Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Packing and covering a unit equilateral triangle with equilateral triangles. | Packing and covering a unit equilateral triangle with equilateral triangles Yuqin Zhang1 Yonghui Fan2 department of Mathematics Beijing Institute of Technology 100081 Beijing China email yuqinzhang@ 2College of Mathematics and Information Science Hebei Normal University 050016 Shijiazhuang China Submitted Jun 7 2005 Accepted Oct 20 2005 Published Oct 25 2005 Abstract Packing and covering are elementary but very important in combinatorial geometry they have great practical and theoretical significance. In this paper we discuss a problem on packing and covering a unit equilateral triangle with smaller triangles which is originated from one of Erdos favorite problems. Keywords packing minimal covering Mathematics Subject Classification 2000 52C15 1 Introduction Packing and covering are elementary but very important in combinatorial geometry they have great practical and theoretical significance. In 1932 Erdos posed one of his favorite problems on square-packing which was included in 2 Let S be a unit square. Inscribe n squares with no common interior point. Denote by e1 e2 . en the sides length of these n squares. Put f n max ei. In 3 P. Erdos and Soifer gave some results of f n . i 1 In 1 Connie Campbell and William Staton considered this problem again. Because packing and covering are usually dual to each other we discussed a problem of a minimal square-covering in 5 . In this paper we generalize this kind of problem to the case of using equilateral triangles to pack and cover a unit equilateral triangle and obtain corresponding results. Foundation items This work is supported by the Doctoral Funds of Hebei Province in China B2004114 . THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R55 1 2 Packing a unit equilateral triangle Firstly we give the definition of the packing function Definition . Let T be a unit equilateral triangle. Inscribe n equilateral triangles T1 T2 . Tn with no common interior point in such a way which satisfies T has side of length t 0
