Convex pentagons and convex hexagons that can form rotationally symmetric tilings
CConvex pentagons and convex hexagons that can formrotationally symmetric tilings
Teruhisa SUGIMOTO , The Interdisciplinary Institute of Science, Technology and Art Japan Tessellation Design AssociationE-mail: [email protected]
Abstract
In this study, the properties of convex hexagons that can generate rotationally symmet-ric edge-to-edge tilings are discussed. Since the convex hexagons are equilateral convexparallelohexagons, convex pentagons generated by bisecting the hexagons can form ro-tationally symmetric non-edge-to-edge tilings. In addition, under certain circumstances,tiling-like patterns with an equilateral convex polygonal hole at the center can be formedusing these convex hexagons or pentagons.
Keywords: pentagon, hexagon, tiling, rotationally symmetry, monohedral
In [3], Klaassen has demonstrated that there exist countless rotationally symmetric non-edge-to-edge tilings with convex pentagonal tiles. , The convex pentagonal tiles are consideredto be equivalent to bisecting equilateral convex parallelohexagons, which are hexagons wherethe opposite edges are parallel and equal in length. Thus, there exist countless rotationallysymmetric edge-to-edge tilings with convex hexagonal tiles.Figure 1 shows a five-fold rotationally symmetric edge-to-edge tiling by a convex hexagonaltile (equilateral convex parallelohexagon) that satisfies the conditions, “ A = D = 72 ◦ , B = C = E = F = 144 ◦ , a = b = c = d = e = f .” Figure 2 shows examples of five-fold rotation-ally symmetric non-edge-to-edge tilings with convex pentagonal tiles generated by bisectingthe equilateral convex parallelohexagon in Figure 1. Since the equilateral convex parallelo-hexagons have two-fold rotational symmetry, the number of ways to form convex pentagonsgenerated by bisecting the convex hexagons (the dividing line needs to be a straight line thatpasses through the rotational center of the convex hexagon and intersects the opposite edge)is countless. The bisecting method can be divided into three cases: Case (i) the dividingline intersects edges c and f , as shown in Figures 2(a) and 2(b); Case (ii) the dividing lineintersects edges a and d , as shown in Figure 2(c); Case (iii) the dividing line intersects the A tiling (or tessellation ) of the plane is a collection of sets that are called tiles, which covers a planewithout gaps and overlaps, except for the boundaries of the tiles. The term “tile” refers to a topological disk,whose boundary is a simple closed curve. If all the tiles in a tiling are of the same size and shape, then thetiling is monohedral [1, 6]. In this paper, a polygon that admits a monohedral tiling is called a polygonaltile [4]. Note that, in monohedral tiling, it admits the use of reflected tiles. A tiling by convex polygons is edge-to-edge if any two convex polygons in a tiling are either disjoint orshare one vertex or an entire edge in common. Then other case is non-edge-to-edge [1, 4]. a r X i v : . [ m a t h . M G ] M a y onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure. Rotationally symmetric tiling with D symmetry that formed by a convex hexagonal tilewith D symmetry (Note that the gray area in the figure is used to clearly depict the structure) edges b and e , as shown in Figure 2(d). If the dividing line is selected such that the oppo-site vertices of the equilateral convex parallelohexagons are connected, the parallelohexagonscontain two congruent convex quadrangles. In such cases, as shown in Figure 3, there existfive-fold rotationally symmetric edge-to-edge tilings with convex quadrangles.The equilateral convex parallelohexagons that satisfy “ A = D (cid:54) = B = C = E = F, a = b = c = d = e = f ,” as demonstrated in Figure 1, have two-fold rotational symmetry and twoaxes of reflection symmetry passing through the center of the rotational symmetry (hereafter,this property is described as D symmetry ). Therefore, the parallelohexagon and reflectedparallelohexagon have identical outlines. (In the parallelohexagon with D symmetry, Cases(ii) and (iii) can be regarded as having a reversible relationship.) If two convex pentagons aregenerated in the equilateral convex parallelohexagon with D symmetry, which is bisectedby a dividing line that does not overlap with the axis of reflection symmetry, as shown inFigures 2(b), 2(c), and 2(d), the reflected parallelohexagon has the same outline. However, thearrangement of the inner convex pentagons is different. By using this property, the reflectedconvex pentagons can be freely incorporated into the tiling. Figure 4 shows a random tiling “ D ” is based on the Schoenflies notation for symmetry in a two-dimensional point group [7, 8]. “ D n ”represents an n -fold rotation axes with n reflection symmetry axes. The notation for symmetry is based onthat presented in [3]. onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure. Five-fold rotationally symmetric tilings with convex pentagons based on the five-fold rotationally symmetric tiling structure of a convex pentagonal tilecreated using this property.In this paper, the properties of convex hexagons and pentagons that can generate ro-tationally symmetrical tilings presented by Klaassen are explored. The convex hexagonsand pentagons can form rotationally symmetric tilings and rotationally symmetric tiling-likepatterns with an equilateral convex polygonal hole at the center. Note that the tiling-likepatterns are not considered tilings due to the presence of a gap, but are simply called tilingsin this paper. Herein, the various types of convex hexagons and pentagons are introducedand explored.
In [3], the theorem and proof that countless non-edge-to-edge tilings can be formed withconvex pentagonal tiles are presented, and the figures of five- and seven-fold rotationally onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure. Five-fold rotationally symmetric tilings with convex quadranglesFigure. Random tiling based on the five-fold rotationally symmetric tiling structure of a convexpentagonal tile onvex pentagons and convex hexagons that can form rotationally symmetric tilings n -fold rotationally symmetric edge-to-edge tiling are expressed in (1), whileconsidering the pentagonal tilings as convex hexagonal tilings. Note that the vertices andedges of the convex hexagon will be referred to using the nomenclature shown in Figure 1. A = D = 360 ◦ /n,B = E,C = F,A + B + C = 360 ◦ ,a = b = c = d = e = f, (1)where n is an integer greater than or equal to three, since 0 ◦ < A < ◦ . In (1), the A and D pair are selected as the vertices for 360 ◦ /n . However, due to the symmetry of theequilateral convex parallelohexagon, the B and E pair, as well as the C and F pair, are alsopossible vertex symbol starting points.When convex hexagonal tiles satisfy (1), where B = C = E = F , they become equilateralconvex parallelohexagons with D symmetry, as shown in Figure 1. The conditions of theconvex hexagonal tiles, in this case, are expressed in (2). A = D = 360 ◦ /n,B = C = E = F = 180 ◦ − A/ ◦ − ◦ /n,a = b = c = d = e = f. (2)The convex hexagons that satisfy (1) belong to the Type 1 family. In contrast, theconvex hexagons that satisfy (2) belong to the Type 1 and 2 families. When n = 3 in (2),a regular hexagon has A = B = C = D = E = F = 120 ◦ , which belongs to Type 1, 2, and3 families. The convex pentagons generated by bisecting the convex hexagons belong to theType 1 family. Table 1 presents some of the relationships between the interior angles of convex hexagonssatisfying (2) that can form the n -fold rotationally symmetric edge-to-edge tilings. (For n = 3 −
8, tilings with convex hexagonal tiles and convex pentagonal tiles generated bybisecting them are drawn. For further details, Figures 1, 2, 15–19.) As mentioned above,when n = 3, the convex hexagonal tile becomes a regular hexagon. Owing to the D symmetryof the regular hexagon, the pentagonal pair corresponding to the bisected regular hexagoncan be arranged by freely combining operations of 120 ◦ or 240 ◦ turnings and their reflections.(In [2], there are figures that depict a few tilings with such operations).For n = 3, the convex hexagonal tile that satisfies (2) is a regular hexagon, and thus, itstiling has D symmetry. For n ≥ n -fold, rotationally symmetric edge-to-edge tilingby a convex hexagonal tile that satisfies (2) has D n symmetry [3]. Therefore, the tiling inFigure 1 has D symmetry. The n -fold rotationally symmetric tilings with convex pentagonsor convex quadrangles generated by bisecting the convex hexagons that satisfy (2) along theaxis of reflection symmetry have D n symmetry. Therefore, the tilings of Figures 2(a) and3(a) have D symmetry. It is known that convex hexagonal tiles belong to at least one of the three families referred to as a“Type” [1, 9]. To date, fifteen families of convex pentagonal tiles, each of them referred to as a “Type,” are known [1,4,6].For example, if the sum of three consecutive angles in a convex pentagonal tile is 360 ◦ , the pentagonal tilebelongs to the Type 1 family. Convex pentagonal tiles belonging to some families also exist. In May 2017,Micha¨el Rao declared that the complete list of Types of convex pentagonal tiles had been obtained (i.e., theyhave only the known 15 families), but it does not seem to be fixed as of March 2020 [6]. onvex pentagons and convex hexagons that can form rotationally symmetric tilings n -fold rotationallysymmetric edge-to-edge tilings n Value of interior angle (degree) Figurenumber
A B C D E F .
43 154 .
29 154 .
29 51 .
43 154 .
29 154 .
29 188 45 157 . . . . .
73 163 .
64 163 .
64 32 .
73 163 .
64 163 . .
69 166 .
15 166 .
15 27 .
69 166 .
15 166 . .
71 167 .
14 167 .
14 25 .
71 167 .
14 167 . . .
75 168 .
75 22 . .
75 168 . .
18 169 .
41 169 .
41 21 .
18 169 .
41 169 . ... ... ... ... ... ... When the convex hexagonal tile that satisfies (1) has B = E (cid:54) = C = F , the equilateralconvex parallelohexagon has two-fold rotational symmetry but no axis of reflection symmetry(hereafter, this property is described as C symmetry ). The n -fold rotationally symmetricedge-to-edge tilings by equilateral convex parallelohexagons with C symmetry have C n sym-metry because they have rotational symmetry but no axis of reflection symmetry [3]. Forexample, the five-fold rotationally symmetric edge-to-edge tiling by the convex hexagonaltile, where “ A = D = 72 ◦ , B = E = 134 ◦ , C = F = 154 ◦ , a = b = c = d = e = f ” shown inFigure 5(a), has C symmetry. Additionally, the five-fold rotationally symmetric non-edge-to-edge tiling of the convex pentagonal tile generated by bisecting the convex hexagon, shownin Figure 5(b), has C symmetry. As described above, for n ≥ n -fold, rotationallysymmetric edge-to-edge tiling by a convex hexagonal tile that satisfies (2) has D n symmetry.Conversely, the n -fold rotationally symmetric tilings by a convex pentagonal tile or a convexquadrangle generated from the convex hexagonal tile, which satisfies (2) and is bisected by adividing line that does not overlap with the axis of reflection symmetry, have C n symmetry.Therefore, the tilings of Figures 2(b), 2(c), 2(d), and 3(b) have C symmetry.Here, the formation of rotationally symmetric edge-to-edge tiling with convex hexagonaltiles is briefly explained. First, as shown in STEP 1 in Figure 6, create a unit connecting theconvex hexagonal tiles so that they form n -fold rotationally symmetric edge-to-edge tiling inone direction so that B + D + F = 360 ◦ and A + C + E = 360 ◦ . The tiles can then be assembled “ C ” is based on the Schoenflies notation for symmetry in a two-dimensional point group [7, 8]. “ C n ”represents an n -fold rotation axes without reflection. onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure. Rotationally symmetric edge-to-edge tiling with C symmetry by a convex hexagonal tilethat satisfies “ A = D = 72 ◦ , B = E = 134 ◦ , C = F = 154 ◦ , a = b = c = d = e = f ” and five-foldrotationally symmetric non-edge-to-edge tiling of the convex pentagonal tile generated by bisectingthe convex hexagon in such a way as to increase the number of pieces from one to two to three, and so on, in order.Then create a similar unit with the reflected hexagons. Next, connect the two units createdin STEP 1, so that A + E + F = 360 ◦ as in STEP 2 in Figure 6. Subsequently, take the unitfrom STEP 2 and rotate it by the value of the inner angle of vertex A . When the originalunit and the rotated unit are arranged, so that A + B + F = 360 ◦ and A + B + C = 360 ◦ asshown in STEP 3 in Figure 6, 2 /n parts in the n -fold rotationally symmetric tiling can beformed. Then, by repeating this process as many times as necessary, an n -fold rotationallysymmetric edge-to-edge tiling with convex hexagonal tiles can be formed. When the convexhexagons are bisected as depicted in Figure 2, it will result in an n -fold rotationally symmetricnon-edge-to-edge tiling with convex pentagonal tiles. Note that when the hexagons with the D symmetry are used, as presented in Table 1, a unit of reflected hexagons is not required.In this case, the units are arranged so that A + B + E = 360 ◦ and A + C + F = 360 ◦ as inSTEP 2, and A + B + F = 360 ◦ , A + C + F = 360 ◦ , and A + B + E = 360 ◦ as in STEP 3. In [2], there are figures of rotationally symmetric tilings with a hole in the center of a regularhexagon, octagon, or 12-gon using convex pentagons (or elements that can be regarded asconvex hexagons). The regular hexagonal hole can be filled with convex pentagons. However,since the other holes cannot be filled with convex pentagons, they are not exactly tilings. Theconvex pentagons that can form rotationally symmetric tilings with a regular m -gonal holeat the center can be generated using the convex hexagons that satisfy (1). The conditionsof the convex hexagonal tiles that can be formed with a rotationally symmetric tiling with a onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure. Formation method of rotationally symmetric edge-to-edge tiling with convex hexagonaltiles regular m -gonal hole are expressed in (3). A = D = 720 ◦ /m,B = C = E = F = 180 ◦ − A/ ◦ − ◦ /m,a = b = c = d = e = f, (3)where m is an integer greater than or equal to five, since 0 ◦ < A < ◦ . The convex hexagonsthat satisfy (3) are equilateral convex parallelohexagons with D symmetry. Note that, since“180 ◦ − ◦ /m ” corresponds to one interior angle of a regular m -gon, the value of “ A + F ”in (3) is equal to the outer angle (180 ◦ + 360 ◦ /m ) of one vertex of a regular m -gon.Table 2 presents some of the relationships between the interior angles of convex hexagonssatisfying (3) that can form the rotationally symmetric tilings with a regular m -gonal holeat the center. (For m = 5 − , , , , tilings with regular m -gonal holes at thecenter by convex hexagons and convex pentagons generated by bisecting them are drawn.For further details, Figures 20–28.) As mentioned above, if these elements are consideredto be convex hexagons, the connection is edge-to-edge, and if they are considered to beconvex pentagons created by bisecting the convex hexagon, the connection is non-edge-to-edge. These rotationally symmetric tilings with regular m -gonal holes with D m symmetry atthe center have C m symmetry. If convex hexagons satisfying (3) have m that is divisible bytwo, they are also convex hexagonal tiles that satisfy (2). That is, the convex hexagonal tiles onvex pentagons and convex hexagons that can form rotationally symmetric tilings n -fold rotationallysymmetric tilings with a regular m -gonal hole at the center m Value of interior angle (degree) n ofTable 1 Figurenumber A B C D E F .
86 128 .
57 128 .
57 102 .
86 128 .
57 128 .
57 228 90 135 135 90 135 135 4 239 80 140 140 80 140 140 2410 72 144 144 72 144 144 5 2511 65 .
45 147 .
27 147 .
27 65 .
45 147 .
27 147 . .
38 152 .
31 152 .
31 55 .
38 152 .
31 152 . .
43 154 .
29 154 .
29 51 .
43 154 .
29 154 .
29 7 2715 48 156 156 48 156 15616 45 157 . . . . .
35 158 .
82 158 .
82 42 .
35 158 .
82 158 . .
89 161 .
05 161 .
05 37 .
89 161 .
05 161 . .
29 162 .
86 162 .
86 34 .
29 162 .
86 162 . .
73 163 .
64 163 .
64 32 .
73 163 .
64 163 .
64 1123 31 .
30 164 .
35 164 .
35 31 .
30 164 .
35 164 . . . . . . . ... ... ... ... ... ... that satisfy (2) can form a rotationally symmetric tiling with C n symmetry with a regular2 n -gonal hole with D n symmetry at the center and a rotationally symmetric tiling with D n symmetry. Then, the convex pentagonal tiles generated by bisecting convex hexagonsthat satisfy (2) can form a rotationally symmetric tiling with C n symmetry with a regular2 n -gonal hole with D n symmetry at the center, a rotationally symmetric tiling with D n symmetry, and rotationally symmetric tilings with C n symmetry. Here, the formation of rotationally symmetric tiling with a regular m -gonal hole at thecenter is briefly explained. First, as shown in STEP 1 in Figure 7, create a unit connecting theconvex hexagons satisfying (3) in one direction so that B + D + F = 360 ◦ and A + C + E = 360 ◦ .The hexagons can then be assembled in such a way as to increase the number of pieces fromone to two to three, and so on, in order. Then, copy the unit in STEP 1, and rotate it bythe half value of the inner angle of vertex A . Subsequently, connect the two units of STEP1, so that A + B + E = 360 ◦ and A + C + F = 360 ◦ as in STEP 2 in Figure 7. The seriesof two edges, AF , of the unit in STEP 2 are edges of the contour of a regular m -gon. Then, As shown in [3], tilings with D or D symmetry are possible; however, the explanation for them hasbeen omitted from this paper. onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure. Formation method of rotationally symmetric tiling with a regular m -gonal hole at thecenter by convex hexagons in Table 2 by repeating this process as many times as necessary, a rotationally symmetric tiling with aregular m -gonal hole at the center with convex hexagons can be formed. STEP 3 in Figure 7is 4 /
10 parts of a regular 10-gon (The completed state is shown in Figure 25). As shown inTable 2 and Figure 21, the hexagon, m = 6, is a regular hexagon, so it is possible to fit theregular hexagon into the center hole. C symmetry The conditions of the equilateral convex parallelohexagons with C symmetry are expressedin (4). A = D (cid:54) = B,B = E (cid:54) = C,C = F (cid:54) = A,A + B + C = 360 ◦ ,a = b = c = d = e = f. (4)As mentioned in Section 2, the convex hexagonal tiles that satisfy (4) and have interiorangles of 360 ◦ /n , where n is an integer greater than or equal to three, can form rotationallysymmetric tilings with C n symmetry.In [2], Iliev presents some tilings using a convex pentagon that bisects an equilateral convexparallelohexagon with C symmetry that satisfies “ A = D = 90 ◦ , B = E = 120 ◦ , C = onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure. Rotationally symmetric tilings with C or C symmetry by a convex pentagon based on aconvex hexagon with C symmetry that satisfies “ A = D = 90 ◦ , B = E = 120 ◦ , C = F = 150 ◦ , a = b = c = d = e = f ”) F = 150 ◦ , a = b = c = d = e = f .” They are a rotationally symmetric tiling with C symmetry, as shown in Figure 8(a), a rotationally symmetric tiling with C symmetry withan equilateral convex octagonal hole with D symmetry at the center, as shown in Figure 8(b),and a rotationally symmetric tiling with C symmetry with an equilateral convex hexagonalhole with D symmetry at the center, as shown in Figure 8(c). Notably, those tilings canbe formed with convex octagonal and hexagonal holes that are equilateral but not regularpolygons at the center, as shown in Figure 8(b) and 8(c). Thus, the convex hexagonal tilesthat satisfy (4) and have interior angles of 360 ◦ /n can form rotationally symmetric tilingswith C n symmetry with an equilateral convex 2 n -gonal hole with D n symmetry at the center.For example, Figure 9(a) presents a rotationally symmetric tiling with C symmetry with anequilateral convex 10-gonal hole with D symmetry at the center using an equilateral convexparallelohexagon with C symmetry that satisfies “ A = D = 360 ◦ / ◦ , B = E = onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure. Rotationally symmetric tiling with C symmetry with an equilateral convex 10-gonal holewith D symmetry at the center using a convex hexagon with C symmetry that satisfies “ A = D =72 ◦ , B = E = 134 ◦ , C = F = 154 ◦ , a = b = c = d = e = f ,” and the version of the tiling withconvex pentagons based on the convex hexagons ◦ , C = F = 154 ◦ , a = b = c = d = e = f ,” and Figure 9(b) is a version of the tiling withconvex pentagons bisecting the convex hexagons. Note that the convex hexagon in Figure 9is the same as the convex hexagon in Figure 5. Then, Figures 10(a) shows a rotationallysymmetric tiling with C symmetry, and Figures 10(b) shows a rotationally symmetric tilingwith C symmetry with an equilateral convex 14-gonal hole with D symmetry at the center,using an equilateral convex parallelohexagon with C symmetry that satisfies “ A = D =360 ◦ / ≈ . ◦ , B = E ≈ . ◦ , C = F = 165 ◦ , a = b = c = d = e = f .”Let us supplement the reason why the number of edges of the central polygonal hole, i.e.,the equilateral convex polygonal hole at the center that is formed by an equilateral convexparallelohexagon with C symmetry as described above, is even. For example, when m = 5in Figure 20 (the five-fold rotationally symmetric tilings with a regular pentagonal hole by aconvex hexagonal tile that satisfies (3) and A = D = 720 ◦ / ◦ ), it can be observed thatthe five units created in STEP 1 in Figure 7 are used. Conversely, when forming a rotationallysymmetric tiling by an equilateral convex parallelohexagon with the C symmetry, as shownin Figures 5, 6, 8, 9 and 10, the units with hexagons and units with reflected hexagons shouldbe connected alternately. Therefore, if the number of edges of the central polygonal holeis odd, the convex hexagons with C symmetry cannot form the central polygon shown inFigure 11 (when trying to form a nonagonal hole by an equilateral convex parallelohexagonwith C symmetry that satisfies A = D = 720 ◦ / ◦ ). Since an equilateral convex 2 n -gonwith D n symmetry has two types of internal angles in vertices, there are two types of outerangles in vertices. If A = D = 360 ◦ /n in (4), values of “ A + B, A + F ” are equal to the twotypes of outer angles (see Figure 10(b)).Figure 12 shows a rotationally symmetric tiling with C symmetry using a convex hexago-nal tile that satisfies “ A = D = 90 ◦ , B = E = 120 ◦ , C = F = 150 ◦ , a = b = c = d = e = f ” onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure.
Rotationally symmetric tiling with C symmetry and rotationally symmetric tiling with C symmetry with an equilateral convex 14-gonal hole with D symmetry at the center using anequilateral convex parallelohexagon with C symmetry that satisfies “ A = D = 360 ◦ / ≈ . ◦ , B = E ≈ . ◦ , C = F = 165 ◦ , a = b = c = d = e = f in Figure 8. Then, Figures 13(a) shows a three-fold rotationally symmetric tiling generatedby the convex hexagonal tile that satisfies “ A = D = 80 ◦ , B = E = 120 ◦ , C = F =160 ◦ , a = b = c = d = e = f ” shown in Figure 11. As shown in Figure 13(b), the convexhexagons can also form a rotationally symmetric tiling with C symmetry with an equilateralconvex hexagonal hole at the center. This arrangement indicates that the convex hexagonsthat satisfy (4), where one of the vertices A , B , or C of which is equal to 360 ◦ /n , can forman n -fold rotationally symmetric tiling and a tiling with C n symmetry with an equilateralconvex 2 n -gonal hole with D n symmetry at the center. (It is a matter of starting the vertexsymbol from somewhere. As shown in Figure 13, if “ A → F (cid:48) , B → A (cid:48) , . . . ” is replaced, then A (cid:48) = D (cid:48) = 360 ◦ / ◦ , which corresponds to (1).)Since the convex hexagonal tile that satisfies “ A = D = 90 ◦ , B = E = 120 ◦ , C = F = 150 ◦ , a = b = c = d = e = f ” shown in Figures 8 and 12 is the equilateral convexparallelohexagon with 360 ◦ / ◦ and 360 ◦ / ◦ , it can form rotationally symmetrictilings with C or C symmetry, a rotationally symmetric tiling with C symmetry with anequilateral convex hexagonal hole with D symmetry, and a rotationally symmetric tiling with C symmetry with an equilateral convex octagonal hole with D symmetry at the center. Asan equilateral convex parallelohexagons with C symmetry that has the same properties asabove, there is a convex hexagonal tile that satisfies “ A = D = 72 ◦ , B = E = 120 ◦ , C = F =168 ◦ , a = b = c = d = e = f .” This convex hexagonal tile can form rotationally symmetrictilings with C or C symmetry (see Figures 14(a), 14(b)), a rotationally symmetric tiling with C symmetry with an equilateral convex 10-gonal hole with D symmetry (see Figure 14(c)),and a rotationally symmetric tiling with C symmetry with an equilateral convex hexagonal onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure.
State in which a nonagonal hole cannot be formed using convex hexagons that satisfy” A = D = 720 ◦ / ◦ , B = E (cid:54) = C = F, A + B + C = 360 ◦ , a = b = c = d = e = f ”Figure. Three-fold rotationally symmetric tiling by a convex hexagon that satisfies “ A = D =90 ◦ , B = E = 120 ◦ , C = F = 150 ◦ , a = b = c = d = e = f ” and the version of the tiling with convexpentagons based on the convex hexagons hole with D symmetry at the center (see Figure 14(d)). Note that Figure 14 shows thetilings with convex pentagons based on the convex hexagons.The convex hexagonal tiles that satisfy (4) and have particular angles (i.e., angles corre-sponding to 360 ◦ /n ) of two or more types can form multiple rotationally symmetric tilingswith an equilateral polygonal hole at the center, and multiple rotationally symmetric tilings.Such equilateral convex parallelohexagons with C symmetry that can form multiple ro-tationally symmetric tilings with an equilateral polygonal hole at the center and multiple onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure.
Three-fold rotationally symmetric tiling and rotationally symmetric tiling with an equi-lateral convex hexagonal hole at the center by a convex hexagon that satisfies “ A = D = 80 ◦ , B = E = 120 ◦ , C = F = 160 ◦ , a = b = c = d = e = f ” rotationally symmetric tilings are the above two cases with angles of “90 ◦ , ◦ , ◦ ” and“72 ◦ , ◦ , ◦ .” It is because that interior angles of the convex hexagons that satisfy (4)can be selected up to two different values corresponding to 360 ◦ /n when n is an integergreater than or equal to three. Therefore, the two particular angles of this special case canbe selected from “72 ◦ , ◦ , ◦ .” This paper summarizes how the convex pentagons and hexagons can form rotationally sym-metric tilings based on the information in [2] and [3]. The n -fold rotationally symmetricedge-to-edge tilings formed of convex hexagonal tiles that satisfy (1) can be divided into C n symmetry and D n symmetry depending on the selection of the angles of the vertices in theconvex hexagon. The convex pentagonal tiles generated by bisecting the convex hexagonaltiles can form rotationally symmetric non-edge-to-edge tilings with C n or D n symmetry de-pending on the shape or division method of the convex hexagon. In contrast, the n -foldrotationally symmetric edge-to-edge tilings with the convex pentagonal tiles shown in [5] canform only C n symmetry. In addition, this paper demonstrated a convex hexagons (convexpentagons) that can form rotationally symmetric tilings with an equilateral convex polygonalhole at the center.In [2], there are cases where m = 6 , ,
12 present in Table 2 and tilings as shown inFigure 8 are described. However, there is no description of properties (tiles conditions suchas (1), (2), and (3)) as discussed in this paper. In [3], there is no description of tilings with anequilateral convex polygonal hole at the center, and there is insufficient description regarding onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure.
Five- or three-fold rotationally symmetric tilings and rotationally symmetric tilings withan equilateral convex 10-gonal or hexagonal hole at the center by a convex pentagon based on a convexhexagon that satisfies “ A = D = 72 ◦ , B = E = 120 ◦ , C = F = 168 ◦ , a = b = c = d = e = f ” onvex pentagons and convex hexagons that can form rotationally symmetric tilings References [1] B. Gr¨unbaum, G.C. Shephard,
Tilings and Patterns . W. H. Freeman and Company, New York,1987. pp.15–35 (Chapter 1), pp.471–518 (Chapter 9).[2] I. Iliev,
Mosaics from parquet forming irregular convex pentagons . Alliance print, Sofia, 2018.[3] B. Klaassen, Rotationally symmetric tilings with convex pentagons and hexagons,
Elemente derMathematik , (2016) 137–144. doi:10.4171/em/310. Available online: https://arxiv.org/abs/1509.06297 (accessed on 23 February 2020).[4] T. Sugimoto, Tiling Problem: Convex pentagons for edge-to-edge tiling and convex polygons foraperiodic tiling (2015). https://arxiv.org/abs/1508.01864 (accessed on 23 February 2020).[5] —, Convex pentagons and concave octagons that can form rotationally symmetric tilings (2020). https://arxiv.org/abs/2005.08470 (accessed on 19 May 2020).[6] Wikipedia contributors, Pentagon tiling, Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Pentagon_tiling (accessed on 31 March 2020).[7] Wikipedia contributors, Point group, Wikipedia, Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Point_group (accessed on 23 February 2020).[8] Wikipedia contributors, Schoenflies notation, Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Schoenflies_notation (accessed on 23 February 2020).[9] Wolfram MathWorld, Hexagon Tiling, Wolfram, https://mathworld.wolfram.com/HexagonTiling.html (accessed on 23 February 2020).Figure. Three-fold rotationally symmetric tilings by a convex hexagon and a convex pentagon onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure.
Four-fold rotationally symmetric tilings by a convex hexagon and a convex pentagonFigure.
Six-fold rotationally symmetric tilings by a convex hexagon and a convex pentagonFigure.
Seven-fold rotationally symmetric tilings by a convex hexagon and a convex pentagon onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure.
Eight-fold rotationally symmetric tilings by a convex hexagon and a convex pentagonFigure.
Rotationally symmetric tiling with C symmetry with a regular convex pentagonal holeat the center by a convex pentagon onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure.
Rotationally symmetric tiling with C symmetry with a regular convex hexagonal hole atthe center by a convex pentagonFigure. Rotationally symmetric tiling with C symmetry with a regular convex heptagonal holeat the center by a convex pentagon onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure.
Rotationally symmetric tiling with C symmetry with a regular convex octagonal hole atthe center by a convex pentagonFigure. Rotationally symmetric tiling with C symmetry with a regular convex nonagonal holeat the center by a convex pentagon onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure.
Rotationally symmetric tiling with C symmetry with a regular convex 10-gonal hole atthe center by a convex pentagonFigure. Rotationally symmetric tiling with C symmetry with a regular convex 12-gonal hole atthe center by a convex pentagon onvex pentagons and convex hexagons that can form rotationally symmetric tilings Figure.
Rotationally symmetric tiling with C symmetry with a regular convex 14-gonal hole atthe center by a convex pentagonFigure. Rotationally symmetric tiling with C16