GeoGebra勾股树制作

效果预览

GeoGebra简介

GeoGebra（音译：几何不难）
是数形结合的神器，个人使用完全免费，并且可跨平台使用

官网https://www.geogebra.org/
经典版本下载 http://download.geogebra.org/installers/5.0/
跨平台的网页版说明
https://wiki.geogebra.org/en/Reference:GeoGebra_Apps_Embedding
跨平台的网页版下载
https://download.geogebra.org/package/geogebra-math-apps-bundle

勾股树的本质是几何分形，为了方便理解，我们从GeoGebra的代数例子入手

分形数阵详解

问题

要求做出下面的分形数阵

1
11, 12
111, 112, 121, 122
1111, 1112, 1121, 1122, 1211, 1212, 1221, 1222
...

求解

指令：

IterationList(join(zip({10*k+1,10*k+2}, k, p)),p,{{1}},2)

结果：

{{1}, {11, 12}, {111, 112, 121, 122}}

这个一大串的东西，着实复杂，不过没关系，我们可以看下面的分析

过程分析

上面的指令用到了GeoGebra内置的三个函数，点击下面三个函数名字，可以直接跳转到官网说明

我们只需要知道每个函数可以输入什么对象，输出什么对象。接地气的说法就是：吃进什么，吐出什么。

IterationList
iterationList(p*10+1, p, {1}, 2)结果为 {1, 11, 111}.

Join
Join({{1, 2, 3}, {3, 4}, {8, 7}}) 结果为 {1, 2, 3, 3, 4, 8, 7}.
Join({5, 4, 3}, {1, 2, 3}) 结果为 {5, 4, 3, 1, 2, 3}.

Zip
Zip(f(2), f, {x+1,x+3}) 结果为 {3, 5}.
Zip(Degree(a), a, {x^2, x^3, x^6}) 结果为 {2, 3, 6}.
zip({10*k+1,10*k+2}, k, {1,2,3}) 结果为 {{11,12}, {21,22}, {31,32}}.

上面的函数，至于怎么理解，怎么翻译，全看个人！在作者看来：

代代相传：IterationList
天人合一：Join
雨露均沾：Zip

理解了GeoGebra这三个内置的函数，我们再分析原来的分形数阵，这里我们从天人合一Join + 雨露均沾Zip开始

先回顾下Zip的最后一个例子zip({10*k+1,10*k+2}, k, {1,2,3}) ，它最后一个参数是一个列表{1,2,3}，雨露均沾之后，每个元素都有了一个新的小家庭 {{11,12}, {21,22}, {31,32}}，并且各个家庭之间都有一堵坚实的{}围墙，当然需要外修天人合一功法Join去除不必要的杂念纷争，去掉{}围墙合为一家，因此下面的命令

Join(zip({10*k+1,10*k+2}, k, {1,2,3}))

不难想到，其结果是

{11,12,21,22,31,32}

类似有
Join(zip({10*k+1,10*k+2}, k, {1}))结果是 {11,12}
Join(zip({10*k+1,10*k+2}, k, {11,12}))结果是 {111, 112, 121, 122}
…
这些指令明显符合代代相传的特点，总不能一直纯手动一步步输入这些指令吧，因此修行第三层代代相传IterationList功法为上策！再回头看看

IterationList(join(zip({10*k+1,10*k+2}, k, p)),p,{{1}},2)

修行代代相传IterationList功法当然要知道：祖师爷是谁，神功传了几代，每两代之间如何继承和发扬这个神功。很明显这里的祖师爷是{{1}}，因为IterationList最后一个参数为2，因此神功只传了两代，这里神功继承和发扬的方法不是别的，正是前面一直提到的天人合一Join + 雨露均沾Zip即join(zip({10*k+1,10*k+2}, k, p))。

为了进一步理解三层神功，读者可以试试把祖师爷{{1}}改写成{{11,12}}。也可以修改下神功传的代数2。

勾股树详解

对前面的代数分形做个总结

雨露均沾Zip，出现很多不融合的家庭，需要天人合一功法Join去除杂念纷争，让各个家庭合成一家，最后把这种优秀的神功代代相传IterationList，必然子子孙孙无穷匮也。

对勾股树的绘制做个分析

不难知道制作勾股树最简单的做法是只绘制矩形，中间的直角三角形自然会通过矩形刻画出来。

我们需要完成下面的步骤：

先绘制一个矩形
完成能由1个矩形得到2个矩形的指令，
结合天人合一 Join + 雨露均沾Zip就可以在上步基础上实现由2个矩形得到4个矩形的指令，以及4个矩形得到8个矩形…
最后用代代相传IterationList，大功告成

具体实现

绘制矩形p1

p1 = Polygon((-1,0), (1,0), 4)

通过矩形p1得到矩形p11与p12

先绘制半圆

c1 = Semicircle(Vertex(p1,4), Vertex(p1,3))

再半圆上取点，保证直角三角形，0.5表示点的位置，后面将会用滑条代替
```
A = Point(c1,0.5)
```

得到矩形p11与p12

p11 = Polygon(Vertex(p1,4), A, 4)
p12 = Polygon(A, Vertex(p1,3), 4)

结合天人合一 Join + 雨露均沾Zip

我们先把上一步的所有指令写成一块
要注意下面的一大块有缩进的代码，其实就是一行代码，写成这样的格式为了提高可读性，在使用的时候，需要整块指令复制Ctrl+C后，按快捷键Ctrl+V粘贴到GeoGebra的命令行中
```
{Polygon(Vertex(p1,4),  
	     Point(Semicircle(Vertex(p1,4), Vertex(p1,3)) ,0.5), 
	     4),  
 Polygon(Point(Semicircle(Vertex(p1,4), Vertex(p1,3)) ,0.5), 
         Vertex(p1,3), 
         4)
 }
```

再套用前面的代数功法天人合一 Join + 雨露均沾Zip

Join(zip({10*k+1,10*k+2}, k, {1,2,3}))

把代数功法运用到几何里面，就是

p2 = Join(zip(
		{Polygon(Vertex(k,4),  
		         Point(Semicircle(Vertex(k,4), Vertex(k,3)) ,0.5), 
		         4),  
		 Polygon(Point(Semicircle(Vertex(k,4), Vertex(k,3)) ,0.5),
		         Vertex(k,3), 
		         4)
		 }, 
	     k, 
	     {p1}
	     )
	)

从而有

p3 = Join(zip(
		{Polygon(Vertex(k,4),  
		         Point(Semicircle(Vertex(k,4), Vertex(k,3)) ,0.5), 
		         4),  
		 Polygon(Point(Semicircle(Vertex(k,4), Vertex(k,3)) ,0.5),
		         Vertex(k,3), 
		         4)
		 }, 
	     k, 
	     p2
	     )
	)

p4 = Join(zip(
		{Polygon(Vertex(k,4),  
		         Point(Semicircle(Vertex(k,4), Vertex(k,3)) ,0.5), 
		         4),  
		 Polygon(Point(Semicircle(Vertex(k,4), Vertex(k,3)) ,0.5),
		         Vertex(k,3), 
		         4)
		 }, 
	     k, 
	     p3
	     )
	)

p5 = Join(zip(
		{Polygon(Vertex(k,4),  
		         Point(Semicircle(Vertex(k,4), Vertex(k,3)) ,0.5), 
		         4),  
		 Polygon(Point(Semicircle(Vertex(k,4), Vertex(k,3)) ,0.5),
		         Vertex(k,3), 
		         4)
		 }, 
	     k, 
	     p4
	     )
	)

…

依次做下去，可以得到美丽的勾股树，这种繁琐的做法当然是不可取!

要注意小细节： 从{p1}到p2，参数{p1}是有围墙{}的，而从p2到p3，参数p2没有围墙，并且后面的都没围墙。

最后用神功代代相传IterationList可得到如下指令

IterationList(
	Join(zip(
		{Polygon(Vertex(k,4),  
		         Point(Semicircle(Vertex(k,4), Vertex(k,3)), 0.5), 
		         4),  
		 Polygon(Point(Semicircle(Vertex(k,4), Vertex(k,3)), 0.5),
		         Vertex(k,3), 
		         4)
		 }, 
	     k, 
	     p
	     )
	),
	 p,
	 {{p1}},
	 6  
)

这就意味着，先输入

p1 = Polygon((-1,0), (1,0), 4)

再输入上面那大一串的代代相传相关指令，就可以直接得到5层勾股树！
但是我们还希望用滑动条控制勾股树的层数，以及直角三角形的形状，为此只需要稍加修改，先输入两个滑条指令，

t = slider(0, 1, 0.01)
n = slider(1, 4, 1)

把上面两段代码中的0.5都换成t, 并且把6换成n

全部指令

p1 = Polygon((-1,0), (1,0), 4)
t = slider(0, 1, 0.01)
n = slider(1, 4, 1)
IterationList(Join(zip({Polygon(Vertex(k,4), Point(Semicircle(Vertex(k,4),Vertex(k,3)), t), 4), Polygon(Point(Semicircle(Vertex(k,4), Vertex(k,3)), t), Vertex(k,3), 4) }, k, p)),p,{{p1}},n)

上面的指令，需要逐行复制Ctrl+C，再快捷键Ctrl+V粘贴到GeoGebra的命令行中

学以致用

在线试试

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温故知新

本次教程学会了三大神功

代代相传：IterationList
天人合一：Join
雨露均沾：Zip

并且学会了代数分形，几何分形相关制图

更学会了很实用的方法：复杂的几何绘图问题可以从代数开始

如果有所尝试，一定会感受到数形结合的美

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本文来源 https://kz16.top/geogebra/ptree

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作者：Zhou Bingzhen
2019.10.19

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