四面体的棱长组合计数

hujunhua

【问题】以长度为11，12，13，14，15，16的六条线段为棱，能够搭建多少个不同的四面体？
（镜像对称而不能旋转重合的两个四面体亦视为不同）。

mathe

镜像对称的（也就是一个顶点不变，余下三个改变方向的）先看成一样，那么显然总数为${6!}/{4!}=30$种
如果考虑镜像对称的看成不同，每个就可以有两个不同的取向，所以应该$60$种。

hujunhua

@mathe 边长置换数 / 顶点置换数，妙！

有一个同学是这么计数的：边长置换数为6！，一个三棱锥有3个旋转，所以一个四面体有12个旋转，故除去旋转相同的共是6！/12=60种。

两者一样很巧妙啊。

hujunhua

一个同学用的分类统计法：

11，12，13三条边的位置关系可分为以下3种：
一、构成三角形，这类共有2*3！=12种
二、构成Y字形，这是三角形的对偶形，所以也是2*3！=12种
三、构成Z字形，共有2*3*3！=36种
所以总共是12+12+36=60种。

hujunhua

另一个同学的分类统计法：

考虑11边的对边，共有5种选择，剩下的四条边构成一个回路，圆排列数=3！共有5*3！=30种。
再将11与对边互换位置，也有30种。
总计60种。
感觉这个分类比上面的简明。但是好像有问题，跟圆排列有什么关系呢？

nyy

6！=720
怎么判断两种排列具有等价的空间结构呢，看它们的体积!
按凯莱行列式计算体积，发现每24个排列共一个体积值，共有30个不同的体积：\[
\left\{21 \sqrt{159},\frac{39 \sqrt{699}}{4},\frac{7 \sqrt{10031}}{3},\frac{5 \sqrt{20135}}{3},\frac{5 \sqrt{23999}}{3},\frac{17 \sqrt{27011}}{12},\frac{13 \sqrt{56459}}{12},\sqrt{73079},\frac{3 \sqrt{111471}}{4},\frac{3 \sqrt{114351}}{4},\frac{5 \sqrt{314135}}{12},\frac{5 \sqrt{385199}}{12},\frac{5 \sqrt{393839}}{12},\frac{5 \sqrt{403079}}{12},\frac{\sqrt{614759}}{3},\frac{\sqrt{1136459}}{4},\frac{\sqrt{7930751}}{12},\frac{\sqrt{8073251}}{12},\frac{\sqrt{9043319}}{12},\frac{\sqrt{9156551}}{12},\frac{\sqrt{9199319}}{12},\frac{\sqrt{9233831}}{12},\frac{\sqrt{9690071}}{12},\frac{\sqrt{9844511}}{12},\frac{\sqrt{9934019}}{12},\frac{\sqrt{10057031}}{12},\frac{\sqrt{10100471}}{12},\frac{\sqrt{10314911}}{12},\frac{\sqrt{10391471}}{12},\frac{\sqrt{10450151}}{12}\right\}
\]考虑镜像对称，因此有30*2=60种不同的四面体。

nyy

我已经用穷举法计算过了，对于a b c d e f六条棱，
计算出720种排列，然后计算出所有的排列对应的体积，
然后体积值进行合并，总共得到30种不同的体积！
每种体积都对应24种置换情况

王守恩

【问题】以长度为11，12，13，14，15，16的六条线段为棱，能够搭建多少个不同的四面体？
（镜像对称而不能旋转重合的两个四面体亦视为不同）。

把四面体看作一个平放在桌面上的三角形ABC(P是内部一点),

BC=a,CA=b,AB=c,PA=A,PB=B,PC=C

我们总可以让 11=a。

12有3种可能：

12=b，可以有24种变化。

12=c，可以有24种变化。

12=A，可以有24种变化，其中有12种=另外12种。

合计能够搭建24+24+24/2=60个不同的四面体。

王守恩

【问题】以长度为11，12，13，14，15，16的六条线段为棱，能够搭建多少个不同的四面体？
（镜像对称而不能旋转重合的两个四面体亦视为不同）。

把四面体看作一个平放在桌面上的三角形ABC(P是内部一点),

BC=a,CA=b,AB=c,PA=A,PB=B,PC=C

我们总可以让 11=a。

12有3种可能。

第1种可能: 12=b，可以有24种变化。
1: a,b,c,A,B,C=11,12,13,14,15,16,  
2: a,b,c,A,B,C=11,12,13,14,16,15,
3: a,b,c,A,B,C=11,12,13,15,14,16,
4: a,b,c,A,B,C=11,12,13,15,16,14,
5: a,b,c,A,B,C=11,12,13,16,14,15,
6: a,b,c,A,B,C=11,12,13,16,15,14,
1: a,b,c,A,B,C=11,12,14,13,15,16,  
2: a,b,c,A,B,C=11,12,14,13,16,15,
3: a,b,c,A,B,C=11,12,14,15,13,16,
4: a,b,c,A,B,C=11,12,14,15,16,13,
5: a,b,c,A,B,C=11,12,14,16,13,15,
6: a,b,c,A,B,C=11,12,14,16,15,13,
1: a,b,c,A,B,C=11,12,15,13,14,16,  
2: a,b,c,A,B,C=11,12,15,13,16,14,
3: a,b,c,A,B,C=11,12,15,14,13,16,
4: a,b,c,A,B,C=11,12,15,14,16,13,
5: a,b,c,A,B,C=11,12,15,16,13,14,
6: a,b,c,A,B,C=11,12,15,16,14,13,
1: a,b,c,A,B,C=11,12,16,13,14,15,  
2: a,b,c,A,B,C=11,12,16,13,15,14,
3: a,b,c,A,B,C=11,12,16,14,13,15,
4: a,b,c,A,B,C=11,12,16,14,15,13,
5: a,b,c,A,B,C=11,12,16,15,13,14,
6: a,b,c,A,B,C=11,12,16,15,14,13,

第2种可能: 12=c，可以有24种变化。
1: a,b,c,A,B,C=11,13,12,14,15,16,  
2: a,b,c,A,B,C=11,13,12,14,16,15,
3: a,b,c,A,B,C=11,13,12,15,14,16,
4: a,b,c,A,B,C=11,13,12,15,16,14,
5: a,b,c,A,B,C=11,13,12,16,14,15,
6: a,b,c,A,B,C=11,13,12,16,15,14,
1: a,b,c,A,B,C=11,14,12,13,15,16,  
2: a,b,c,A,B,C=11,14,12,13,16,15,
3: a,b,c,A,B,C=11,14,12,15,13,16,
4: a,b,c,A,B,C=11,14,12,15,16,13,
5: a,b,c,A,B,C=11,14,12,16,13,15,
6: a,b,c,A,B,C=11,14,12,16,15,13,
1: a,b,c,A,B,C=11,15,12,13,14,16,  
2: a,b,c,A,B,C=11,15,12,13,16,14,
3: a,b,c,A,B,C=11,15,12,14,13,16,
4: a,b,c,A,B,C=11,15,12,14,16,13,
5: a,b,c,A,B,C=11,15,12,16,13,14,
6: a,b,c,A,B,C=11,15,12,16,14,13,
1: a,b,c,A,B,C=11,16,12,13,14,15,  
2: a,b,c,A,B,C=11,16,12,13,15,14,
3: a,b,c,A,B,C=11,16,12,14,13,15,
4: a,b,c,A,B,C=11,16,12,14,15,13,
5: a,b,c,A,B,C=11,16,12,15,13,14,
6: a,b,c,A,B,C=11,16,12,15,14,13,

第3种可能: 12=A，可以有24种变化，其中有12种=另外12种。
1: a,b,c,A,B,C=11,13,14,12,15,16,  
2: a,b,c,A,B,C=11,13,14,12,16,15,
3: a,b,c,A,B,C=11,13,15,12,14,16,
4: a,b,c,A,B,C=11,13,15,12,16,14,
5: a,b,c,A,B,C=11,13,16,12,14,15,
6: a,b,c,A,B,C=11,13,16,12,15,14,
1: a,b,c,A,B,C=11,14,13,12,15,16,  
2: a,b,c,A,B,C=11,14,13,12,16,15,
3: a,b,c,A,B,C=11,14,15,12,13,16,
4: a,b,c,A,B,C=11,14,15,12,16,13,
5: a,b,c,A,B,C=11,14,16,12,13,15,
6: a,b,c,A,B,C=11,14,16,12,15,13,

合计能够搭建24+24+24/2=60个不同的四面体。

nyy

{{{11,12,13,14,15,16,(39 Sqrt[699])/4},{11,13,12,14,16,15,(39 Sqrt[699])/4},{11,15,16,14,12,13,(39 Sqrt[699])/4},{11,16,15,14,13,12,(39 Sqrt[699])/4},{12,11,13,15,14,16,(39 Sqrt[699])/4},{12,13,11,15,16,14,(39 Sqrt[699])/4},{12,14,16,15,11,13,(39 Sqrt[699])/4},{12,16,14,15,13,11,(39 Sqrt[699])/4},{13,11,12,16,14,15,(39 Sqrt[699])/4},{13,12,11,16,15,14,(39 Sqrt[699])/4},{13,14,15,16,11,12,(39 Sqrt[699])/4},{13,15,14,16,12,11,(39 Sqrt[699])/4},{14,12,16,11,15,13,(39 Sqrt[699])/4},{14,13,15,11,16,12,(39 Sqrt[699])/4},{14,15,13,11,12,16,(39 Sqrt[699])/4},{14,16,12,11,13,15,(39 Sqrt[699])/4},{15,11,16,12,14,13,(39 Sqrt[699])/4},{15,13,14,12,16,11,(39 Sqrt[699])/4},{15,14,13,12,11,16,(39 Sqrt[699])/4},{15,16,11,12,13,14,(39 Sqrt[699])/4},{16,11,15,13,14,12,(39 Sqrt[699])/4},{16,12,14,13,15,11,(39 Sqrt[699])/4},{16,14,12,13,11,15,(39 Sqrt[699])/4},{16,15,11,13,12,14,(39 Sqrt[699])/4}},{{11,12,13,14,16,15,Sqrt[9844511]/12},{11,13,12,14,15,16,Sqrt[9844511]/12},{11,15,16,14,13,12,Sqrt[9844511]/12},{11,16,15,14,12,13,Sqrt[9844511]/12},{12,11,13,16,14,15,Sqrt[9844511]/12},{12,13,11,16,15,14,Sqrt[9844511]/12},{12,14,15,16,11,13,Sqrt[9844511]/12},{12,15,14,16,13,11,Sqrt[9844511]/12},{13,11,12,15,14,16,Sqrt[9844511]/12},{13,12,11,15,16,14,Sqrt[9844511]/12},{13,14,16,15,11,12,Sqrt[9844511]/12},{13,16,14,15,12,11,Sqrt[9844511]/12},{14,12,15,11,16,13,Sqrt[9844511]/12},{14,13,16,11,15,12,Sqrt[9844511]/12},{14,15,12,11,13,16,Sqrt[9844511]/12},{14,16,13,11,12,15,Sqrt[9844511]/12},{15,11,16,13,14,12,Sqrt[9844511]/12},{15,12,14,13,16,11,Sqrt[9844511]/12},{15,14,12,13,11,16,Sqrt[9844511]/12},{15,16,11,13,12,14,Sqrt[9844511]/12},{16,11,15,12,14,13,Sqrt[9844511]/12},{16,13,14,12,15,11,Sqrt[9844511]/12},{16,14,13,12,11,15,Sqrt[9844511]/12},{16,15,11,12,13,14,Sqrt[9844511]/12}},{{11,12,13,15,14,16,(5 Sqrt[385199])/12},{11,13,12,15,16,14,(5 Sqrt[385199])/12},{11,14,16,15,12,13,(5 Sqrt[385199])/12},{11,16,14,15,13,12,(5 Sqrt[385199])/12},{12,11,13,14,15,16,(5 Sqrt[385199])/12},{12,13,11,14,16,15,(5 Sqrt[385199])/12},{12,15,16,14,11,13,(5 Sqrt[385199])/12},{12,16,15,14,13,11,(5 Sqrt[385199])/12},{13,11,12,16,15,14,(5 Sqrt[385199])/12},{13,12,11,16,14,15,(5 Sqrt[385199])/12},{13,14,15,16,12,11,(5 Sqrt[385199])/12},{13,15,14,16,11,12,(5 Sqrt[385199])/12},{14,11,16,12,15,13,(5 Sqrt[385199])/12},{14,13,15,12,16,11,(5 Sqrt[385199])/12},{14,15,13,12,11,16,(5 Sqrt[385199])/12},{14,16,11,12,13,15,(5 Sqrt[385199])/12},{15,12,16,11,14,13,(5 Sqrt[385199])/12},{15,13,14,11,16,12,(5 Sqrt[385199])/12},{15,14,13,11,12,16,(5 Sqrt[385199])/12},{15,16,12,11,13,14,(5 Sqrt[385199])/12},{16,11,14,13,15,12,(5 Sqrt[385199])/12},{16,12,15,13,14,11,(5 Sqrt[385199])/12},{16,14,11,13,12,15,(5 Sqrt[385199])/12},{16,15,12,13,11,14,(5 Sqrt[385199])/12}},{{11,12,13,15,16,14,(5 Sqrt[403079])/12},{11,13,12,15,14,16,(5 Sqrt[403079])/12},{11,14,16,15,13,12,(5 Sqrt[403079])/12},{11,16,14,15,12,13,(5 Sqrt[403079])/12},{12,11,13,16,15,14,(5 Sqrt[403079])/12},{12,13,11,16,14,15,(5 Sqrt[403079])/12},{12,14,15,16,13,11,(5 Sqrt[403079])/12},{12,15,14,16,11,13,(5 Sqrt[403079])/12},{13,11,12,14,15,16,(5 Sqrt[403079])/12},{13,12,11,14,16,15,(5 Sqrt[403079])/12},{13,15,16,14,11,12,(5 Sqrt[403079])/12},{13,16,15,14,12,11,(5 Sqrt[403079])/12},{14,11,16,13,15,12,(5 Sqrt[403079])/12},{14,12,15,13,16,11,(5 Sqrt[403079])/12},{14,15,12,13,11,16,(5 Sqrt[403079])/12},{14,16,11,13,12,15,(5 Sqrt[403079])/12},{15,12,14,11,16,13,(5 Sqrt[403079])/12},{15,13,16,11,14,12,(5 Sqrt[403079])/12},{15,14,12,11,13,16,(5 Sqrt[403079])/12},{15,16,13,11,12,14,(5 Sqrt[403079])/12},{16,11,14,12,15,13,(5 Sqrt[403079])/12},{16,13,15,12,14,11,(5 Sqrt[403079])/12},{16,14,11,12,13,15,(5 Sqrt[403079])/12},{16,15,13,12,11,14,(5 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Sqrt[23999])/3},{12,13,14,11,16,15,(5 Sqrt[23999])/3},{12,14,13,11,15,16,(5 Sqrt[23999])/3},{12,15,16,11,14,13,(5 Sqrt[23999])/3},{12,16,15,11,13,14,(5 Sqrt[23999])/3},{13,11,15,16,12,14,(5 Sqrt[23999])/3},{13,12,14,16,11,15,(5 Sqrt[23999])/3},{13,14,12,16,15,11,(5 Sqrt[23999])/3},{13,15,11,16,14,12,(5 Sqrt[23999])/3},{14,11,16,15,12,13,(5 Sqrt[23999])/3},{14,12,13,15,11,16,(5 Sqrt[23999])/3},{14,13,12,15,16,11,(5 Sqrt[23999])/3},{14,16,11,15,13,12,(5 Sqrt[23999])/3},{15,11,13,14,12,16,(5 Sqrt[23999])/3},{15,12,16,14,11,13,(5 Sqrt[23999])/3},{15,13,11,14,16,12,(5 Sqrt[23999])/3},{15,16,12,14,13,11,(5 Sqrt[23999])/3},{16,11,14,13,12,15,(5 Sqrt[23999])/3},{16,12,15,13,11,14,(5 Sqrt[23999])/3},{16,14,11,13,15,12,(5 Sqrt[23999])/3},{16,15,12,13,14,11,(5 Sqrt[23999])/3}},{{11,13,16,12,14,15,Sqrt[8073251]/12},{11,14,15,12,13,16,Sqrt[8073251]/12},{11,15,14,12,16,13,Sqrt[8073251]/12},{11,16,13,12,15,14,Sqrt[8073251]/12},{12,13,15,11,14,16,Sqrt[8073251]/12},{12,14,16,11,13,15,Sqrt[8073251]/12},{12,15,13,11,16,14,Sqrt[8073251]/12},{12,16,14,11,15,13,Sqrt[8073251]/12},{13,11,16,14,12,15,Sqrt[8073251]/12},{13,12,15,14,11,16,Sqrt[8073251]/12},{13,15,12,14,16,11,Sqrt[8073251]/12},{13,16,11,14,15,12,Sqrt[8073251]/12},{14,11,15,13,12,16,Sqrt[8073251]/12},{14,12,16,13,11,15,Sqrt[8073251]/12},{14,15,11,13,16,12,Sqrt[8073251]/12},{14,16,12,13,15,11,Sqrt[8073251]/12},{15,11,14,16,12,13,Sqrt[8073251]/12},{15,12,13,16,11,14,Sqrt[8073251]/12},{15,13,12,16,14,11,Sqrt[8073251]/12},{15,14,11,16,13,12,Sqrt[8073251]/12},{16,11,13,15,12,14,Sqrt[8073251]/12},{16,12,14,15,11,13,Sqrt[8073251]/12},{16,13,11,15,14,12,Sqrt[8073251]/12},{16,14,12,15,13,11,Sqrt[8073251]/12}},{{11,13,16,12,15,14,Sqrt[9233831]/12},{11,14,15,12,16,13,Sqrt[9233831]/12},{11,15,14,12,13,16,Sqrt[9233831]/12},{11,16,13,12,14,15,Sqrt[9233831]/12},{12,13,14,11,15,16,Sqrt[9233831]/12},{12,14,13,11,16,15,Sqrt[9233831]/12},{12,15,16,11,13,14,Sqrt[9233831]/12},{12,16,15,11,14,13,Sqrt[9233831]/12},{13,11,16,15,12,14,Sqrt[9233831]/12},{13,12,14,15,11,16,Sqrt[9233831]/12},{13,14,12,15,16,11,Sqrt[9233831]/12},{13,16,11,15,14,12,Sqrt[9233831]/12},{14,11,15,16,12,13,Sqrt[9233831]/12},{14,12,13,16,11,15,Sqrt[9233831]/12},{14,13,12,16,15,11,Sqrt[9233831]/12},{14,15,11,16,13,12,Sqrt[9233831]/12},{15,11,14,13,12,16,Sqrt[9233831]/12},{15,12,16,13,11,14,Sqrt[9233831]/12},{15,14,11,13,16,12,Sqrt[9233831]/12},{15,16,12,13,14,11,Sqrt[9233831]/12},{16,11,13,14,12,15,Sqrt[9233831]/12},{16,12,15,14,11,13,Sqrt[9233831]/12},{16,13,11,14,15,12,Sqrt[9233831]/12},{16,15,12,14,13,11,Sqrt[9233831]/12}}}
这个是我的计算，最后一个元素对应的是体积！