Properties of Tilings by Convex Pentagons

Transcription

1 eview Foma, 1, , 006 Popeties of Tilings by Convex Pentagons Teuhisa SUGIMOTO 1 * and Tohu OGAWA,3 1 The Institute of Statistical Mathematics, Minami-Azabu, Minato-ku, Tokyo , Japan (Emeitus Pofesso of Univesity of Tsukuba), Tennodai, Tsukuba-shi, Ibaaki , Japan 3 The Intedisciplinay Institute of Science, Technology and At, GanadoB-10, Kitahaa, Asaka-shi, Saitama , Japan * addess: sugimoto@ism.ac.jp (eceived Septembe 10, 005; Accepted Apil 6, 006) Keywod: Pentagon, Tile, Tiling, Patten, Tessellation Abstact. Let us conside an edge-to-edge and stongly balanced tiling of plane by pentagons. A node of valence s ( 3) in an edge-to-edge tiling is a point that is the common vetex of s tiles. Let W 1 be a finite closed disk satisfying the popety that the aveage valence of nodes in W 1 is nealy equal to 10/3. Then, let T denote the union of the set of pentagons meeting the bounday of W 1 but not contained in W 1 and the set of pentagons contained in W 1, and let V s denote the numbe of s-valent nodes in T. If the tiling in T is fomed of only 3- and k-valent nodes, then V 3 : V k 3k 10 : 1 whee k 4. On the othe hand, if the tiles in edge-to-edge tiling ae conguent convex pentagons, then at least two of the edges (of this conguent convex pentagon) ae of equal length. 1. Intoduction Tiling efes to coveage of the plane with polygons (tiles) without gaps o ovelapping. Especially a single conguent polygon that tiles the Euclidean plane is called a pototile o a polygonal tile, and the plane tiling with convex polygons has pimaily been studied in an attempt to exhaust all the conditions of pototile. The cuent state of knowledge in the tilings by conguent polygons can be summaized as follows. Any single tiangle and quadilateal, including concave quadilateals, is tileable (i.e., all pototiles), since the sums of the (inteio) angles of tiangles and quadilateals divide evenly. On the othe hand, tiling with othe convex polygons is not necessaily possible because of constaints on angles and edge-lengths. In the case of convex hexagons, pototiles can be categoized into thee types (BOLLOBÁS, 1963; GÜNBAUM and SHEPHAD, 1987; SUGIMOTO, 1999). Fo the convex polygons with seven o moe edges, no pototiles exist. Fo the convex pentagons, thee ae at pesent 14 types (see Figs. 1 and ), but it emains unpoven whethe this is the pefect list of such pentagons (KESHNE, 1968; GADNE, 1975; KLANE, 1981; GÜNBAUM and SHEPHAD, 1987; SCHATTSCHNEIDE, 1987; WELLS, 1991; SUGIMOTO, 1999; SUGIMOTO and OGAWA, 000, 003c, 005). As shown in Figs. 1 and, each of convex pentagonal tiles is defined by some conditions between lengths of edges and 113

2 114 T. SUGIMOTO and T. OGAWA Fig. 1. Convex pentagonal tiles of type 1 10.

3 Popeties of Tilings by Convex Pentagons 115 Fig.. Convex pentagonal tiles of type magnitudes of angles; but some degees of feedom emain. (Howeve, only the pentagonal tile of type 14 has one degee of feedom, that of size. Fo example, the exact value of C in pentagon of type 14 is cos 1 (( )/16) ad 69.3, and the values of angle B, D, and E can be obtained by C.) Then, unless a convex pentagonal tile is a new pototile, any convex pentagonal tile belongs to one o moe of 14 types. In the study of tiling the plane with conguent convex polygons, the case of pentagon is the only unsolved poblem. Then we believe that the poblem has yet to be appoached fom a new point of view. Fo example, the tilings themselves can be distinguished into two kinds by the connecting method: edge-to-edge tilings and non-edge-to-edge tilings. In edge-to-edge tilings, the vetices and edges of polygons coincide with the vetices and edges of the tiling. In non-edge-to-edge tilings, the vetices of polygons may contact the edges of adjoining polygons, that is, thee is no estiction on how adjoining polygons meet. As to the pentagonal tilings, though the edge-to-edge tilings ae still pentagonal even in topological point of view, the non-edge-to-edge tilings ae not. Howeve, the distinction between these two connecting methods is seldom done in the pevious studies of convex pentagonal tiling poblem. Ou inteest lies moe in edge-to-edge tiling since it is moe essential (OGAWA et al., 001; SUGIMOTO and OGAWA, 000, 003a, 003c, 005). Theefoe, thoughout the epot, we will conside only edge-to-edge tiling. Heeafte, as long as cautions ae unnecessay, an edge-to-edge tiling is witten simply a tiling. On the othe hand, so fa the classification and exhaustive studies of tilings ae not vey much noticed to the pesent. Howeve, fo convex pentagonal tiles, it will be impossible to expess the necessay and sufficient conditions fo identifying pototiles without classifying tilings. Theefoe, we should pay attention to the popeties of tilings. Fist, in this epot,

4 116 T. SUGIMOTO and T. OGAWA we show the popeties which become impotant and famous in tiling with polygons (see Poposition in Subsection.). Next, the popeties about pentagonal tiling ae shown (see Theoem 1 in Subsection.3, Theoem in Subsection.4, and Theoem 4 in Section 3) and the classification of convex pentagons with equal edges is also shown (see Table in Section 3).. Popeties of Nodes in Stongly Balanced Tilings.1. Definition A node of valence κ in a tiling is a point that is the common vetex of κ polygons (tiles). That is, in this epot, heeafte a vetex of a tiling is called a node. Note that the valence κ of a node is at least thee. Given a tiling, the fist coona of a tile is the set of all tiles that have a common bounday point with that tile (including the oiginal tile itself). Two tiles ae called adjacent if they have an edge in common, and then each is called an adjacent of the othe. Theefoe, in an edge-to-edge tiling by polygons, the numbe of adjacents of a polygon is equal to the numbe of edges of that. Then, a polygon with h edges (and theefoe h vetices) will be called a h-gon. A tiling by polygons is called nomal if thee ae positive numbes and such that any polygon contains a cetain disk of adius and is contained in a cetain disk of adius. Given a nomal tiling by polygons, let W be a closed disk of adius ρ (>0) on the plane. Then, let F 1 and F denote the set of the polygons contained in W and the set of polygons meeting the bounday of W but not contained in W, espectively. Hee, define F := F 1 F (i.e., F is the set of polygons geneated by W). We denote by P(F) the numbe of polygons in F. In addition, let E(F) and N(F) denote the numbe of edges and nodes in F, espectively. The tiling is balanced if it is nomal and satisfies the following condition: the limits lim NF PF EF and lim 1 PF () exist and ae finite (GÜNBAUM and SHEPHAD, 1987). Now, let P h (F) and N κ (F) be the numbe of polygons with h adjacents in F and the numbe of κ-valent nodes in F, espectively. The tiling is stongly balanced if it is nomal and satisfies the following condition: all the limits lim Ph F Nκ F and lim PF PF exist (GÜNBAUM and SHEPHAD, 1987). Note that, in the following Subsection., the stongly balanced condition is not needed.

5 Popeties of Tilings by Convex Pentagons Aveage valence of nodes in balanced tilings by polygons with h edges Given a nomal tiling of plane by polygons and let P(F ) denote the numbe of polygons in F. Since is nomal, then (see GÜNBAUM and SHEPHAD, 1987; BAGINA, 004): PF lim 0. 3 PF = () Heeafte, let N(W) be the numbe of nodes of the tiling contained in W. Lemma 1. Given a nomal tiling, if the adius ρ of the disk W tends to infinity then = NW lim 1. 4 NF Poof. Fist, we conside the uppe bound fo the numbe of edges in a polygon in. Let p 0 be any polygon. We suppose that m polygons contact p 0. So, the fist coona of p 0 in tiling is contained in some disk of adius 3. Since any polygon of contains a disk of adius and these disks do not ovelap, we have ( m+ 1) π < π ( 3) o m< 9 1. Theefoe the uppe bound fo the numbe of edges in a polygon is 9( / ) 1. Thus, fo the numbe N(F) N(W) we have: NF NW < 9 PF ( ). ( 5) Next, we conside the uppe bound of valence of nodes in. We take a node G of valence κ, daw a cicle of adius centeed at a node G. It contains fully the sta of node G in the tiling. Since this sta contains κ polygons, we get Hence fo any valence κ then: κ π < π( ). Now we will pove that κ < 4. ( 6)

6 118 T. SUGIMOTO and T. OGAWA NF > 3 PF ( 7 4 ). Let u i denote the numbe of vetices in the i-th polygon in F (i = 1,..., P(F)). Since u i 3 fo any i {1,..., P(F)}, then PF ui 3 P( F). i= 1 On the othe hand, fom (6) and N(F) (the numbe of nodes of the tiling in F), at most 4( / ) N(F) vetices of polygons exist in F. Theefoe, fo the numbe of vetices of polygons fom F, we have PF 4 NF > ui 3 PF. Theefoe, we obtain the inequality (7). Given and, fom (5) and (7), we have NW = 1 NF NF NW NF i= 1 PF 4 PF PF = PF < Thus, if ρ, the elation (4) is deived fom (3) and (8). Now, let K(W) and K(F) be the sum of valences of N(W) nodes of the tiling contained in W and the sum of valences of N(F) nodes of the tiling in F, espectively; NW KW: κ and KF: N( F) = = j j = 1 j = 1 whee κ j is the valence of j-th nodes of the tiling in F (j = 1,,..., N(W),..., N(F)). Lemma. Given a nomal tiling, if the adius ρ of the disk W tends to infinity then κ j KW lim 1. 9 KF () = Poof. Fom (6) and the fact that the valence of a node is at least thee, we have 4 > κ j 3.

7 Popeties of Tilings by Convex Pentagons 119 Theefoe: 4 NF KF 3 NF > and 4 ( NF NW )> ( KF KW ) 3 ( NF NW ). Hence, we have 4 ( NF NW ) KF KW > 3 NF KF 3 NF NW > 4 NF o 4 3 KW 3 NW 1 10 > KF > 4. NF NW 1 1 NF Given and, if ρ, the elation (9) follows fom (10) and Lemma 1. Fom the statement ( a nomal tiling in which evey tile has the same numbe of adjacents is balanced ) in GÜNBAUM and SHEPHAD (1987) p. 133, a nomal tiling by h-gons (h 3) is balanced. (Note that, in this epot, we conside the edge-to-edge tiling.) Heeafte, in this Subsection, we will conside a balanced tiling h of plane by h-gons. Then, we deive the limits lim EF PF = h/, lim NF PF = (h/) 1, and ( ) lim Ph F P F = 1 (i.e., the values of limits in (1) and in one of () ae able to be obtained). The tiling h satisfies Lemmas 1 and. Hee, denote κ := lim ( KF NF ), and it is called the aveage valence of nodes in h. Poposition. Given a balanced tiling h by h-gons (h 3), the κ is found as follows: h h κ =. ( 11) Poof. Fix a disk W with lage adius ρ in h. Then the sum of angles of P(F) h-gons in

8 10 T. SUGIMOTO and T. OGAWA a set F of h-gons geneated by W and the sum of angles of P(F) P(F ) h-gons, of the tiling contained in W, ae (h )π P(F) and (h )π (P(F) P(F )), espectively. Since the total sum of angles meeting at the N(W) nodes, of the tiling contained in W, is π N(W), we have o > > h π PF π NW h π PF PF 1 > Theefoe, due to (3), as ρ, NW 1 ( ) > PF h P F PF. NW 1 1 ( ). h P F On the othe hand, the total numbe of edges fo P(F) h-gons in F and the total numbe of edges fo P(F) P(F ) h-gons, of the tiling contained in W, ae h P(F) and h (P(F) P(F )), espectively. Let κ W denote the aveage valence of nodes contained in W; κ W := K(W)/N(W). Then we have o h PF NW h PF PF κ W κ 1 W NW 1 h P( F) Theefoe, as ρ, fom (3), we have PF. PF κ W NW h P F 1. ( 13) Fom Lemmas 1 and the limits κ = exists and is finite. KF κ κ = KW = W NW lim lim lim = lim W ( 14) NF NW NW ρ ρ ρ ρ

9 Popeties of Tilings by Convex Pentagons 11 Table 1. Aveage valence κ of nodes in balanced tiling by h-gons. h-gon κ (aveage valence of nodes) Tiangles (h = 3) 6 Quadilateals (h = 4) 4 Pentagons (h = 5) 10/3 Hexagons (h = 6) 3 Thus, the elation (11) follows fom (1), (13), and (14). Fom Poposition we have the aveage valences fo the cases of tiangles (h = 3), of quadilateals (h = 4), of pentagons (h = 5), and of hexagons (h = 6) as in Table 1. These esults fo h = 3, 4, and 6 ae well known. (Note that, in ode to pove othe theoems in this epot, we poved the Poposition exactly accoding to the definition of this eseach.) Fo h = 5, the aveage valence is 10/ Since the aveage valence is not an intege then thee must be nodes with valences smalle than But the smallest valence at node is thee. So in any nomal tiling of plane by pentagon thee must be nodes with valence 3. In addition, fo the same eason (the aveage numbe is not an intege), thee ae no tilings with all nodes of the same valence. If Eq. (11) is applied to the convex polygons with seven o moe edges (i.e., h 7), thei tilings ae impossible, since the aveage valence of nodes is smalle than 3 when h 7. Theefoe, in the case of convex polygons with seven o moe edges, we can undestand that no balanced tiling exists..3. ates of 3- and k-valent nodes in stongly balanced tilings by pentagons The following aguments ae elevant to stongly balanced tilings by pentagons. Then, evey stongly balanced tiling is necessaily balanced. Let 5 be a stongly balanced tiling of plane by pentagons. Heeafte, we denote N s (s 3) as the numbe of s-valent nodes in F of pentagons geneated by a closed disk W on a pentagonal tiling 5. Lemma 3. Given a pentagonal tiling 5, if 5 is fomed of only 3- and k-valent nodes, then lim N N k = k whee k 4. Poof. When the adius ρ of the disk W tends to infinity (i.e., ρ ), the limit lim N 3 N k exists since 5 is stongly balanced. The total numbe of nodes in F is N 3 + N k and the sum of valences of nodes in F is 3N 3 + k N k. Theefoe, fom Poposition, we have

10 1 T. SUGIMOTO and T. OGAWA N3 3 + k 3N3 + k Nk Nk 10 lim = lim. N3 + N N k N ρ k = ( 16) Hence, fom (16), we obtain (15). Hee, we conside a closed disk W 1 of finite adius ρ 1 ( << ρ 1 < ) on the stongly balanced pentagonal tiling 5. In addition, we assume that W 1 satisfies the popety that the aveage valence of nodes in W 1 is nealy equal to 10/3. Then, let T denote the finite set of pentagons geneated by W 1. Theefoe, the each numbe of pentagons and nodes in T is finite. Note that the set T needs to be finite since we discuss about the numbe of nodes in a tiling such as Theoem 1. Heeafte, we denote V s (s 3) as the numbe of s-valent nodes in T. Theoem 1. If the tiling in T is fomed of only 3- and k-valent nodes, then whee k 4. Poof. It is clea fom Lemma 3. V 3 : V k 3k 10 : 1 (17) Coollay 1. If the tiling in T is fomed of only 3- and 4-valent nodes, then V 3 : V 4 : 1. (18) Coollay. If the tiling in T is fomed of only 3- and 6-valent nodes, then V 3 : V 6 8 : 1. (19) Poof. When k = 4 and 6 in (17) of Theoem 1, (18) and (19) ae obtained, espectively. Now, Coollaies 1 and can be applied to tilings with conguent convex pentagons. If a tiling is edge-to-edge, it satisfies Theoem 1. As a esult, fo convex pentagonal tiles in Figs. 1 and, we find that tilings of type 4, 6, 7, 8, and 9 satisfy (18), and tiling of type 5 satisfies (19) (SUGIMOTO, 1999; SUGIMOTO and OGAWA, 003c, 004). Note that, although the tilings of type 1 and in the list of 14 types ae geneally non-edge-to-edge (see Fig. 1), tilings by convex pentagonal tiles which belong to type 1 and can be edgeto-edge in special cases. Fo example, a convex pentagonal tile with A + B + C = 360, a = d belongs to type 1 accoding to the pesent classification scheme has an edge-to-edge tiling (see Fig. 3(a)). Similaly, a convex pentagonal tile satisfying the condition A + B + D = 360, a = d, c = e is of type, and has an edge-to-edge tiling (see Fig. 3(b)). Then, the tilings by these two pentagons satisfy (18) togethe. Like these examples, when the tilings by conguent convex pentagonal tiles which belong to type 1 o ae edge-toedge, in the ange which we know, the tilings satisfy (18).

11 Popeties of Tilings by Convex Pentagons 13 Fig. 3. Tilings by conguent convex pentagons. (a) Convex pentagonal tiles belong to type 1. (b) Convex pentagonal tiles belong to type. Among 14 convex pentagonal tilings in Figs. 1 and, the tiling which satisfies (17) is not found in the case of k = 5 o k emaks Fo the tilings (especially peiodic tilings) by conguent convex pentagons, the popeties κ = 10/3 and (17) should be ealized not only globally but also locally. That is, the popeties should be included in the fundamental egions of thei peiodic tilings. Hee, let us investigate the popeties of tilings by conguent convex pentagons fom the point of ou analysis in pevious subsections. Fist, the tilings of type 4, 6, 7, 8, and 9 (see Fig. 1) satisfy (18). They have thee kinds of nodes. Especially, the tilings of type 6, 7, 8, and 9 ae fomed of two kinds of 3-valent nodes and one kind of 4-valent node (SUGIMOTO, 1999; SUGIMOTO and OGAWA, 003a, 003b, 004). Fo example, the tiling of type 6 in Fig. 1 is fomed of thee kinds of nodes satisfying A + B + D = 360, E + A = 360, and C + B + D = 360. In the thee (= + 1) kinds of nodes, each of the five vetices A, B, C, D, and E of pentagon appeas twice; i.e., the total numbe of vetices is 10 (= 5 = ). As a esult, we see the elations 10/3 and (two kinds of 3- valent nodes) : (one kind of 4-valent node) = : 1. In addition, fo the pentagonal tilings that have two-kinds of 3-valent nodes and one-kind of 4-valent node, the even numbe of pentagonal tiles ae necessay in ode to fom the fundamental egion since each of the five vetices of pentagon is used twice in the thee kinds of nodes. On the othe hand, the tiling of type 4 in Fig. 1 is fomed of thee kinds of nodes satisfying A + B + D = 360, 4C = 360, and 4E = 360. That is, thee ae one-kind of 3-valent nodes and two-kind of 4-valent nodes in the tiling. When thee ae fou vetices C and fou vetices E (i.e., 4C = 360 and 4E = 360 ) in the tiling, the numbe of vetices A, B, and D also need to be fou, espectively. Theefoe, one 4C = 360 and one 4E = 360 have to coespond to fou A + B + D = 360. Hence, (total fou 3-valent nodes) : (two kinds of 4-valent nodes) = 4 : (1 + 1) = : 1, and six (= ) nodes ae constucted at 0 (= 5 4 =

12 14 T. SUGIMOTO and T. OGAWA ) vetices of pentagons; namely 0/6 = 10/3. Thus, fo the tiling of types 4 in Fig. 1, the fundamental egion is fomed of the fou pentagonal tiles. Next, the tiling of type 5 satisfies (19) and is fomed of thee kinds of nodes satisfying B + D + E = 360, 3A = 360, and 6C = 360 (see Fig. 1). In ode to fom the fundamental egion of the tiling of type 5, six pentagonal tiles ae necessay. Theefoe, in the combinations by using the node conditions B + D + E = 360, 3A = 360, and 6C = 360, each of the five vetices A, B, C, D, and E of pentagon has to appea six times. Thus, six B + D + E = 360 and two 3A = 360 have to coespond to one 6C = 360 ; i.e., (total eight 3-valent nodes) : (one kind of 6-valent node) = (6 + ) : 1 = 8 : 1. Futhemoe, since nine (= ) nodes ae constucted at 30 (= 5 6 = ) vetices of pentagons, we see the elation 30/9 = 10/3. Now, based on the popeties that ae locally ealized, we conside the cases that the tiling in T is fomed of the nodes of thee o moe kinds of valence. Note that the 3-valent nodes in T exist necessaily. If the tiling in T is fomed of only 3, 4, and 5-valent nodes, then V 3 : V 4 : V 5 (x + 5y) : x : y, (0) whee x = 1,, 3,... and y = 1,, 3,.... We will pove this elation late. The total numbe of pentagonal vetices on nodes in T is (3V 3 + 4V 4 + 5V 5 ), and the total numbe of nodes in T is (V 3 + V 4 + V 5 ). Since T is the finite set of pentagons geneated by W 1 and the aveage valence of node in W 1 is nealy equal to 10/3, (3V 3 + 4V 4 + 5V 5 )/(V 3 + V 4 + V 5 ) 10/3. Theefoe, V 3 is nealy equal to x + 5y fo V 4 = x and V 5 = y. Hee, we conside the concete chaacte of tiling that satisfies (0). Fo example, if it is possible that the pentagonal tiling satisfies V 3 : V 4 : V 5 1 : 1 :, the pentagons of numbe of the multiple of 10 ae necessay in ode to fom the fundamental egion. It is because that, when each of the five vetices A, B, C, D, and E of pentagon appea ten times, 15 (= ) nodes ae constucted at 50 (= 5 10 = ) vetices of pentagons (i.e., 50/15 = 10/3). Fom simila consideation, we can obtain the following elations. If the tiling in T is fomed of only 3-, 4-, and 6-valent nodes, then V 3 : V 4 : V 6 (x + 8y) : x : y, (1) whee x = 1,, 3,... and y = 1,, 3,.... If the tiling in T is fomed of only 3-, 5-, and 6-valent nodes, then V 3 : V 5 : V 6 (5x + 8y) : x : y, () whee x = 1,, 3,... and y = 1,, 3,.... If the tiling in T is fomed of only 3-, 4-, 5-, and 6-valent nodes, then V 3 : V 4 : V 5 : V 6 (x + 5y + 8z) : x : y : z, (3) whee x = 1,, 3,..., y = 1,, 3,..., and z = 1,, 3,....

13 Popeties of Tilings by Convex Pentagons 15 Fom the above consideation, we find the following Theoem. Theoem. If the valence numbe of all nodes in T is finite, then V3 3k 10 V k. 4 k 4 Poof. The total numbe of pentagonal vetices on nodes in T is 3V 3 + numbe of nodes in T is V 3 + to 10/3, V k k 4 k 4 k V k, and the total. Since the aveage valence of node in T is nealy equal 3V 3 V + k V 3 + k 4 V k k 4 k 10. ( 5) 3 Hence, fom (5), we obtain (4). Besides the elations (0) (3), it is possible to seach fo simila elations. Howeve, in the pentagonal tilings of 14 types in Figs. 1 and, we see that thee ae only tilings which satisfy (18) o (19). 3. Classification of Convex Pentagons with Equal Edges and Popeties of Thei Tiling Pentagons can be categoized by the numbe of equal-length edges and thei positions, fom figues with five unequal edges to those with five equal edges. Hee, the edge-lengths ae designated symbolically in anticlockwise ode, with identical symbols fo edges of identical lengths, and desciptions of conguent shapes with diffeent stating points o mio-eflections ae excluded. Beginning with equilateal pentagons, followed by those with fou equal-length edges, etc., thee ae a total of 1 unique combinations (see Table ) (SUGIMOTO, 1999; SUGIMOTO and OGAWA, 000, 003a, 003c, 005). Fo example, the combination [11111] in Table is the pentagon with all the identical edge-lengths; i.e., it is equilateal pentagon. On othe hand, the combinations [111] and [111] in Table ae the pentagons that have the edge-lengths of two kinds, but the aangements of five edge-lengths ae diffeent. Hee, we summaize the study of tilings by conguent convex pentagons with combination [11111] (i.e., equilateal convex pentagons) (HISCHHON and HUNT, 1985; SCHATTSCHNEIDE, 1987; BAGINA, 004). Hischhon and Hunt gave the following theoem. Theoem 3 (HISCHHON and HUNT, 1985). An equilateal convex pentagon tiles the plane if and only if it has two angles adding to 180, o it is the unique equilateal convex pentagon with angles A, B, C, D, E satisfying B + C = D + A = E + A + C = 360 (A 89.6, B , C 70.88, D , E ).

14 16 T. SUGIMOTO and T. OGAWA Table. Twelve combination of edges of pentagon. Edges Combination Example Equilateal [11111] a = b = c = d = e Two kinds [1111] a = b = c = d e [111] a = b = c d = e [111] a = b = d c = e Thee kinds [1113] a = b = c, d e, d a e [1113] a = b = d c e, a e [113] e a = b c = d e [113] d a = b c = e d [113] e a = c b = d e Fou kinds [1134] a = b c d e, c e, a d, a e [1134] a = c, b d e, b e, a b, a d, a e Five kinds [1345] a b, a c, a d, a e, b c, b d, b e, c d, c e, d e Fig. 4. Pentagons P 1, P, and P 3. Theefoe, the tiles of tilings by conguent convex pentagons with combination [11111] belong to type 1,, o 7. On the othe hand, we give the following Theoem 4. Theoem 4. If the tiles in tiling ae conguent convex pentagons, then at least two of the edges (of this conguent convex pentagon) ae of equal length. Poof. Given a tiling c by convex pentagons, let P 1 be the pentagon of edges EA = a, AB = b, BC = c, CD = d, and DE = e (see Fig. 4). Since the nodes of valence 3 ae suely necessay in the pentagonal tiling, we suppose that the vetex A of P 1 exists on a 3-valent node. Then, let P and P 3 be the pentagons which shae edges EA and AB, espectively. In addition, we denote AF as the common edge of P and P 3. If all pentagons in c ae conguent, the edge AF in P is equal to b o e, since P 1 and

15 Popeties of Tilings by Convex Pentagons 17 P ae adjacent with thei common edge EA. Similaly, if all pentagons in c ae conguent, the edge AF in P is equal to a o c, since P 1 and P have thei common edge AB. Theefoe, any 3-valent node of tiling by conguent convex pentagons with combination [1345] can not exist. Thus, the tiling by conguent convex pentagons with combination [1345] is impossible. On the othe hand, the tiling by conguent convex pentagon with combination [1134] exists. Fo example, see the tiling by conguent convex pentagon with A + B + C = 360, a = d in Fig. 3(a). 4. Conclusion In the convex pentagonal tiling poblem, it is impotant to conside popeties of tilings and tiles both. Fist, when a tiling is edge-to-edge and stongly balanced, we obseve the numbe and kinds of nodes in tiling and gave Lemma 3 and Theoem 1 (see Subsection.3). Next, fo tiles, we find that the pentagons can be classified into 1 kinds by the numbe of equal-length edges and thei positions (see Table ) (SUGIMOTO, 1999; SUGIMOTO and OGAWA, 000, 003a, 003c, 005). Then, we see that the tiling by conguent convex pentagon is impossible when all edges of convex pentagon ae of diffeent length (see Theoem 4) (SUGIMOTO and OGAWA, 003c). On the othe hand, it is known that the equilateal convex pentagons which can tile the plane have to belong to type 1,, o 7 in the pesent list. Among 1 cases of pentagons of Table, the two cases wee solved. But, the investigations about othe 10 cases have not been completely finished yet. The popeties of nodes in pentagonal tiling have not been hadly noticed o fomally discussed until now. That is, befoe this epot, thee wee no pevious epots descibing the concete nodes popeties (Lemma 3, Theoem 1 and, and Coollay 1 and ). In addition, the elations between the combinations of pentagonal edges and thei pentagonal tilings had not been discussed in as much detail. Theoem 4 also had not been shown fomally until now. Hence, fo the fist time, we have shown and poven thei popeties explicitly. The convex pentagonal tilings poblem has yet to be fully appoached scientifically. The solution to this poblem equies a systematic appoach. Thus, although ou esults may be elementay, we asset that the popeties in this epot ae impotant in ode to conside the convex pentagonal tiling poblem. We actually tackled the convex pentagonal tiling poblem fom the popeties achieved in this epot. Specifically, fo the tilings by pentagons with combination [1111] which ae fomed of two kinds of 3-valent nodes (including the case when the two kinds ae identical) and one kind of 4-valent node (i.e., the tilings satisfy (18)), we investigated in SUGIMOTO and OGAWA (003a, 003b, 004, 005). The esults of SUGIMOTO and OGAWA (003a, 003b, 004, 005) ae summaized as follows. Accoding to the pesent classification (see Figs. 1 and ), the convex pentagonal tiles which ae diectly yielded by ou investigation belong to one o moe of type 1,, 4, 7, 8, o 9. Theefoe, as mentioned in the intoduction, a new pototile was not found in SUGIMOTO and OGAWA (003a, 003b, 004, 005). Howeve, ou investigation on the basis of the popeties shown in this epot should be applicable to ceate a pefect list of convex pentagonal tiles. Incidentally, we found some new convex pentagonal tilings in SUGIMOTO and OGAWA (004).

16 18 T. SUGIMOTO and T. OGAWA The authos would like to thank Pofesso N. Dolbilin, Steklov Math. Inst., fo his mathematical helpful comments and guidance, and Pofesso H. Maehaa, Univ. of yukyu, and Pofesso M. Tanemua, ISM, fo helpful suggestions and obsevations and a citical eading of the manuscipt. The comments of the efeee also helped to claify the aticle. The eseach was suppoted by the Gant-in-Aid fo Scientific eseach (the Gant-in-Aid fo JSPS Fellows) fom the Ministy of Education, Cultue, Spots, Science, and Technology (MEXT) of Japan. EFEENCES BAGINA, O. (004) Tiling the plane with conguent equilateal convex pentagons, Jounal of Combinatoial Theoy, Seies A, 105, 1 3. BOLLOBÁS, B. (1963) Filling the plane with conguent convex hexagons without ovelapping, Annales Univesitatis Scientaum Budapestinensis, GADNE, M. (1975) On tessellating the plane with convex polygon tiles, Scientific Ameican, July, GÜNBAUM, B. and SHEPHAD, G. C. (1987) Tiling and Pattens, W.H. Feeman and Company, New Yok, pp (Chapte 1), pp (Chapte 3) and pp (Chapte 9). HISCHHON, M. D. and HUNT, D. C. (1985) Equilateal convex pentagons which tile the plane, Jounal of Combinatoial Theoy, Seies A, 39, KESHNE,. B. (1968) On paving the plane, Ameican Mathematical Monthly, 75, KLANE, D. (ed.) (1981) The Mathematical Gadne: In Paise of Amateus, Wadswoth Intenational, Belmont, pp OGAWA, T., WATANABE, Y., TESHIMA, Y. and SUGIMOTO, T. (001) Tiling, packing and tessellation, in SYMMETY 000 Pat 1 (eds. I. Hagittai and T. C. Lauent), pp , Potland Pess Ltd, London. EINHADT, K. (1918) Übe die Zelegung de Ebene in Polygone, Dissetation, Univesität Fankfut. SCHATTSCHNEIDE, D. (1987) Tiling the plane with conguent pentagons, Mathematics Magazine, 51, SUGIMOTO, T. (1999) Poblem of tessellating convex pentagons, Maste Thesis, Univesity of Tsukuba (in Japanese). SUGIMOTO, T. and OGAWA, T. (000) Tiling poblem of convex pentagon, Foma, 15, SUGIMOTO, T. and OGAWA, T. (003a) Systematic study of tessellating convex pentagons and thei tiling pattens, I Convex pentagons with fou edges of the equal length. 1: The plan and the fist step of appoach on the whole poblem, Bulletin of the Society fo Science on Fom, 18, (in Japanese). SUGIMOTO, T. and OGAWA, T. (003b) Systematic study of tessellating convex pentagons and thei tiling pattens, II Convex pentagons with fou edges of the equal length. : Exhaustion of the tessellating pentagons within the simplest categoy of vetex conditions, Bulletin of the Society fo Science on Fom, 18, (in Japanese). SUGIMOTO, T. and OGAWA, T. (003c) Chaacteistic fo tilings of tessellating convex pentagons, eseach Memoandum, ISM, 900. SUGIMOTO, T. and OGAWA, T. (004) Systematic study of tessellating convex pentagons and thei tiling pattens, III Convex pentagons with fou edges of the equal length. 3: Tilings unde the vetical concentation conditions, Bulletin of the Society fo Science on Fom, 19, (in Japanese). SUGIMOTO, T. and OGAWA, T. (005) Systematic study of convex pentagonal tilings, I: Case of convex pentagons with fou equal-length edges, Foma, 0, WELLS, D. (1991) The Penguin Dictionay of Cuious and Inteesting Geomety, Penguin Books, London, pp