Convex Pentagons that can Tessellate the Plane

Transcription

1 Casey Mann (University of Washington Bothell) Jennifer McLoud (University of Washington Bothell) David Von Derau (University of Washington Bothell) 2016 PNACP Meeting Portland Community College April 1, 2016

2 Tale of Contents I 1 Physical Motivation and Background 2

3 Why might physicists e interested in tilings? Tilings, or tessellations, can e thought of as geometric idealizations of structures with large scale symmetry. Tilings are arrangements of tiles (cells) that fill a region of the plane (space) without overlaps or gaps. A few physical examples: Crystals/Quasicrystals Theoretical Physics (string theory, rane theory) Self-Assemly

4 Crystals and Quasicrystals Crystals are those structures whose unit cells fill space in a periodic fashion. The symmetry is reflected in the diffraction images as periodic 2-dimensional tilings. Quasicrystals have long-range order, ut are not periodic. Their diffraction images have symmetries not possile in periodic tilings. Mention Luca Bindi. Diffraction image of a crystal Diffraction image of a quasicrystal

5 Brane tilings

6 Self-Assemly: The Toacco Mosaic Virus

7 Symmetry Tiling theory provides a mathematical framework for applications involving lots of things fitting together. There are many aspects of tilings that are studied mathematically, ut two of the ig aspects of interest are symmetry and complexity. A symmetry of a tiling is any rigid motion (of the plane or space) that maps the tiling onto itself. The collection of all symmetries of a tiling is called the symmetry group of the tiling. Symmetry properties of special interest are periodicity and tile-transitivity.

8 Periodic vs. Nonperiodic Tilings A periodic tiling A nonperiodic tiling

9 Symmetry and Monohedral Tilings A tiling in which all of the tiles are congruent to each other is a monohedral tiling. Tile-transitive tilings are highly symmetric monohedral tilings. The defining property of a tile-transitive tiling is that for any two tiles in the tiling, there is a symmetry of the tiling that takes the first tile to the second tile. All tile-transitive tilings are periodic, so in the case of monohedral tilings, tile-transitivity is a finer way to discuss symmetry. It turns out that there are exactly 81 types of tile-transitive tilings (up to topological equivalence) (Grunaum-Shephard)

10 Tile-Transitive Tiling A tile-transitive tiling. There exists a symmetry of the tiling taking the red tile to the lue tile.

11 Adjacency symols for tile-transitivity There is a great notation that serves as a sort of recipe for how a tile-transitive tiling tiles the plane. For example, consider the adjacency symol [ a + c + dc ; d + c + a ]. This descries how copies surround a centrally placed copy. c a d c c a d c c a d c c a d c c a d c c a d c c a d c

12 Convex polygons Triangle Quadrilateral Nonconvex Quadrilateral Pentagon Hexagon Septagon (7 sides) 13-gon (tridecagon) Convex means that the corners all point out. Polygons may e irregular (not all angles and sides are necessarily the same).

13 Convex polygons that tile the plane: triangles and quadrilaterals All triangles tile the plane All quadrilaterals tile the plane (convex or not)

14 Convex polygons that tile the plane: hexagons and n-gons with n 7 K. Reinhardt (1918) proved that there are exactly 3 kinds of hexagons that tile the plane. Ivan Niven (1977) proved that no convex n-gons with n 7 can tile the plane.

15 Convex hexagons that tile the plane a A F f E B e c C d D (a) A + B + C = 360 ; a = d () A + B + D = 360 ; a = d; c = e (c) A = C = E = 120 ; a = ; c = d; e = f Reinhardt s three types of hexagons that tile the plane

16 What Aout Pentagons? K. Reinhardt (1918) proved that there are exactly 5 types of pentagons that give rise to tile-transitive tilings. Reinhardt elieved that his 5 pentagons represented a complete classification of convex pentagons that tile the plane. Why did he elieve this? David Hilert, that s why!

17 Reinhardt Discovered Pentagon Types 1-5 Type 1 Type 2 A + B = 180 C + E = 180, a = d Type 3 A = C = D = 120 a =, d = c + e Type 4 A = C = 90 a =, c = d Type 5 C = 2A = 120 a =, c = d

18 Hilert s 18th Prolem David Hilert (1900) posed 23 prolems that shaped 20th century mathematics. Prolem #18 asked if there exists a non-tile-transitive 3-D shape that tiles space. Hilert elieved a 2-D analog does not exist. Reinhardt solved Hilert s 18th prolem in 1928 y discovering a 3-D anisohedral" tile. Reinhardt was a one-time assistant to Hilert, and like Hilert, he elieved that a 2-D analog does not exist. That is why he thought his 5 pentagons represented a complete list.

19 But maye there are more pentagons! Heinrich Heesch (1935) discovered a 2-D analog solution to Hilert s 18th prolem. That is, he discovered a 2-D tile that admits only non-tile-transitive tilings. This means there might e more pentagons eyond Reinhardt s Types 1-5!

20 Kershner finds 3 new types of pentagons! Roert Kershner (1968) discovered 3 new types of pentagons (Types 6-8). His pentagons give rise to only non-tile-transitive tilings. Kershner claims that Types 1-8 complete the classification. Type 6 C + E = 180 A = 2C a = = e, c = d Type 7 2B + C = 360 2D + A = 360 a = = c = d Type 8 2A + B = 360 2D + C = 360 a = = c = d

21 Open the flood gates! Richard James, III (1975), a computer scientist, after reading aout Kershner s work in Scientific American, discovers a new type of pentagon (Type 10), despite Kershner s claims! Marjorie Rice ( ), a homemaker in California, reads the same Scientific American article and proceeds to find 4 new types of pentagons over a period of two years. Type 10 E = 90, A + D = 180 2B D = 180, 2C + D = 360 a = e = + d Marjorie Rice in 1998

22 Marjorie Rice s Pentagons - Types 9, 11, 12, and 13 Type 9 2E + B = 360 2D + C = 360 a = = c = d Type 11 A = 90, C + E = 180 2B + C = 360 d = e = 2a + c stained glass window ased on an M. Rice s pentagon Type 12 A = 90, C + E = 180 2B + C = 360 2a = c + e = d Type 13 A = C = 90 2B = 2E = 360 D c = d, 2c = e Foyer at the MAA headquarters in Washington D.C.

23 M. Rice Artwork

24 Two more to go. Rolf Stein (1985), a graduate student working under Ludwig Danzer (noted tiling theorist), discovers a 14th type of convex pentagon that tiles the plane. Casey Mann, Jennifer McLoud, and David Von Derau (2015) find Type 15. Type 14 2B + C = 360 C + E = 180 2a = 2c = d = e Type 15 A = 150, B = 60, C = 135 D = 105, E = 90 a = c = e, = 2a

25 Types 6-15 are all i-lock transitive" While pentagon Types 6-15 do not admit any tile-transitive tilings, they do admit i-lock transitive tilings. A monohedral tiling y convex pentagons is i-lock transitive if the pentagons clump" into groups of i pentagons, and those clumps form a tile-transitive tiling.

26 f e d c f e d c Physical Motivation and Background Types 6-15 are all i-lock transitive" Example: Clusters of two Type 12 pentagons can tile the plane in a tile-transitive manner according to tile-transitive type 6: [a + + c + d + e + f + ; a + e c + f d ] Type 12 pentagons admit 2-lock transitive tilings. c f c e d d a f a a a e d e f c f c c f e d a e a a d a c d f e

27 Hypothetical Tile-Transitive Pentagons Our idea for searching for new pentagons was to create hypothetical" i-locks of made out of pentagons having a certain numer of flat nodes. We then checked what happened when we required these hypothetical locks to have a certain tile-transitivity type. A ig facet of this research was some theoretical cominatorial results we proved that we will not ore you with now. But, a ig upshot of the theory is that any i-lock has at most i flat nodes. This means there are only finitely many hypothetical i-locks to consider for each i!

28 All hypothetical 3-locks A shape laeled 5 has no flat nodes. A shape laeled 6 has one flat node. 7 s have two flat nodes, and 8 s have three flat nodes.

29 1 Choose a hypothetical lock and lael its corners and sides in every possile way. 2 Partition the oundary into 3, 4, 5, or 6 pieces in every possile way. C aa E e B c D d d E e A D B B C A a c c C A hypothetical 3-lock with laeled corners and sides. The white dots indicate the partition of the oundary. d D e π E

30 3 Choose a tile-transitivity type (from among the 81 possile types) for the hypothetical lock and apply the symol to the lock in every possile way. ε + E A E A A B a + C B C A A β + γ + D π D a + φ + φ + B C A A D B B C C E δ - E E A ε + β + δ + D D π D π D E C C The hypothetical lock laeled according to tile-transitivity type #5 with adjacency symol [a + + c + d + e + f +, a + e + d c + f + ]. γ -

31 4 Read off the required equations on the angles and sides forced y this picture. A A C B c c B C A A a a E E A e D π D d d e B C A A a a E E A e D B B c c C C E d e D π D a d D D π E C e C 2A + B + C = 360 2E + A = 360 2D = 360 2C + E = 360 2B + D = 360 = d = e a = e + d

32 5 Simplify equations and see if the resulting simplified equations match any known types. A = 60 B = 135 C = 105 D = 90 E = 150 a = 2 = 2d = 2e

33 Types 1-5 Type 1 Type 2 A + B = 180 C + E = 180, a = d Type 3 A = C = D = 120 a =, d = c + e Type 4 A = C = 90 a =, c = d Type 5 C = 2A = 120 a =, c = d

34 Types 6-10 Type 6 C + E = 180 A = 2C a = = e, c = d Type 7 2B + C = 360 2D + A = 360 a = = c = d Type 8 2A + B = 360 2D + C = 360 a = = c = d Type 9 2E + B = 360 2D + C = 360 a = = c = d Type 10 E = 90, A + D = 180 2B D = 180, 2C + D = 360 a = e = + d

35 Types Type 11 A = 90, C + E = 180 2B + C = 360 d = e = 2a + c Type 12 A = 90, C + E = 180 2B + C = 360 2a = c + e = d Type 13 A = C = 90 2B = 2E = 360 D c = d, 2c = e Type 14 2B + C = 360 C + E = 180 2a = 2c = d = e

36 Is this a known type? Solution: A = 60, B = 135, C = 105, D = 90, E = 150, a = 2 = 2d = 2e Oservation 1: There is not a pair of supplementary angles. Eliminates Types 1, 2, 4, 6, 10, 11, 12, 13, and 14 Oservation 2: There is not a 120 angle. Eliminates Types 3 and 5 Oservation 3: No four sides are equal. Eliminates Types 7, 8, and 9

37 6 Determine if the set of equations can actually form a convex pentagon. C d D e E c a B A Yes, we found type 15!

38 7 Make a pretty picture!

39 What else did we accomplish? What is left to do? Our theory along with our computer algorithm have verified that, among pentagons admitting 1-, 2-, 3-, and 4-lock transitive tilings, Types 1-15 are the only types possile. We will soon search through the pentagons that admit 5-lock transitive tiiings. It is likely that if there are any other types to e found, they will e i-lock transitive for some i 5. But, there is also the (remote) possiility that there exist so-called aperiodic pentagons, and our search would never detect such pentagons.

40 Are there any more pentagons out there? We don t know!