※ 引述《chaogold (新扇不扇不擅訕新)》之銘言： : ※ 引述《darkseer (公假中)》之銘言： : : As title, : : 給定一個n邊形,一個操作 = 把該n邊形對某一條邊做鏡射 : : 問是否對任意n邊形,都存在一個由n種操作所構成的無窮數列 : : 使得在將該n邊形依數列操作時該n邊形會跑遍整個平面 : : ------------------------------------------------------------------------------ : : 剛看到以為不難 : : 結果三角形的case我都弄不出來XD : : 不過已經證出若在三角形成立則原命題成立 : 每一次操作的正n邊形是同一個? : 鏡射出來的正n邊形是一直在的嗎? 我的表達實在不好XD 直接po原文好了 Let P be an n-gon , lying on a plane. We name its edges 1,2,3,..........n. If S = s_1, s_2,.............. be a finite or infinite sequence such that for each i s_i is in {1,2........n}. We move P in accordance with the sequence S such that we first reflect P through s_1 ( the number s_i corresponds to the s_i th edge of the polygon P ) and then through s_2 and so on. Show the following holds : a) Show that there exists a infinite sequence S such that if we move P according to S then then we can cover the whole plane. b) Prove that the sequence S is'nt periodic. c) Assume that P is a regular pentagon with the radius of its circumcircle as 1 and let D be another circle with radius 1.00001 lying in the plane arbitrary. Does there exist a sequence S such that we move P accordingly then P reside in D completely. 原題中的(a)就是我問的 (b)(c)簡單的多(好奇怪的配置XD) --

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