microsoft power point 遞迴關係-雪花曲線 [相容模式]
TRANSCRIPT
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遞迴關係-雪花曲線
![Page 2: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/2.jpg)
次數段數
每段長度
全長
0 1 1 1
1 4 (1/3) 4 1 × (1/3) 1
2 42 (1/3) 2 4 2 × (1/3) 2
3 43 (1/3) 3 4 3 × (1/3) 3
4 44 (1/3) 4 4 4 × (1/3) 4
5 45 (1/3) 5 4 5 × (1/3) 5
�
n 4n (1/3) n 4 n × (1/3) n
von Koch Curve
第 0 迴
![Page 3: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/3.jpg)
次數段數
每段長度
全長
0 1 1 1
1 4 (1/3) 4 1 × (1/3) 1
2 42 (1/3) 2 4 2 × (1/3) 2
3 43 (1/3) 3 4 3 × (1/3) 3
4 44 (1/3) 4 4 4 × (1/3) 4
5 45 (1/3) 5 4 5 × (1/3) 5
�
n 4n (1/3) n 4 n × (1/3) n
von Koch Curve
第 1 迴
![Page 4: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/4.jpg)
次數段數
每段長度
全長
0 1 1 1
1 4 (1/3) 4 1 × (1/3) 1
2 42 (1/3) 2 4 2 × (1/3) 2
3 43 (1/3) 3 4 3 × (1/3) 3
4 44 (1/3) 4 4 4 × (1/3) 4
5 45 (1/3) 5 4 5 × (1/3) 5
�
n 4n (1/3) n 4 n × (1/3) n
von Koch Curve
第 2 迴
![Page 5: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/5.jpg)
次數段數
每段長度
全長
0 1 1 1
1 4 (1/3) 4 1 × (1/3) 1
2 42 (1/3) 2 4 2 × (1/3) 2
3 43 (1/3) 3 4 3 × (1/3) 3
4 44 (1/3) 4 4 4 × (1/3) 4
5 45 (1/3) 5 4 5 × (1/3) 5
�
n 4n (1/3) n 4 n × (1/3) n
von Koch Curve
第 3 迴
![Page 6: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/6.jpg)
次數段數
每段長度
全長
0 1 1 1
1 4 (1/3) 4 1 × (1/3) 1
2 42 (1/3) 2 4 2 × (1/3) 2
3 43 (1/3) 3 4 3 × (1/3) 3
4 44 (1/3) 4 4 4 × (1/3) 4
5 45 (1/3) 5 4 5 × (1/3) 5
�
n 4n (1/3) n 4 n × (1/3) n
von Koch Curve
第 4 迴
![Page 7: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/7.jpg)
次數段數
每段長度
全長
0 1 1 1
1 4 (1/3) 4 1 × (1/3) 1
2 42 (1/3) 2 4 2 × (1/3) 2
3 43 (1/3) 3 4 3 × (1/3) 3
4 44 (1/3) 4 4 4 × (1/3) 4
5 45 (1/3) 5 4 5 × (1/3) 5
�
n 4n (1/3) n 4 n × (1/3) n
von Koch Curve
第 5 迴
![Page 8: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/8.jpg)
Snow Flake
次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ 3 ×××× 40 ×××× (1/9) 1 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ 3 ×××× 41 ×××× (1/9) 2 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ 3 ×××× 42 ×××× (1/9) 3 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ 3 ×××× 43 ×××× (1/9) 4 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ 3 ×××× 44 ×××× (1/9) 5 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-
1 (1/9) n ∆ 3 ×××× 4(n-1) ∆ ×××× (1/9) n ∆第 0 迴
![Page 9: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/9.jpg)
Snow Flake
次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ 3 ×××× 40 ×××× (1/9) 1 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ 3 ×××× 41 ×××× (1/9) 2 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ 3 ×××× 42 ×××× (1/9) 3 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ 3 ×××× 43 ×××× (1/9) 4 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ 3 ×××× 44 ×××× (1/9) 5 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-1 (1/9) n ∆ 3 ×××× 4(n-1) ∆ ×××× (1/9) n ∆第 1 迴
![Page 10: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/10.jpg)
Snow Flake
次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ 3 ×××× 40 ×××× (1/9) 1 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ 3 ×××× 41 ×××× (1/9) 2 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ 3 ×××× 42 ×××× (1/9) 3 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ 3 ×××× 43 ×××× (1/9) 4 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ 3 ×××× 44 ×××× (1/9) 5 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-1 (1/9) n ∆ 3 ×××× 4(n-1) ∆ ×××× (1/9) n ∆
第 2 迴
![Page 11: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/11.jpg)
Snow Flake
次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ 3 ×××× 40 ×××× (1/9) 1 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ 3 ×××× 41 ×××× (1/9) 2 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ 3 ×××× 42 ×××× (1/9) 3 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ 3 ×××× 43 ×××× (1/9) 4 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ 3 ×××× 44 ×××× (1/9) 5 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-1 (1/9) n ∆ 3 ×××× 4(n-1) ∆ ×××× (1/9) n ∆
第 3 迴
![Page 12: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/12.jpg)
Snow Flake
次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ 3 ×××× 40 ×××× (1/9) 1 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ 3 ×××× 41 ×××× (1/9) 2 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ 3 ×××× 42 ×××× (1/9) 3 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ 3 ×××× 43 ×××× (1/9) 4 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ 3 ×××× 44 ×××× (1/9) 5 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-1 (1/9) n ∆ 3 ×××× 4(n-1) ∆ ×××× (1/9) n ∆
第 4 迴
![Page 13: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/13.jpg)
Snow Flake
第 4 迴
![Page 14: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/14.jpg)
Snow Flake
第 4 迴
![Page 15: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/15.jpg)
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![Page 17: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/17.jpg)
Snow Flake次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ (1/9) 1 ∆ ×××× 3 ×××× 40 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ (1/9) 2 ∆ ×××× 3 ×××× 41 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ (1/9) 3 ∆ ×××× 3 ×××× 42 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ (1/9) 4 ∆ ×××× 3 ×××× 43 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ (1/9) 5 ∆ ×××× 3 ×××× 44 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-1 (1/9) n ∆ (1/9) n ∆ ×××× 3 ×××× 4(n-1) ∆
第 0 迴
![Page 18: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/18.jpg)
Snow Flake次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ (1/9) 1 ∆ ×××× 3 ×××× 40 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ (1/9) 2 ∆ ×××× 3 ×××× 41 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ (1/9) 3 ∆ ×××× 3 ×××× 42 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ (1/9) 4 ∆ ×××× 3 ×××× 43 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ (1/9) 5 ∆ ×××× 3 ×××× 44 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-1 (1/9) n ∆ (1/9) n ∆ ×××× 3 ×××× 4(n-1) ∆
第 1 迴
![Page 19: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/19.jpg)
Snow Flake次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ (1/9) 1 ∆ ×××× 3 ×××× 40 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ (1/9) 2 ∆ ×××× 3 ×××× 41 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ (1/9) 3 ∆ ×××× 3 ×××× 42 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ (1/9) 4 ∆ ×××× 3 ×××× 43 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ (1/9) 5 ∆ ×××× 3 ×××× 44 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-1 (1/9) n ∆ (1/9) n ∆ ×××× 3 ×××× 4(n-1) ∆
第 2 迴
![Page 20: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/20.jpg)
Snow Flake次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ (1/9) 1 ∆ ×××× 3 ×××× 40 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ (1/9) 2 ∆ ×××× 3 ×××× 41 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ (1/9) 3 ∆ ×××× 3 ×××× 42 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ (1/9) 4 ∆ ×××× 3 ×××× 43 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ (1/9) 5 ∆ ×××× 3 ×××× 44 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-1 (1/9) n ∆ (1/9) n ∆ ×××× 3 ×××× 4(n-1) ∆
第 3 迴
![Page 21: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/21.jpg)
Snow Flake次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ (1/9) 1 ∆ ×××× 3 ×××× 40 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ (1/9) 2 ∆ ×××× 3 ×××× 41 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ (1/9) 3 ∆ ×××× 3 ×××× 42 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ (1/9) 4 ∆ ×××× 3 ×××× 43 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ (1/9) 5 ∆ ×××× 3 ×××× 44 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-1 (1/9) n ∆ (1/9) n ∆ ×××× 3 ×××× 4(n-1) ∆
第 4 迴
![Page 22: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/22.jpg)
Snow Flake次數段數
增加面數
每面面積
增加面積
0 3 1 1 ∆ ∆
1 3 ×××× 4 3 (1/9) ∆ (1/9) 1 ∆ ×××× 3 ×××× 40 ∆
2 3 ×××× 42 3 ×××× 4 (1/9) 2 ∆ (1/9) 2 ∆ ×××× 3 ×××× 41 ∆
3 3 ×××× 43 3 ×××× 42 (1/9) 3 ∆ (1/9) 3 ∆ ×××× 3 ×××× 42 ∆
4 3 ×××× 44 3 ×××× 43 (1/9) 4 ∆ (1/9) 4 ∆ ×××× 3 ×××× 43 ∆
5 3 ×××× 45 3 ×××× 44 (1/9) 5 ∆ (1/9) 5 ∆ ×××× 3 ×××× 44 ∆
�
n 3 ×××× 4n 3 ×××× 4 n-1 (1/9) n ∆ (1/9) n ∆ ×××× 3 ×××× 4(n-1) ∆
第 4 迴
![Page 23: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/23.jpg)
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問題 民國84年大學聯考
將邊長為1的等邊三角形之三邊分別三等分,取中間段為一邊向外側做正三角形,並且將中間這一段擦去,其次將剩下的每一邊再三等分,取中間段為一邊向外做正三角形,再將中間這段擦去。仿此程序繼續做下去,得一系列的碎形。求(1)第6次之碎形的周長。(2)第n次碎形的周長。
![Page 27: Microsoft power point 遞迴關係-雪花曲線 [相容模式]](https://reader036.vdocuments.site/reader036/viewer/2022081720/558a1620d8b42ad7028b468c/html5/thumbnails/27.jpg)
問題
設第n次碎形的周長為an,則
a2 =4= 3×(4/3)
a3 =16/3= 3×(4/3)2
a4 =64/9= 3×(4/3)3
a5 =256/27= 3×(4/3)4
an之遞迴關係式
a1 =3
民國84年大學聯考
將邊長為1的等邊三角形之三邊分別三等分,取中間段為一邊向外側做正三角形,並且將中間這一段擦去,其次將剩下的每一邊再三等分,取中間段為一邊向外做正三角形,再將中間這段擦去。仿此程序繼續做下去,得一系列的碎形。求(1)第6次之碎形的周長。(2)第n次碎形的周長。
a6 =1024/81= 3×(4/3)5
第n+1個碎形的周長 =
第n個碎形的周長去掉每一邊的中間段,再增加中間段的 2倍。
an
= an-1−(1/3)a
n -1+(2/3)an -1=(4/3)a
n -1
an之一般式
an
= 3×(4/3)n-1
a1 =3