\u5b9f\u6570 \\(a , b\\) \u306b\u5bfe\u3057\u5e73\u9762\u4e0a\u306e\u70b9 \\(\\text{P} {} _ n \\ ( x _ n , y _ n )\\) \u3092\r\n\\[\\begin{align}\r\n( x _ 0 , y _ 0 ) & = (1,0) \\\\\r\n( x _ {n+1} , y _ {n+1} ) & = ( ax _ n -by _ n , bx _ n +ay _ n )\r\n\\end{align}\\]\r\n\u306b\u3088\u3063\u3066\u5b9a\u3081\u308b. \u3053\u306e\u3068\u304d, \u6b21\u306e\u6761\u4ef6 (i)<\/strong> , (ii)<\/strong> \u304c\u3068\u3082\u306b\u6210\u308a\u7acb\u3064\u3088\u3046\u306a \\((a,b)\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p>\r\n (i)<\/strong>\u3000\\(\\text{P} {} _ 0 = \\text{P} {} _ 6\\)<\/p><\/li>\r\n (ii)<\/strong>\u3000\\(\\text{P} {} _ 0 , \\text{P} {} _ 1 , \\text{P} {} _ 2 , \\text{P} {} _ 3 , \\text{P} {} _ 4 , \\text{P} {} _ 5\\) \u306f\u76f8\u7570\u306a\u308b.<\/p><\/li>\r\n<\/ol>\r\n \\(A = \\left( \\begin{array}{cc} a & -b \\\\ b & a \\end{array} \\right)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\left( \\begin{array}{c} x _ {n+1} \\\\ y _ {n+1} \\end{array} \\right) = A \\left( \\begin{array}{c} x _ {n} \\\\ y _ {n} \\end{array} \\right) \\quad ... [1]\r\n\\]\r\n\u3055\u3089\u306b, \\(a = r \\cos \\theta\\) , \\(b = r \\sin \\theta\\) \uff08 \\(r \\geqq 0\\) , \\(0 \\leqq \\theta \\lt 2 \\pi\\) \uff09, \u539f\u70b9\u4e2d\u5fc3 \\(\\theta\\) \u56de\u8ee2\u3092\u8868\u3059\u884c\u5217\u3092 \\(R( \\theta )\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nA = r \\left( \\begin{array}{cc} \\cos \\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{array} \\right) = r R( \\theta ) \\quad ... [2]\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] [2] \u3068\u56de\u8ee2\u884c\u5217\u306e\u6027\u8cea\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n\\left( \\begin{array}{c} x _ {n} \\\\ y _ {n} \\end{array} \\right) = A^n \\left( \\begin{array}{c} x _ {0} \\\\ y _ {0} \\end{array} \\right) = r^n R( n \\theta ) \\left( \\begin{array}{c} x _ {0} \\\\ y _ {0} \\end{array} \\right)\r\n\\]\r\n\u6761\u4ef6 (i)<\/strong> \u3088\u308a\r\n\\[\\begin{align}\r\nr^6 R( 6 \\theta ) \\left( \\begin{array}{c} x _ {0} \\\\ y _ {0} \\end{array} \\right) & = \\left( \\begin{array}{c} x _ {0} \\\\ y _ {0} \\end{array} \\right) \\\\\r\n\\text{\u2234} \\quad \\left\\{ r^6 R( 6 \\theta ) -E \\right\\} & \\left( \\begin{array}{c} x _ {0} \\\\ y _ {0} \\end{array} \\right) = 0\r\n\\end{align}\\]\r\n\\(\\left( \\begin{array}{c} x _ {0} \\\\ y _ {0} \\end{array} \\right) \\neq \\left( \\begin{array}{c} 0 \\\\ 0 \\end{array} \\right)\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nr^6 R( 6 \\theta ) -E & = O \\\\\r\n\\text{\u2234} \\quad r^6 R( 6 \\theta ) & = E\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(E = R(0)\\) \u306a\u306e\u3067\r\n\\[\r\nr = 1 , \\ \\theta = \\dfrac{k \\pi}{3} \\quad \\left( k = 0, 1, \\cdots , 5 \\right)\r\n\\]\r\n\u6761\u4ef6 (ii)<\/strong> \u306b\u3064\u3044\u3066\u8003\u3048\u308b\u3068,\r\n\\[\r\nA^m \\neq E \\quad \\left( k = 0, 1, \\cdots , 5 \\right)\r\n\\]\r\n\u3092\u6e80\u305f\u3059\u5fc5\u8981\u304c\u3042\u308b. \\(k=0\\) \u306e\u3068\u304d, \\(A=E\\) \u3068\u306a\u308a\u4e0d\u9069<\/p><\/li>\r\n \\(k=2\\) \u306e\u3068\u304d, \\(A^3=E\\) \u3068\u306a\u308a\u4e0d\u9069<\/p><\/li>\r\n \\(k=3\\) \u306e\u3068\u304d, \\(A^2=E\\) \u3068\u306a\u308a\u4e0d\u9069<\/p><\/li>\r\n \\(k=4\\) \u306e\u3068\u304d, \\(A^3=E\\) \u3068\u306a\u308a\u4e0d\u9069<\/p><\/li>\r\n<\/ul>\r\n \u3057\u305f\u304c\u3063\u3066, \u6761\u4ef6\u3092\u6e80\u305f\u3059 \\(k\\) \u306f\r\n\\[\r\nk = 1 , 5\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\((a,b)\\) \u306e\u7d44\u306f\r\n\\[\r\n(a,b) = \\underline{\\left( \\dfrac{1}{2} , \\ \\pm \\dfrac{\\sqrt{3}}{2} \\right)}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570 \\(a , b\\) \u306b\u5bfe\u3057\u5e73\u9762\u4e0a\u306e\u70b9 \\(\\text{P} {} _ n \\ ( x _ n , y _ n )\\) \u3092 \\[\\begin{align} ( x _ 0 , y _ 0 ) & = (1,0) \\\\ … \u7d9a\u304d\u3092\u8aad\u3080 \r\n
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\u5404 \\(k\\) \u306e\u5024\u306b\u3064\u3044\u3066<\/p>\r\n\r\n