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--- 수학:코흐_곡선 [2022/01/16 22:15] – yong
+++ 수학:코흐_곡선 [2022/01/17 16:23] – yong
@@ Line 76: / Line 76: @@
 $d_{\rm sq}\,, d_{\rm dead}\,, d_{\rm cor}$는 0, $d_{\rm para}$는 1, $d_{\rm line}$는 11, $d_{\rm empty}$는 25의 고윳값을 가지는 것을 확인할 수 있다.
-고유방정식의 의미를 생각해보면, $d_{\rm sq}\,, d_{\rm dead}\,, d_{\rm cor}$는 irrelevant하며, $d_{\rm para}$는 marginal, $d_{\rm line}$와 $d_{\rm empty}$는 relevant하다는 것을 알 수 있다.
+고유방정식의 의미를 생각해보면, $d_{\rm sq}\,, d_{\rm dead}\,, d_{\rm cor}$는 irrelevant하며, $d_{\rm para}$는 marginal, $d_{\rm line}$와 $d_{\rm empty}$는 relevant하다.
-다음으로 정수로 구성되는 고유벡터는,
-\begin{align}
+한편 각 고유값에 대응되는 고유벡터는,
-	\mathbf{d}_{\rm{sq}} 	=& \left( -1 ~~ 4 ~~ -4 ~~ 0 ~~ 0 ~~ 1 \right) \\
+\begin{equation*}
-	\mathbf{d}_{\rm{dead}} 	=& \left( 0 ~~ 1 ~~ -2 ~~ 0 ~~ 1 ~~ 0 \right) \\
+\begin{aligned}
-	\mathbf{d}_{\rm{cor}} 		=& \left( 1 ~~ -2 ~~ 0 ~~ 1 ~~ 0 ~~ 0 \right) \\
+	\mathbf{d}_{\rm{sq}} 	=& \, \left( -1 ~~ 4 ~~ -4 ~~ 0 ~~ 0 ~~ 1 \right) \\
-	\mathbf{d}_{\rm{para}} 	=& \left( 0 ~~ 1 ~~ -1 ~~ 1 ~~ -3 ~~ 2 \right) \\
+	\mathbf{d}_{\rm{dead}} 	=& \, \left( 0 ~~ 1 ~~ -2 ~~ 0 ~~ 1 ~~ 0 \right) \\
-	\mathbf{d}_{\rm{line}} 	=& \left( 0 ~~ -98 ~~ -112 ~~ -28 ~~ -196 ~~ 379 \right) \\
+	\mathbf{d}_{\rm{cor}} 	=& \, \left( 1 ~~ -2 ~~ 0 ~~ 1 ~~ 0 ~~ 0 \right)
-	\mathbf{d}_{\rm{empty}}	=& \left( 0 ~~ 0 ~~ 0 ~~ 0 ~~ 0 ~~ 1 \right)
+\end{aligned}
-\end{align}
+\qquad
+\begin{aligned}
+	\mathbf{d}_{\rm{para}} 	=& \, \left( 0 ~~ 1 ~~ -1 ~~ 1 ~~ -3 ~~ 2 \right) \\
+	\mathbf{d}_{\rm{line}} 	=& \, \left( 0 ~~ -98 ~~ -112 ~~ -28 ~~ -196 ~~ 379 \right) \\
+	\mathbf{d}_{\rm{empty}}	=& \, \left( 0 ~~ 0 ~~ 0 ~~ 0 ~~ 0 ~~ 1 \right)
+\end{aligned}
+\end{equation*}
+고유벡터로 구성되는 행렬 Q를 $Q = \left( \mathbf{d}_{\rm{sq}}\,, \mathbf{d}_{\rm{dead}}\,, \mathbf{d}_{\rm{cor}}\,, \mathbf{d}_{\rm{para}}\,, \mathbf{d}_{\rm{line}}\,, \mathbf{d}_{\rm{empty}} \right)$
+으로 구성하면 대각화의 성질을 이용하여 n번의 변환을 거친 프랙탈 기본 구성 사각형의 총 개수를 얻을 수 있다. 즉,
+\begin{equation*}
+	SI^{n} = \left( QDQ^{-1} \right)^{n} = QD^{n}Q^{-1}
+\end{equation*}
+python을 이용하여 그래프를 그리면 다음을 얻을 수 있다.
+{{:수학:kochgraph.png?600 |}}
+<code Python | fractalstep.py>
+import numpy as np
+import numpy.linalg as lin
+import matplotlib.pyplot as plt
+matrix = np.array(
+        [[0,0,0,0,0,0],
+        [8,5,3,2,1,0],
+        [4,4,3,4,2,0],
+        [4,2,1,0,0,0],
+        [4,6,5,8,4,0],
+        [1,5,11,9,17,25]]
+)
+steps = 10
+eigval, eigvec = lin.eig(matrix)
+diagmat = np.zeros((6,6))
+np.fill_diagonal(diagmat,eigval)
+diagmat0 = diagmat
+sum_mat = np.zeros((6,steps))
+for i in range(steps):
+        A = eigvec @ diagmat @ lin.inv(eigvec)
+        for j in range(6):
+                sum_mat[j][i] = np.sum(A[j])
+        diagmat = diagmat0 @ diagmat
+x = np.arange(1,steps+1,1)
+label = ['square', 'deadend', 'corner', 'parallel', 'line', 'empty']
+for i in range(6):
+        plt.plot(x, sum_mat[i], label=label[i])
+plt.yscale('log')
+plt.plot((2,8),(5**5*10,5**17*10),'--',label='2')
+log_5_11=np.log(11)/np.log(5)
+plt.plot((2,8),(5**(log_5_11*2)*10,5**(log_5_11*8)*10),'--',label=r'$\log 11 / \log 5$')
+plt.xlabel('steps')
+plt.ylabel('n (steps)')
+plt.legend()
+plt.savefig('kochgraph.png', dpi=300, transparent=True)
+</code>
+====== 함께 보기 ======
+  * [[물리:프랙탈 차원]]
+  * [[http://events.kias.re.kr/h/statphys.winterschool/?pageNo=4526|The 19th KIAS-APCTP Winter School on Statistical Physics]]