1 00:00:05,000 --> 00:00:10,000 Darts and kites in the sky of mathematics 2 00:00:10,000 --> 00:00:13,000 The Penrose tessellation 3 00:00:13,000 --> 00:00:16,000 (http://frecceaquiloni.dmf.unicatt.it/) 4 00:00:18,000 --> 00:00:23,000 A. Musesti - M. Paolini 5 00:00:28,000 --> 00:00:31,000 Department of Mathematics and Physics 6 00:00:31,000 --> 00:00:34,000 Catholic University - Brescia 7 00:00:38,000 --> 00:00:42,000 Made with www.povray.org 8 00:00:55,000 --> 00:01:01,000 Penrose tiling is a NON-PERIODIC way to fill a plane 9 00:01:01,000 --> 00:01:07,000 using two tiles: the KITE (light tiles) and the DART (dark tiles) 10 00:01:07,000 --> 00:01:13,000 Actually the main feature of this tiling is that 11 00:01:13,000 --> 00:01:19,000 the two tiles are APERIODIC: they cannot be used to perform PERIODIC tilings 12 00:01:19,000 --> 00:01:23,000 Now we will see the meaning of APERIODIC TILING 13 00:01:23,000 --> 00:01:27,000 and we will have to introduce ADJACENCY CONSTRAINTS between the tiles 14 00:01:27,000 --> 00:01:30,000 in order to ensure that they are indeed aperiodic 15 00:01:30,000 --> 00:01:35,000 the making of darts and kites 16 00:01:36,000 --> 00:01:42,000 The two tiles, the 'kite' and the 'dart', can be built 17 00:01:42,000 --> 00:01:48,000 starting from a regular decagon with side of 1 and radius of 1.618 (the golden ratio) 18 00:01:48,000 --> 00:01:52,000 according to the rules described in the animation 19 00:01:55,000 --> 00:02:01,000 Surprisingly, we meet the ubiquitous GOLDEN RATIO 20 00:02:05,000 --> 00:02:10,000 this is the DART 21 00:02:13,000 --> 00:02:18,000 and this is the KITE 22 00:02:20,000 --> 00:02:25,000 periodic tessellation 23 00:02:25,000 --> 00:02:31,000 It is quite simple to fill the whole plane with the two tiles: 24 00:02:31,000 --> 00:02:37,000 by construction, they can be easily joined together 25 00:02:37,000 --> 00:02:43,000 forming a rhombus (with a side of 1.618...) 26 00:02:43,000 --> 00:02:49,000 But in this way we obtain a PERIODIC tessellation: 27 00:02:49,000 --> 00:02:55,000 as you can see, the tiles can be shifted in many directions 28 00:02:55,000 --> 00:03:01,000 without affecting the initial pattern. 29 00:03:05,000 --> 00:03:11,000 A periodic tessellation requires translations in at least two different directions 30 00:03:11,000 --> 00:03:17,000 with the initial pattern maintained. 31 00:03:25,000 --> 00:03:31,000 There are other symmetries for a rhomboidal tiling, e.g. rotations by 180 degrees 32 00:03:35,000 --> 00:03:41,000 by the way, rotations are not symmetries of our periodic tessellation 33 00:03:45,000 --> 00:03:51,000 indeed, the colors move and the initial pattern is not recovered 34 00:04:06,000 --> 00:04:11,000 Note that the symmetries have to move ALL the tessellation, not only a few tiles 35 00:04:11,000 --> 00:04:16,000 In this case we obtain the symmetry group named "cm" in crystallography 36 00:04:16,000 --> 00:04:20,000 We can obtain all possible symmetries by combining two translations 37 00:04:20,000 --> 00:04:24,000 and a reflection about a vertical axis 38 00:04:25,000 --> 00:04:30,000 Of course, other "generators" can be chosen 39 00:04:30,000 --> 00:04:35,000 A rotation of 180 degrees about a rhombus does not work! 40 00:04:35,000 --> 00:04:40,000 Unless we consider a tessellation where only rhombi are present, 41 00:04:40,000 --> 00:04:45,000 achieving the group "cmm" 42 00:04:55,000 --> 00:05:00,000 enforcing adjacency constraints 43 00:05:00,000 --> 00:05:06,000 In order to prevent periodic tiling, we can shape the tiles as in a jigsaw puzzle 44 00:05:06,000 --> 00:05:09,000 as you can see in the animation 45 00:05:12,000 --> 00:05:18,000 From now on, we will presume that the tiles are this shape, even if they will not be shown 46 00:05:24,000 --> 00:05:29,000 trial and error 47 00:05:30,000 --> 00:05:36,000 It is no longer possible to match kites and darts to obtain a rhombus, 48 00:05:36,000 --> 00:05:42,000 however it still seems possible (at least by hand) to fill the entire plane. 49 00:05:59,000 --> 00:06:05,000 How can we be sure that it is REALLY possible to fill the entire plane without holes? 50 00:06:07,000 --> 00:06:13,000 Well, we are going to describe an operational procedure which leaves no holes. 51 00:06:13,000 --> 00:06:18,000 breaking the tiles 52 00:06:18,000 --> 00:06:23,000 In the first step we SPLIT the tiles, obtaining two isosceles triangles 53 00:06:23,000 --> 00:06:29,000 KITE: two acute isosceles triangles 54 00:06:29,000 --> 00:06:35,000 DART: two obtuse isosceles triangles 55 00:06:38,000 --> 00:06:43,000 golden triangles 56 00:06:43,000 --> 00:06:49,000 The animation shows that the sides of the triangles are in the golden ratio 57 00:06:51,000 --> 00:06:57,000 Acute triangles: base = 1, side = 1.618... 58 00:07:03,000 --> 00:07:08,000 Obtuse triangles: side = 1, base = 1.618... 59 00:07:08,000 --> 00:07:13,000 subdivision 60 00:07:26,000 --> 00:07:32,000 The main idea is to split up the triangles into smaller copies of themselves 61 00:07:43,000 --> 00:07:47,000 Acute triangles: two acute triangles and one obtuse triangle 62 00:07:47,000 --> 00:07:55,000 Obtuse triangles: one obtuse triangle and one acute triangle 63 00:07:58,000 --> 00:08:03,000 deflation inflation 64 00:08:04,000 --> 00:08:07,000 Let us carry out the procedure... 65 00:08:07,000 --> 00:08:12,000 We begin with whatever initial configuration we want (the axiom) 66 00:08:12,000 --> 00:08:18,000 for instance, a decagon made up by five kites 67 00:08:19,000 --> 00:08:25,000 Then we apply the subdivision procedure on each triangle, obtaining a more refined subdivision 68 00:08:27,000 --> 00:08:33,000 Look: the new triangles pair together to form kites and darts 69 00:08:35,000 --> 00:08:39,000 Finally, we blow up all the tiles to the original size 70 00:08:39,000 --> 00:08:42,000 and repeat the procedure once more 71 00:08:43,000 --> 00:08:47,000 Now we also have to split some obtuse triangles 72 00:08:49,000 --> 00:08:55,000 The procedure complies with the adjacency constraints between the tiles 73 00:08:56,000 --> 00:08:59,000 we blow up the tiles again to the original size 74 00:09:00,000 --> 00:09:03,000 Third step... 75 00:09:20,000 --> 00:09:23,000 Fourth step... 76 00:09:33,000 --> 00:09:39,000 A central decagon exactly alike the initial configuration has appeared! 77 00:09:40,000 --> 00:09:43,000 Fifth step... 78 00:10:00,000 --> 00:10:03,000 Sixth step... 79 00:10:15,000 --> 00:10:18,000 Seventh step... 80 00:10:27,000 --> 00:10:30,000 Eighth step... 81 00:10:33,000 --> 00:10:39,000 We can go on, tiling a larger and larger part of the plane 82 00:10:44,000 --> 00:10:50,000 This is the result after nine steps 83 00:10:50,000 --> 00:10:56,000 In conclusion, we have a procedure which tiles the whole plane and respects the adjacency constaints. 84 00:10:59,000 --> 00:11:04,000 tessellation symmetries 85 00:11:05,000 --> 00:11:10,000 Given the initial configuration (a decagon made up by five kites), the resulting tiling 86 00:11:10,000 --> 00:11:16,000 will have a rotational symmetry (by 72 degrees) 87 00:11:22,000 --> 00:11:28,000 and also axial symmetries (reflections about five axes through the center) 88 00:11:28,000 --> 00:11:34,000 as suggested by some of the movements that you can see 89 00:11:40,000 --> 00:11:46,000 There are precisely ten symmetries, forming the dihedral group "D5" (symmetries of a pentagon) 90 00:11:48,000 --> 00:11:55,000 The group can be generated by a rotation of 72 degrees 91 00:11:55,000 --> 00:12:02,000 ...and a reflection about a suitable axis through the center 92 00:12:12,000 --> 00:12:17,000 birds... and reptiles 93 00:12:17,000 --> 00:12:23,000 We can modify the silhouette of the DART and the KITE, obtaining new tiles 94 00:12:23,000 --> 00:12:26,000 in the spirit of Escher's art 95 00:12:32,000 --> 00:12:35,000 We employ the new forms in the tessellation... 96 00:12:36,000 --> 00:12:42,000 The result has the same structure as the KITE and DART tiling 97 00:13:10,000 --> 00:13:16,000 Here is the result after five steps of deflation/inflation 98 00:13:16,000 --> 00:13:21,000 fat rhombi thin rhombi 99 00:13:21,000 --> 00:13:26,000 A remarkable option can be found by a suitable choice of two different rhombi 100 00:13:27,000 --> 00:13:31,000 However, there is a close relationship with the Kite and Dart, as you can see in the animation 101 00:13:32,000 --> 00:13:36,000 The rhombi are split into two isosceles triangles 102 00:13:36,000 --> 00:13:42,000 which are like those we have already seen, apart from the proportions 103 00:14:00,000 --> 00:14:06,000 We can perform a deflation/inflation procedure starting from a suitable subdivision of the golden triangles 104 00:14:06,000 --> 00:14:10,000 similar to the subdivision of Dart and Kite 105 00:14:10,000 --> 00:14:14,000 After six steps of the new procedure of deflation/inflation 106 00:14:14,000 --> 00:14:20,000 starting from a suitable initial configuration 107 00:14:20,000 --> 00:14:26,000 we obtain the tiling with rhombi... 108 00:14:55,000 --> 00:15:00,000 the original Penrose tiles 109 00:15:00,000 --> 00:15:06,000 Roger Penrose tried to split a large regular pentagon into six smaller pentagons 110 00:15:06,000 --> 00:15:10,000 scaled by the square of the inverse of the golden ratio 111 00:15:10,000 --> 00:15:14,000 Repeating the subdivision a few times, some holes remain 112 00:15:14,000 --> 00:15:18,000 Some of them have the shape of a regular pentagon, hence they can be included in the subdivision procedure 113 00:15:18,000 --> 00:15:24,000 Some other holes have different shapes: rhombus, crown, star 114 00:15:24,000 --> 00:15:30,000 The three different colors of the pentagons are useful to mark adjacency constraints 115 00:15:30,000 --> 00:15:33,000 that induce a non periodic tiling. 116 00:15:33,000 --> 00:15:39,000 In such a way we get a set of six tiles which are aperiodic 117 00:15:39,000 --> 00:15:48,000 To date it is not known if there is a single tile having this property of aperiodicity! 118 00:15:50,000 --> 00:15:54,000 Darts and kites in the sky of mathematics 119 00:15:54,000 --> 00:15:58,000 In 1974 Roger Penrose conceived the fascinating tilings named after him 120 00:15:58,000 --> 00:16:02,000 In them we find a mix of mathematics, geometry, physics and art in a surprising blend 121 00:16:02,000 --> 00:16:06,000 Making this short animation was a demanding yet enjoyable challenge.