Tiling cm ou M Lattice and fundamental region for cm (M1) Tiling pm ou M1g Lattice and fundamental region for pm (M1g) Tiling pmg ou M1R Lattice and fundamental region for pgm (M1R2) Tiling pmm ou M Lattice and fundamental region for pmm (M2) Tiling cmm ou M2R Lattice and fundamental region for cmm (M2R2) Tiling p4g ou M2R Lattice and fundamental region for p4g (M2R4) Tiling p4m ou M Lattice and fundamental region for p4m (M4) Tiling p31m ou M3R Lattice and fundamental region for p31m (M3R3) Tiling p3m1 or M Lattice and fundamental region for p3m1 (M3) Tiling p6m ou M Lattice and fundamental region for p6m (M6) The hyperbolictilings... 77
3 Tiling, tessellations by hand 3 Tiling, tessellation, periodic or non periodic tiling: fulfilling a picture by some figure was used by all the civilizations to decorate walls, carpets, potteries This book can help everybody to produces and drawn his own tiling, without any knowledge in mathematics. After some introduction to the subject with elementary notions of geometry, we introduce the seventeen groups of tiling of the plane. I have developed software that helps us to draw a picture for each group of tiling. Our aim is to introduce people to: 1. Recognize a tiling, and the basic figure, 2. Describe the tiling group, i.e. the transformation used to fulfill the plane, which rotation, symmetries. 3. How realize a tiling by drawing, cutting and gluing without using difficult techniques. 4. For each tiling s group a complete sequence of realization is done, explaining and allowing realizing it quickly. 5. Every tiling group is illustrated by one picture. The software has been developed by Marcel Morales, but the author has learned a lot from the high school class of Alice Morales. The software has been used by school students during many years, in all the degrees of schools. A joint work with the classroom of Alice Morales has been presented in the international exposition Exposciences International 2001 in Grenoble. The software has been introduced in many scientific expositions in France and outside France: Mexico, Peru, Colombia, Iran, Turkey, and Vietnam. Moreover, after the expositions and collaborations with high schools, it appears that doing tiling can be a kind of game, but also improves the knowledge in mathematics, without any formal course. We introduce some tiling founded in old civilizations. Par Marcel Morales Professeur à l IUFM de Lyon, Université Claude Bernard Lyon I Rattaché pour la recherche à Institut Fourier Université de Grenoble I
4 Tiling, tessellations by hand 4
5 Tiling, tessellations by hand 5 Tiling founded in the old Egypt.
6 Tiling, tessellations by hand 6 Tiling founded on the wall of a mosque.
7 Tiling, tessellations by hand 7 Tiling founded in a Peruvian carpet (Paracas).
8 Tiling, tessellations by hand 8 Tiling decoration of a wall in Mitla, Mexico
9 Tiling, tessellations by hand 9 Potteries, Indians from North America.
10 Tiling, tessellations by hand 11 Tiling realized by a pupil in the High school.
11 Tiling, tessellations by hand 12 Non periodic tiling of an hexagon, by using a rhomb (colored yellow), transformed by using thee transformations (see the cube up): by translations it remains yellow, by rotation of 120 and translations it is colored red, by rotation of 240 and translations it is colored blue.
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20 Rotations by 90, 180, 270 Tiling, tessellations by hand 21 We give now a few on the transformations of the plane in Euclidian Geometry. Translation, rotations, symmetry, and glide symmetry. Translation
21 Tiling, tessellations by hand 22 Symmetry and glide symmetry Doing a pattern for tiling the plane by using a sheet of paper and scissors. We will describe the process for each one of the seventeen groups of tiling. We will use both notations English and French. In each one we have numbered some special points
22 Tiling p1or R0 Tiling, tessellations by hand 23 Take a rectangular piece of paper. Draw a simple curve starting in 1 and going to 2 Cut along the curve and glue it on the right of your rectangle Draw a simple curve starting in 2 and going to 3 Cut along the curve and glue it on the down of your rectangle This is your pattern for this tiling group. We fulfill the plane by using horizontal and vertical translations; we color it in order to distinguish them:
23 Tiling, tessellations by hand 24 Lattice and fundamental region for p1 (R0)
24 Tiling, tessellations by hand 25 The following picture is the lattice associated to the Tiling p1 (R0), any one of the rectangles is a fundamental region and as you can check the arrows are translations.
25 Tiling p2 or R2 Tiling, tessellations by hand 26 Take a square piece of paper. Draw a simple curve starting in 1 and going to 2. Cut along the curve and glue it after doing a half-tour with center in the point 1. Draw a simple curve starting in 2 and going to 3 Cut along the curve, translate it vertically and glue. Draw a simple curve starting in 3 and going to 4. Cut along the curve and glue it after doing a half-tour with center in the point 4. This is your pattern for this tiling group. We fulfill the plane by using rotations of angle 180 and translations.
26 Tiling, tessellations by hand 27 Lattice and fundamental region for p2 (R2)
27 Tiling, tessellations by hand 28 The following picture is the lattice associated to the Tiling p2 (R2), any one of the squares is a fundamental region and as you can check the arrows are translations, and the circles are centers of rotation, also called central symmetry.
28 Tiling, tessellations by hand 29 Tiling p3 ou R3 Take a rhomb piece of paper. (two equilateral triangles) Draw a simple curve starting in 1 and going to 2. Cut along the curve and glue after a rotation of 120 with center in the point 1. Draw a simple curve starting in 2 and going to 3. Cut along the curve and glue after a rotation of 120 with center in the point 3. We fulfill the plane by using rotations of angle 120 and translations: This is your pattern for this tiling group.
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30 Lattice and fundamental region for p3 (R3) Tiling, tessellations by hand 31 The following picture is the lattice associated to the Tiling p2 (R2), the rhombus drawn is a fundamental region and as you can check the arrows are translations, and the small triangles are centers of rotation.
31 Tiling, tessellations by hand 32 Tiling p4 ou R4 Take a square piece of paper. Draw a simple curve starting in 1 and going to 2. Cut along the curve and glue after a rotation of 90 with center in the point 1. Draw a simple curve starting in 2 and going to 3. Cut along the curve and glue after a rotation of 120 with center in the point 3. This is your pattern for this tiling group. We fulfill the plane by using rotations of angles 90, 180, 270 and translations:
32 Tiling, tessellations by hand 33
33 Tiling, tessellations by hand 34 Lattice and fundamental region for p4 (R4) The following picture is the lattice associated to the Tiling p4 (R4), any square drawn is a fundamental region and as you can check the arrows are translations, the circles are centers of rotation and the small squares are centers of rotation.
34 Tiling, tessellations by hand 35 Tiling p6 ou R6 Take an equilateral triangle piece of paper. Draw a simple curve starting in 1 and going to 2. Cut along the curve and glue after a rotation of 60 with center in the point 1. Draw a simple curve starting in 2 and going to 3. Cut along the curve and glue after a rotation of 180 with center in the point 3. This is your pattern for this tiling group. We fulfill the plane by using rotations of angle 60, 120, 180 and translations:
35 Tiling, tessellations by hand 36
36 Tiling, tessellations by hand 37 Lattice and fundamental region for p6 (R6) The following picture is the lattice associated to the Tiling p6 (R6), the triangle drawn is a fundamental region and as you can check the arrows are translations, the circles are centers of rotation, the small hexagons are centers of rotations. And the small triangles are centers of rotation.
37 Tiling pg ou M0 Tiling, tessellations by hand 38 Take a rectangular piece of paper. Draw a simple curve starting in 1 and going to 2. Cut along the curve and glue it on the right side. Draw a simple curve starting in 2 and going to 3. Cut along the curve, turn on and glue it on the top down side. This is your pattern for this tiling group. We fulfill the plane by using glides symmetries and translations:
38 Tiling, tessellations by hand 39 Lattice and fundamental region for pg (M0)
39 Tiling, tessellations by hand 40 The following picture is the lattice associated to the Tiling pg (M0), any rectangle is a fundamental region and as you can check the arrows are translations. The lattice is the same as the tiling p0 (R0).
40 Tiling, tessellations by hand 41 Tiling pgg ou M0R2 Take a rectangular piece of paper. Draw a simple curve starting in 1 and going to 2. Cut along the curve, turn on and glue it on the right side. Draw a simple curve starting in 2 and going to 3. Cut along the curve, turn on and glue it on the top down side. This is your pattern for this tiling group. We fulfill the plane by using glides symmetries, rotations of angle 180 and translations:
41 Tiling, tessellations by hand 42
42 Tiling, tessellations by hand 43 Lattice and fundamental region for pgg (M0R2) The following picture is the lattice associated to the Tiling pgg (M0R2), any rectangle is a fundamental region and as you can check the arrows are translations. The lattice is the same as the tiling p2 (R2).
43 Tiling cm ou M1 Tiling, tessellations by hand 44 Take a rectangular piece of paper. Draw a simple curve starting in 1 and going to 2. Cut along the curve and glue it on the top up side. This is your pattern for this tiling group. We fulfill the plane by using symmetries and translations:
44 Tiling, tessellations by hand 45
45 Tiling, tessellations by hand 46 Lattice and fundamental region for cm (M1) The following picture is the lattice associated to the Tiling cm (M1), the fundamental region is rectangle that you are invited to draw, and as you can check there are vertical and horizontal translations. The lattice is the same as the tiling p0 (R0), and we must add the mirror horizontal lines.
46 Tiling pm ou M1g Tiling, tessellations by hand 47 Take a rectangular piece of paper. Draw a simple curve starting in 1 and going to 2. Cut along the curve, turn on and glue it on the top up side. This is your pattern for this tiling group. We fulfill the plane by using vertical symmetries and glide symmetries:
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48 Tiling, tessellations by hand 49 Lattice and fundamental region for pm (M1g) The following picture is the lattice associated to the Tiling pm (M1g), the fundamental region is a rectangle that you are invited to draw, and as you can check there are vertical and horizontal translations. The lattice is the same as the tiling p0 (R0), and we must add the mirror horizontal lines.
49 Tiling, tessellations by hand 50 Tiling pmg ou M1R2 Take a rectangular piece of paper. Draw a simple curve starting in 1 and going to 2. Cut along the curve and glue it on the right side. Draw a simple curve starting in 2 and going to 3. Cut along the curve, rotate it of 180 and glue it on the top up side. This is your pattern for this tiling group. We fulfill the plane by using rotations of angle 180 and horizontal symmetries:
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51 Tiling, tessellations by hand 52 Lattice and fundamental region for pgm (M1R2) The following picture is the lattice associated to the Tiling pgm (M1R2), the fundamental region is a rectangle that you are invited to draw, and as you can check there are vertical and horizontal translations. The lattice is the same as the tiling p2 (R2), and we must add the mirror vertical lines.
52 Tiling, tessellations by hand 53 Tiling pmm ou M2 Take a rectangular piece of paper. Draw a simple curve starting in 1 and going to 2. Draw a simple curve starting in 2 and going to 3. This is your pattern for this tiling group. (a draw rectangle) We fulfill the plane by vertical and horizontal symmetries :
53 Tiling, tessellations by hand 54
54 Tiling, tessellations by hand 55 Lattice and fundamental region for pmm (M2) The following picture is the lattice associated to the Tiling cm (M1), the fundamental region is a rectangle, and as you can check there are vertical and horizontal translations. The lattice is the same as the tiling p2 (R2), and we must add the mirror horizontal and vertical lines.
55 Tiling cmm ou M2R2 Tiling, tessellations by hand 56 Take a rectangular piece of paper. Draw a simple curve starting in 1 and going to 2. Cut along the curve, rotate it, and glue it on the top up side. This is your pattern for this tiling group. We fulfill the plane by using glides symmetries, rotations of angle 180 and translations:
56 Tiling, tessellations by hand 57
57 Tiling, tessellations by hand 58 Lattice and fundamental region for cmm (M2R2) The following picture is the lattice associated to the Tiling cmm (M2R2), the fundamental region is a rectangle, and as you can check there are vertical and horizontal translations. The lattice is the same as the tiling p2 (R2), and we must add the mirror horizontal and vertical lines.
58 Tiling, tessellations by hand 59 Tiling p4g ou M2R4 Take a square piece of paper. Draw a simple curve starting in 1 and going to 2. Cut along the curve, rotate it, and glue it on the top down side. This is your pattern for this tiling group. We fulfill the plane by using glides symmetries, rotations of angle 90, 180, 270 and translations:
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60 Tiling, tessellations by hand 61 Lattice and fundamental region for p4g (M2R4) The following picture is the lattice associated to the Tiling p4g (M2R4), the fundamental region is a half of a square, that is a right isosceles triangle, and as you can check there are vertical and horizontal translations. The lattice is the same as the tiling p4 (R4), and we must add the mirror horizontal and vertical lines.
61 Tiling, tessellations by hand 62 Tiling p4m ou M4 Take a right isosceles triangular piece of paper. Draw a simple curve starting in 1 and going to 2. Draw a simple curve starting in 2 and going to 3. This is your pattern for this tiling group. (a bicolor triangle) We fulfill the plane by using symmetries with axis the sides of the rectangle triangle:
62 Tiling, tessellations by hand 63
63 Tiling, tessellations by hand 64 Lattice and fundamental region for p4m (M4) The following picture is the lattice associated to the Tiling p4m (M4), the fundamental region is a half of a square, that is a right isosceles triangle, and as you can check there are vertical and horizontal translations. The lattice is the same as the tiling p4 (R4), and we must add the mirror horizontal and vertical lines.
64 Tiling p31m ou M3R3 Tiling, tessellations by hand 65 Take a triangular piece of paper. (isosceles having the principal angle 120 ) Draw a simple curve starting in 1 and going to 2. Cut along the curve, rotate it, and glue it on the right side. This is your pattern for this tiling group. We fulfill the plane by using symmetries and rotations of angle 120 :
65 Tiling, tessellations by hand 66
66 Tiling, tessellations by hand 67 Lattice and fundamental region for p31m (M3R3) The following picture is the lattice associated to the Tiling p31m (M3R3), the fundamental region is equilateral triangle, and as you can check there are translations following the sides of the triangles. The lattice is the same as the tiling p3 (R3), and we must add the mirror lines, that is the sides of the triangles.
67 Tiling, tessellations by hand 68 Tiling p3m1 or M3 Take a triangular piece of paper (equilateral). Draw a simple curve starting in 1 and going to 2. Draw a simple curve starting in 2 and going to 3. This is your pattern for this tiling group. We fulfill the plane by using symmetries with axis the sides of the equilateral triangle:
68 Tiling, tessellations by hand 69
69 Tiling, tessellations by hand 70 Lattice and fundamental region for p3m1 (M3) The following picture is the lattice associated to the Tiling p3m1 (M3), the fundamental region is an equilateral triangle, and as you can check there are translations following the sides of the triangles. The lattice is the same as the tiling p3 (R3), and we must add the mirror lines, that is the sides of the triangles.
70 Tiling, tessellations by hand 71 Tiling p6m ou M6 Take a triangular piece of paper. (The half of an equilateral triangle). Draw a simple curve starting in 1 and going to 2. Draw a simple curve starting in 2 and going to 3. This is your pattern for this tiling group. We fulfill the plane by using symmetries with axis the sides of the rectangle triangle:
71 Tiling, tessellations by hand 72
72 Tiling, tessellations by hand 73 Lattice and fundamental region for p6m (M6) The following picture is the lattice associated to the Tiling p31m (M3R3), the fundamental region is equilateral triangle, and as you can check there are translations following the sides of the triangles. The lattice is the same as the tiling p3 (R3), and we must add the mirror lines, that is the sides of the triangles.
73 Pattern of a tiling of a cube Tiling, tessellations by hand 74 Pattern of a tiling of a dodecahedron
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76 Paper wall Tiling, tessellations by hand 77 The hyperbolictilings We understand by hyperbolic geometry, the drawn of pictures and transformations by using the hyperbolic distance. The Hyperbolic plane is represented by a disk without the border, the border of the disk is the infinite; figures in the hyperbolic plane seem smaller when we approach to infinite. We give some examples of hyperbolic Tiling of the hyperbolic plane, drew by my software. My software is the first in the world to drawn automatically hyperbolic tiling and allows anybody to realize its own hyperbolic tiling. By contrast with Euclidian geometry there are infinitely many hyperbolic tiling groups.
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