论勾股四态、以及正交球心间同构的场方程

On Pythagorean Four-State and Isomorphism Field Equations between Orthogonal Spherical Centers

摘要: 1球至4球正交构成点、线、面、体的勾股4态，对其4态15种正交球心间场，建立了基于各球半径的正交球心间场同构方程行列式，且以此可推广至任意有限高维。

Abstract: A determinant of isomorphic equation of orthogonal spherical interphase field based on radius of each sphere is established for 15 kinds of orthogonal spherical interphase fields of 4 states, which consist of point, line, surface and body, and can be extended to any finite high dimension.

1. 引言

1球至4球正交，构成的点(球)、线(勾股定理)、面(面积勾股定理 [1] )、体(体积勾股定理 [2] )均有各自的定理。那么这些各自的定理间，特别是球心间所围场是否存在同构的公式？

2. 正交球心间存在同构的场方程行列式的证明

1球至4球正交，其球心间所围场方程行列式可分为繁式和简式2种，以及所有球半径均相等公式。

2.1. (繁式)正交球心场方程行列式

类似Cayley-Menger行列式 [3] ，或可称2点间距式， ${ℝ}^n$ 代表球心所围场(含所有子集)，行列式为：

${\left({ℝ}^n\right)}^2={\left(-1\right)}^n{\left(\frac{1}{2}\right)}^{n-1}{\left(\frac{1}{\left(n-1\right)!}\right)}^2\left|\begin{array}{ccccccc}0& 1& 1& 1& 1& \cdots & 1\\ 1& 0& d_{12}^2& d_{13}^2& d_{14}^2& \cdots & d_{1n}^2\\ 1& d_{21}^2& 0& d_{23}^2& d_{24}^2& \cdots & d_{2n}^2\\ 1& d_{31}^2& d_{32}^2& 0& d_{34}^2& \cdots & d_{3n}^2\\ 1& d_{41}^2& d_{42}^2& d_{43}^2& 0& \cdots & d_{4n}^2\\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& d_{n1}^2& d_{n2}^2& d_{n3}^2& d_{n4}^2& \cdots & 0\end{array}\right|$ (1)

$n\in 1,2,3,4$ 表示参与正交球的数量，下标： $ij\in 1,2,3,4$ 表示各球心点，d_ij是连接两个球心连线的长度。

2.1.1. 例：4球正交球心间场为垂心四面体的体积的平方

$\begin{array}{c}{\left({ℝ}_{1234}^4\right)}^2={\left(-1\right)}^4{\left(\frac{1}{2}\right)}^{4-1}{\left(\frac{1}{\left(4-1\right)!}\right)}^2\left|\begin{array}{ccccc}0& 1& 1& 1& 1\\ 1& 0& d_{12}^2& d_{13}^2& d_{14}^2\\ 1& d_{21}^2& 0& d_{23}^2& d_{24}^2\\ 1& d_{31}^2& d_{32}^2& 0& d_{34}^2\\ 1& d_{41}^2& d_{42}^2& d_{43}^2& 0\end{array}\right|\\ =\frac{1}{288}\left|\begin{array}{ccccc}0& 1& 1& 1& 1\\ 1& 0& a^2+b^2& a^2+c^2& a^2+d^2\\ 1& b^2+a^2& 0& b^2+c^2& b^2+d^2\\ 1& c^2+a^2& c^2+b^2& 0& c^2+d^2\\ 1& d^2+a^2& d^2+b^2& d^2+c^2& 0\end{array}\right|\\ =\frac{1}{36}\left(a^2{b}^2{c}^2+a^2{b}^2{d}^2+a^2{c}^2{d}^2+b^2{c}^2{d}^2\right)\end{array}$

各球半径 $\in a,b,c,d$ 。

2.1.2. 例：4个3球正交球心间场为三角形的面积的平方

$\begin{array}{c}{\left({ℝ}_{123}^3\right)}^2={\left(-1\right)}^3{\left(\frac{1}{2}\right)}^{3-1}{\left(\frac{1}{\left(3-1\right)!}\right)}^2\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& d_{12}^2& d_{13}^2\\ 1& d_{21}^2& 0& d_{23}^2\\ 1& d_{31}^2& d_{32}^2& 0\end{array}\right|\\ =\frac{1}{16}\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& a^2+b^2& a^2+c^2\\ 1& b^2+a^2& 0& b^2+c^2\\ 1& c^2+a^2& c^2+b^2& 0\end{array}\right|=\frac{1}{4}\left(a^2{b}^2+a^2{c}^2+b^2{c}^2\right)\end{array}$

下标 $ij\in 1,2,3,4$ 表示各球心点。

或

$\begin{array}{c}{\left({ℝ}_{124}^3\right)}^2={\left(-1\right)}^3{\left(\frac{1}{2}\right)}^{3-1}{\left(\frac{1}{\left(3-1\right)!}\right)}^2\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& d_{12}^2& d_{14}^2\\ 1& d_{21}^2& 0& d_{24}^2\\ 1& d_{41}^2& d_{42}^2& 0\end{array}\right|\\ =\frac{1}{16}\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& a^2+b^2& a^2+d^2\\ 1& b^2+a^2& 0& b^2+d^2\\ 1& d^2+a^2& d^2+b^2& 0\end{array}\right|=\frac{1}{4}\left(a^2{b}^2+a^2{d}^2+b^2{d}^2\right)\end{array}$

或

$\begin{array}{c}{\left({ℝ}_{134}^3\right)}^2={\left(-1\right)}^3{\left(\frac{1}{2}\right)}^{3-1}{\left(\frac{1}{\left(3-1\right)!}\right)}^2\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& d_{13}^2& d_{14}^2\\ 1& d_{31}^2& 0& d_{34}^2\\ 1& d_{41}^2& d_{43}^2& 0\end{array}\right|\\ =\frac{1}{16}\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& a^2+c^2& a^2+d^2\\ 1& c^2+a^2& 0& c^2+d^2\\ 1& d^2+a^2& d^2+c^2& 0\end{array}\right|\\ =\frac{1}{4}\left(a^2{c}^2+a^2{d}^2+c^2{d}^2\right)\end{array}$

或

$\begin{array}{c}{\left({ℝ}_{234}^3\right)}^2={\left(-1\right)}^3{\left(\frac{1}{2}\right)}^{3-1}{\left(\frac{1}{\left(3-1\right)!}\right)}^2\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& d_{23}^2& d_{24}^2\\ 1& d_{32}^2& 0& d_{34}^2\\ 1& d_{42}^2& d_{43}^2& 0\end{array}\right|\\ =\frac{1}{16}\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& b^2+c^2& b^2+d^2\\ 1& c^2+b^2& 0& c^2+d^2\\ 1& d^2+b^2& d^2+c^2& 0\end{array}\right|\\ =\frac{1}{4}\left(b^2{c}^2+b^2{d}^2+c^2{d}^2\right)\end{array}$

2.1.3. 例：6个2球正交球心间场为2点间直线的平方

${\left({ℝ}_{12}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{2}\right)}^{2-1}{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& d_{12}^2\\ 1& d_{21}^2& 0\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& a^2+b^2\\ 1& b^2+a^2& 0\end{array}\right|=a^2+b^2$

或

${\left({ℝ}_{13}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{2}\right)}^{2-1}{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& d_{13}^2\\ 1& d_{31}^2& 0\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& a^2+c^2\\ 1& c^2+a^2& 0\end{array}\right|=a^2+c^2$

或

${\left({ℝ}_{14}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{2}\right)}^{2-1}{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& d_{14}^2\\ 1& d_{41}^2& 0\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& a^2+d^2\\ 1& d^2+a^2& 0\end{array}\right|=a^2+d^2$

或

${\left({ℝ}_{23}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{2}\right)}^{2-1}{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& d_{23}^2\\ 1& d_{32}^2& 0\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& b^2+c^2\\ 1& c^2+b^2& 0\end{array}\right|=b^2+c^2$

或

${\left({ℝ}_{24}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{2}\right)}^{2-1}{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& d_{24}^2\\ 1& d_{42}^2& 0\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& b^2+d^2\\ 1& d^2+b^2& 0\end{array}\right|=b^2+d^2$

或

${\left({ℝ}_{34}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{2}\right)}^{2-1}{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& d_{34}^2\\ 1& d_{43}^2& 0\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& c^2+d^2\\ 1& d^2+c^2& 0\end{array}\right|=c^2+d^2$

2.1.4. 例：4个球正交球心为点的平方

${\left({ℝ}_1^1\right)}^2={\left(-1\right)}^1{\left(\frac{1}{2}\right)}^{1-1}{\left(\frac{1}{\left(1-1\right)!}\right)}^2\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=-1\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=1$

或

${\left({ℝ}_2^1\right)}^2={\left(-1\right)}^1{\left(\frac{1}{2}\right)}^{1-1}{\left(\frac{1}{\left(1-1\right)!}\right)}^2\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=-1\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=1$

或

${\left({ℝ}_3^1\right)}^2={\left(-1\right)}^1{\left(\frac{1}{2}\right)}^{1-1}{\left(\frac{1}{\left(1-1\right)!}\right)}^2\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=-1\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=1$

或

${\left({ℝ}_4^1\right)}^2={\left(-1\right)}^1{\left(\frac{1}{2}\right)}^{1-1}{\left(\frac{1}{\left(1-1\right)!}\right)}^2\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=-1\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=1$

2.2. (简式)正交球心场方程行列式

直接使用各正交球半径的行列式为：

${\left({ℝ}^n\right)}^2={\left(-1\right)}^n{\left(\frac{1}{\left(n-1\right)!}\right)}^2\left|\begin{array}{ccccccc}0& 1& 1& 1& 1& \cdots & 1\\ 1& 0& r_2^2& r_3^2& r_4^2& \cdots & r_n^2\\ 1& r_1^2& 0& r_3^2& r_4^2& \cdots & r_n^2\\ 1& r_1^2& r_2^2& 0& r_4^2& \cdots & r_n^2\\ 1& r_1^2& r_2^2& r_3^2& 0& \cdots & r_n^2\\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& r_1^2& r_2^2& r_3^2& r_4^2& \cdots & 0\end{array}\right|$ (2)

下标： $n\in 1,2,3,4$ 表示参与正交球的数量， $r_n$ 为各正交球半径。

2.2.1. 例：4球正交球心间场为垂心四面体的体积的平方

$\begin{array}{c}{\left({ℝ}_{1234}^4\right)}^2={\left(-1\right)}^4{\left(\frac{1}{\left(4-1\right)!}\right)}^2\left|\begin{array}{ccccc}0& 1& 1& 1& 1\\ 1& 0& r_2^2& r_3^2& r_4^2\\ 1& r_1^2& 0& r_3^2& r_4^2\\ 1& r_1^2& r_2^2& 0& r_4^2\\ 1& r_1^2& r_2^2& r_3^2& 0\end{array}\right|\\ =\frac{1}{36}\left|\begin{array}{ccccc}0& 1& 1& 1& 1\\ 1& 0& b^2& c^2& d^2\\ 1& a^2& 0& c^2& d^2\\ 1& a^2& b^2& 0& d^2\\ 1& a^2& b^2& c^2& 0\end{array}\right|\\ =\frac{1}{36}\left(a^2{b}^2{c}^2+a^2{b}^2{d}^2+a^2{c}^2{d}^2+b^2{c}^2{d}^2\right)\end{array}$

各球半径 $\in a,b,c,d$ 。

2.2.2. 例：4个3球正交球心间场为三角形的面积的平方

${\left({ℝ}_{123}^3\right)}^2={\left(-1\right)}^3{\left(\frac{1}{\left(3-1\right)!}\right)}^2\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& r_2^2& r_3^2\\ 1& r_1^2& 0& r_3^2\\ 1& r_1^2& r_2^2& 0\end{array}\right|=-\frac{1}{4}\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& b^2& c^2\\ 1& a^2& 0& c^2\\ 1& a^2& b^2& 0\end{array}\right|=\frac{1}{4}\left(a^2{b}^2+a^2{c}^2+b^2{c}^2\right)$

下标 $\in 1,2,3,4$ 表示各球心点。

或

${\left({ℝ}_{124}^3\right)}^2={\left(-1\right)}^3{\left(\frac{1}{\left(3-1\right)!}\right)}^2\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& r_2^2& r_4^2\\ 1& r_1^2& 0& r_4^2\\ 1& r_1^2& r_2^2& 0\end{array}\right|=-\frac{1}{4}\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& b^2& d^2\\ 1& a^2& 0& d^2\\ 1& a^2& b^2& 0\end{array}\right|=\frac{1}{4}\left(a^2{b}^2+a^2{d}^2+b^2{d}^2\right)$

或

${\left({ℝ}_{134}^3\right)}^2={\left(-1\right)}^3{\left(\frac{1}{\left(3-1\right)!}\right)}^2\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& r_3^2& r_4^2\\ 1& r_1^2& 0& r_4^2\\ 1& r_1^2& r_3^2& 0\end{array}\right|=-\frac{1}{4}\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& c^2& d^2\\ 1& a^2& 0& d^2\\ 1& a^2& c^2& 0\end{array}\right|=\frac{1}{4}\left(a^2{c}^2+a^2{d}^2+c^2{d}^2\right)$

或

${\left({ℝ}_{234}^3\right)}^2={\left(-1\right)}^3{\left(\frac{1}{\left(3-1\right)!}\right)}^2\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& r_3^2& r_4^2\\ 1& r_2^2& 0& r_4^2\\ 1& r_2^2& r_3^2& 0\end{array}\right|=-\frac{1}{4}\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& c^2& d^2\\ 1& b^2& 0& d^2\\ 1& b^2& c^2& 0\end{array}\right|=\frac{1}{4}\left(b^2{c}^2+b^2{d}^2+c^2{d}^2\right)$

2.2.3. 例：6个2球正交球心间场为2点间直线的平方

${\left({ℝ}_{12}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& r_2^2\\ 1& r_1^2& 0\end{array}\right|=1\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& b^2\\ 1& a^2& 0\end{array}\right|=a^2+b^2$

或

${\left({ℝ}_{13}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& r_3^2\\ 1& r_1^2& 0\end{array}\right|=1\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& c^2\\ 1& a^2& 0\end{array}\right|=a^2+c^2$

或

${\left({ℝ}_{14}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& r_4^2\\ 1& r_1^2& 0\end{array}\right|=1\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& d^2\\ 1& a^2& 0\end{array}\right|=a^2+d^2$

或

${\left({ℝ}_{23}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& r_2^2\\ 1& r_2^2& 0\end{array}\right|=1\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& c^2\\ 1& b^2& 0\end{array}\right|=b^2+c^2$

或

${\left({ℝ}_{24}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& r_4^2\\ 1& r_2^2& 0\end{array}\right|=1\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& d^2\\ 1& b^2& 0\end{array}\right|=b^2+d^2$

或

${\left({ℝ}_{34}^2\right)}^2={\left(-1\right)}^2{\left(\frac{1}{\left(2-1\right)!}\right)}^2\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& r_4^2\\ 1& r_3^2& 0\end{array}\right|=1\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& d^2\\ 1& c^2& 0\end{array}\right|=c^2+d^2$

2.2.4. 例：4个球正交球心为点的平方

${\left({ℝ}_1^1\right)}^2={\left(-1\right)}^1{\left(\frac{1}{\left(1-1\right)!}\right)}^2\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=-1\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=1$

或

${\left({ℝ}_2^1\right)}^2={\left(-1\right)}^1{\left(\frac{1}{\left(1-1\right)!}\right)}^2\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=-1\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=1$

或

${\left({ℝ}_3^1\right)}^2={\left(-1\right)}^1{\left(\frac{1}{\left(1-1\right)!}\right)}^2\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=-1\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=1$

或

${\left({ℝ}_4^1\right)}^2={\left(-1\right)}^1{\left(\frac{1}{\left(1-1\right)!}\right)}^2\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=-1\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|=1$

2.3. 所有正交球半径均等于a时，正交球心场方程可简化为分式型公式为

${\left({ℝ}^n\right)}^2=\frac{n{a}^{2\left(n-1\right)}}{\left(n-1\right){!}^2}$ (3)

例：

${\left({ℝ}^4\right)}^2=\frac{4a^{2\left(4-1\right)}}{\left(4-1\right){!}^2}=\frac{a^6}{9}$

${\left({ℝ}^3\right)}^2=\frac{3a^{2\left(3-1\right)}}{\left(3-1\right){!}^2}=\frac{3a^4}{4}$

${\left({ℝ}^2\right)}^2=\frac{2a^{2\left(2-1\right)}}{\left(2-1\right){!}^2}=2a^2$

${\left({ℝ}^1\right)}^2=\frac{1a^{2\left(1-1\right)}}{\left(1-1\right){!}^2}=1$

3. 场方程行列式方程的非空子集数量，均符合杨辉三角关系 [4]

3.1. 勾股4态

根据表1，我们可以认知，勾股除了线、面、体之外，球属于勾股的点态子集，由此证明了勾股的点、线、面、体4态。

Table 1. Quantitative table of determinant equation subsets of field equation between orthogonal spherical centers

表1. 正交球心间场方程行列式方程子集的数量表

3.2. 正交球心场方程可以推广至任意有限高维

根据正交球心场：公式(1)，公式(2)，公式(3)，不但证明了勾股点、线、面、体4态；更可推广至任意有限高维。