Applied Mathematical Sciences, Vol. 7, 2013, no. 31, 1545-1550 HIKARI Ltd, www.m-hikari.com Regular Hexagon over for Isoperimetric Triangles anyat Sroysang epartment of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121 Thailand banyat@mathstat.sci.tu.ac.th opyright 2013 anyat Sroysang. This is an open access article distributed under the reative ommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A convex region covers a family of arcs if it contains a congruent copy of every arc in the family. In this paper, we search for the smallest regular hexagon which covers the family of all triangles of perimeter two. Mathematics Subject lassification: 5215 Keywords: over, Hexagon, Triangle, Worm Problem 1 Introduction and Preliminaries For a convex region R and a family F of arcs, we say that R covers F or R is a cover for F, if R contains a congruent copy of each arc in F. The Moser s worm problem [2] searches the region of smallest area which contains a congruent copy of every unit arc on the plane. A popular shape of arcs is a triangle. In 1997, Wetzel [6] gave the smallest equilateral triangular cover for the family of all triangles of perimeter two. Moreover, he [5] also gave the smallest rectangular cover for the family of all triangles of perimeter two. In 2000, Furedi and Wetzel [1] gave the smallest convex cover for the family of all triangles of perimeter two. In 2009, Zhang and Yuan [7] gave the smallest regularized parallelogram cover, whose length of the smaller diagonal is not less than one, for the family of all triangles of perimeter two. In 2011, Sroysang [3,4] gave the smallest regularized trapezoid cover, whose length of the smaller diagonal is not less than one, and two smaller angles are opposite, for the family of all triangles of perimeter two.
1546. Sroysang In this paper, we search for the smallest regular hexagon which covers the family of all triangles of perimeter two. The diameter of a convex set X is the maximum of the distance between the two parallel support lines of the set X. The thickness of a convex set X is the minimum of the distance between the two parallel support lines of the set X. Note that (i) the diameter of any triangle is the length of the longest side, (ii) the thickness of any triangle is the length of the altitude to the longest side, (iii) the diameter of any regular hexagon is the length of the diagonal, and (iv) the thickness of any regular hexagon is the distance between two parallel edges of the hexagon. It follows that (i) the diameter of any triangle of perimeter two is at least 2 3, 1 (ii) the thickness of any triangle of perimeter two is at most, 3 (iii) the diameter of any regular hexagon cover for the family of all triangles of perimeter two is at least one, and (iv) the thickness of any regular hexagon cover for the family of all triangles of perimeter two is at least 3 2. 2 Results Theorem The smallest regular hexagon cover for the family of all triangles of perimeter two is the regular hexagon of diameter one (the area is 3 3 ). 8 Proof. Let R be the regular hexagon of diameter one and let T be a triangle of perimeter two. We label the vertices of the triangle T by A, and where the angle A is greater than or equal to the angle, and the angle is greater than or equal to the angle. We divide the triangle T into two cases. ase 1. The diameter of the triangle T is at most 5 6. Let P and Q be two points on the perimeter of the regular hexagon R such that the distance between P and Q is equal to 5 and the segment PQ is parallel to a 6 diagonal of the regular hexagon R as shown in Fig. 1. We see that the distance between the vertex and the segment PQ is equal to 1. Moreover, we note 3 that the thickness of the regular hexagon R is greater than 5 6 as shown in Fig. 1.
Regular hexagon cover for isoperimetric triangles 1547 WLOG, we can put the triangle T into the regular hexagon R where the segment lies on the segment PQ, and = P. This implies that the regular hexagon R contains a congruent copy of the triangle T. Q P Fig. 1. The segment PQ of length 5 6 in the regular hexagon R ase 2. The diameter of the triangle T is greater than 5 6. 1 We note on this case that the thickness of the triangle T is at most. 6 WLOG, we can put the triangle T into the regular hexagon R where the segment lies on a diagonal of the regular hexagon R and the vertex A is above than the segment as shown in Fig. 2 or Fig. 3. Now, the vertex A may be in the regular hexagon R or not in the regular hexagon R. For Fig. 2, we are done. A Fig. 2. A triangle T in the regular hexagon R where the vertex A is in R
1548. Sroysang A E F G Fig. 3. A triangle T in the regular hexagon R where the vertex A is not in R Assume that the vertex A is not in the regular hexagon R as shown in Fig. 3. We rotate the triangle T around the vertex until the vertex meets the segment EF or the segment FG. If the vertex meets the segment FG, then we are done. Thus, we assume that the vertex meets the segment EF. Suppose for a contradiction that the regular hexagon R does not contains the triangle T. Let H be the intersection of the segment A and the segment E as shown in Fig. 4. H E F G Fig. 4. A triangle H in the triangle T and the regular hexagon R Next, we rotate both the triangle T and the regular hexagon R by 180 around the vertex E as shown in Fig. 5.
Regular hexagon cover for isoperimetric triangles 1549 H Z X Y E F G Fig. 5. The 180 -rotation of Fig. 4 Since EF is equal to 120, we obtain that HX H. Then 2 E = Z < H + HX + XZ = H + HX + H + H + = 2. Hence, E < 1. This is a contradiction. Therefore, the regular hexagon R is a cover for the family of all triangles of perimeter two. Next, we note that every cover for the family of all triangles of perimeter two must cover the line segment of length one. Thus, the diagonal of any regular hexagon cover for the family of all triangles of perimeter two must have length at least one. The diagonal of the regular hexagon R has length one. Hence, the regular hexagon R is a smallest cover for the family of all triangles of perimeter two. Acknowledgement. This research was supported by the Thammasat University Research Fund 2012, Thailand. References [1] Z. Furedi and J. E. Wetzel, The smallest convex cover for triangles of perimeter two, Geom. edicata, 81 (2000), 285 293. [2] L. Moser, Poorly formulated unsolved problems of combinatorial geometry, Mimeographed (1996).
1550. Sroysang [3]. Sroysang, Regularized Trapezoid over for Isoperimetric Triangles, Int. J. omput. Appl. Math., 6 (2011), 239 245. [4]. Sroysang, Right Trapezoid cover for triangles of perimeter two, Kasetsart J. (Nat. Sci.), 45 (2011), 756 671. [5] J. E. Wetzel, oxes for isoperimetric triangles, Math. Mag., 73 (2000), 315 319. [6] J. E. Wetzel, The smallest equilateral cover for triangles of perimeter two, Math. Mag., 70 (1997), 125 130. [7] Y. Zhang and L. Yuan, Parallelogram cover for triangles of perimeter two, Southeast Asian ull. Math., 33 (2009), 607 612. Received: January, 2013