多面体欧拉定理 的发现

1、多面体欧拉定理 的发现研究性课题：一、认识欧拉1 1、数学奇才欧拉、数学奇才欧拉欧拉欧拉-瑞士人（瑞士人（Euler,L. 1707-1783）；）；欧拉欧拉-16 岁获得硕士学位；岁获得硕士学位；欧拉欧拉-数学史上数学史上“高产高产”的数学家。在世发表论文的数学家。在世发表论文700多篇，多篇， 去世后还留下去世后还留下100多篇待发表；多篇待发表；欧拉欧拉-首先使用首先使用f(x)表示函数，用表示函数，用e表示自然对数的底，用表示自然对数的底，用a、 b、c 表示表示ABC，用，用表示求和，用表示求和，用i表示虚数单位等；表示虚数单位等；欧拉欧拉-目前数学中有欧拉公式、欧拉常数、欧拉猜想、

2、欧拉目前数学中有欧拉公式、欧拉常数、欧拉猜想、欧拉 方法、欧拉方程、欧拉定理。方法、欧拉方程、欧拉定理。2 2 、欧拉眼中的简单多面体、欧拉眼中的简单多面体棱棱 柱柱棱棱 锥锥正多面体正多面体凸多面体凸多面体简单多面体充气后 在立体几何多面体的研究中欧拉首在立体几何多面体的研究中欧拉首先发现并证明了欧拉定理。今天我们沿先发现并证明了欧拉定理。今天我们沿着他的着他的正多面体正多面体 顶点数顶点数V 面数面数F 棱数棱数E正四面体正四面体正六面体正六面体正八面体正八面体正十二面体正十二面体正二十面体正二十面体 4 4 6 8 6 12 6 8 12 20 12 30 12 20 30二、观察归纳

3、1 1、填表、填表2、结论： V+FE23 3、验证提高、验证提高简单多面体简单多面体 顶点数顶点数V 面数面数F 棱数棱数EN N棱锥棱锥N N棱柱棱柱V、F、E间的关系间的关系 N+1 N+1 2N 2N N+2 3N V+FE2三、欧拉猜想 简单多面体的顶点数简单多面体的顶点数V V、棱数、棱数E E、及、及面数面数F F间有关系：间有关系：V+FV+FE=2E=2 四、尝试证明1、下面我们以四面体、下面我们以四面体ABCD为例为例 体会欧拉的证法体会欧拉的证法：ABCDV+FE的值暂不知道V+F1E变化情况压扁压扁不变V+F1E变化情况不变不变不变V+F1E变化情况不变不变上图中上图中

4、V+F1E2+011那么 V+FE= V+（F1+1）E =1+1=2 简单多面体的顶点数简单多面体的顶点数V V、棱数、棱数E E、及面数及面数F F间有关系：间有关系：V+FV+FE=2E=22、多面体欧拉定理 ：五、总结回顾 欧拉公式的发现和证明是得益于“多面体的表面都是用橡皮薄膜制作的”这一观念上的创新，是得益于“向它们内部冲气”和“底面剪掉，然后其余各面拉开铺平”这一方法的创新。 希望同学们能够象欧拉那样善于思考、乐希望同学们能够象欧拉那样善于思考、乐于创新！于创新！六、作业六、作业： 以正六面体为例验证多面体以正六面体为例验证多面体欧拉定理欧拉定理4C0z)w&s!pYmU

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