Pure Mathematics
Vol. 09  No. 07 ( 2019 ), Article ID: 31967 , 8 pages
10.12677/PM.2019.97100

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

—Application of Pythagorean Theorem of Four Dimensional Volume (Formula 1)

Guowei Cai

Shanghai Huimei Property Co., Ltd., Shanghai

Received: Aug. 5th, 2019; accepted: Aug. 23rd, 2019; published: Aug. 30th, 2019

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.

Keywords:Volume Pythagorean Theorem, Application, Field Equation, Determinant

论勾股四态、以及正交球心间同构的场方程

——四维体积勾股定理的应用(公式一)

蔡国伟

上海汇美房产有限公司,上海

收稿日期:2019年8月5日;录用日期:2019年8月23日;发布日期:2019年8月30日

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。

关键词 :体积勾股定理,应用,场方程,行列式

Copyright © 2019 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

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, n 代表球心所围场(含所有子集),行列式为:

( n ) 2 = ( 1 ) n ( 1 2 ) n 1 ( 1 ( n 1 ) ! ) 2 | 0 1 1 1 1 1 1 0 d 12 2 d 13 2 d 14 2 d 1 n 2 1 d 21 2 0 d 23 2 d 24 2 d 2 n 2 1 d 31 2 d 32 2 0 d 34 2 d 3 n 2 1 d 41 2 d 42 2 d 43 2 0 d 4 n 2 1 d n 1 2 d n 2 2 d n 3 2 d n 4 2 0 | (1)

n 1 , 2 , 3 , 4 表示参与正交球的数量,下标: i j 1 , 2 , 3 , 4 表示各球心点,dij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

( 1234 4 ) 2 = ( 1 ) 4 ( 1 2 ) 4 1 ( 1 ( 4 1 ) ! ) 2 | 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 | = 1 288 | 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 | = 1 36 ( a 2 b 2 c 2 + a 2 b 2 d 2 + a 2 c 2 d 2 + b 2 c 2 d 2 )

各球半径 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

( 123 3 ) 2 = ( 1 ) 3 ( 1 2 ) 3 1 ( 1 ( 3 1 ) ! ) 2 | 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 | = 1 16 | 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 | = 1 4 ( a 2 b 2 + a 2 c 2 + b 2 c 2 )

下标 i j 1 , 2 , 3 , 4 表示各球心点。

( 124 3 ) 2 = ( 1 ) 3 ( 1 2 ) 3 1 ( 1 ( 3 1 ) ! ) 2 | 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 | = 1 16 | 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 | = 1 4 ( a 2 b 2 + a 2 d 2 + b 2 d 2 )

( 134 3 ) 2 = ( 1 ) 3 ( 1 2 ) 3 1 ( 1 ( 3 1 ) ! ) 2 | 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 | = 1 16 | 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 | = 1 4 ( a 2 c 2 + a 2 d 2 + c 2 d 2 )

( 234 3 ) 2 = ( 1 ) 3 ( 1 2 ) 3 1 ( 1 ( 3 1 ) ! ) 2 | 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 | = 1 16 | 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 | = 1 4 ( b 2 c 2 + b 2 d 2 + c 2 d 2 )

2.1.3. 例:6个2球正交球心间场为2点间直线的平方

( 12 2 ) 2 = ( 1 ) 2 ( 1 2 ) 2 1 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 d 12 2 1 d 21 2 0 | = 1 2 | 0 1 1 1 0 a 2 + b 2 1 b 2 + a 2 0 | = a 2 + b 2

( 13 2 ) 2 = ( 1 ) 2 ( 1 2 ) 2 1 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 d 13 2 1 d 31 2 0 | = 1 2 | 0 1 1 1 0 a 2 + c 2 1 c 2 + a 2 0 | = a 2 + c 2

( 14 2 ) 2 = ( 1 ) 2 ( 1 2 ) 2 1 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 d 14 2 1 d 41 2 0 | = 1 2 | 0 1 1 1 0 a 2 + d 2 1 d 2 + a 2 0 | = a 2 + d 2

( 23 2 ) 2 = ( 1 ) 2 ( 1 2 ) 2 1 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 d 23 2 1 d 32 2 0 | = 1 2 | 0 1 1 1 0 b 2 + c 2 1 c 2 + b 2 0 | = b 2 + c 2

( 24 2 ) 2 = ( 1 ) 2 ( 1 2 ) 2 1 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 d 24 2 1 d 42 2 0 | = 1 2 | 0 1 1 1 0 b 2 + d 2 1 d 2 + b 2 0 | = b 2 + d 2

( 34 2 ) 2 = ( 1 ) 2 ( 1 2 ) 2 1 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 d 34 2 1 d 43 2 0 | = 1 2 | 0 1 1 1 0 c 2 + d 2 1 d 2 + c 2 0 | = c 2 + d 2

2.1.4. 例:4个球正交球心为点的平方

( 1 1 ) 2 = ( 1 ) 1 ( 1 2 ) 1 1 ( 1 ( 1 1 ) ! ) 2 | 0 1 1 0 | = 1 | 0 1 1 0 | = 1

( 2 1 ) 2 = ( 1 ) 1 ( 1 2 ) 1 1 ( 1 ( 1 1 ) ! ) 2 | 0 1 1 0 | = 1 | 0 1 1 0 | = 1

( 3 1 ) 2 = ( 1 ) 1 ( 1 2 ) 1 1 ( 1 ( 1 1 ) ! ) 2 | 0 1 1 0 | = 1 | 0 1 1 0 | = 1

( 4 1 ) 2 = ( 1 ) 1 ( 1 2 ) 1 1 ( 1 ( 1 1 ) ! ) 2 | 0 1 1 0 | = 1 | 0 1 1 0 | = 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:

( n ) 2 = ( 1 ) n ( 1 ( n 1 ) ! ) 2 | 0 1 1 1 1 1 1 0 r 2 2 r 3 2 r 4 2 r n 2 1 r 1 2 0 r 3 2 r 4 2 r n 2 1 r 1 2 r 2 2 0 r 4 2 r n 2 1 r 1 2 r 2 2 r 3 2 0 r n 2 1 r 1 2 r 2 2 r 3 2 r 4 2 0 | (2)

下标: n 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

( 1234 4 ) 2 = ( 1 ) 4 ( 1 ( 4 1 ) ! ) 2 | 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 | = 1 36 | 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 | = 1 36 ( a 2 b 2 c 2 + a 2 b 2 d 2 + a 2 c 2 d 2 + b 2 c 2 d 2 )

各球半径 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

( 123 3 ) 2 = ( 1 ) 3 ( 1 ( 3 1 ) ! ) 2 | 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 | = 1 4 | 0 1 1 1 1 0 b 2 c 2 1 a 2 0 c 2 1 a 2 b 2 0 | = 1 4 ( a 2 b 2 + a 2 c 2 + b 2 c 2 )

下标 1 , 2 , 3 , 4 表示各球心点。

( 124 3 ) 2 = ( 1 ) 3 ( 1 ( 3 1 ) ! ) 2 | 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 | = 1 4 | 0 1 1 1 1 0 b 2 d 2 1 a 2 0 d 2 1 a 2 b 2 0 | = 1 4 ( a 2 b 2 + a 2 d 2 + b 2 d 2 )

( 134 3 ) 2 = ( 1 ) 3 ( 1 ( 3 1 ) ! ) 2 | 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 | = 1 4 | 0 1 1 1 1 0 c 2 d 2 1 a 2 0 d 2 1 a 2 c 2 0 | = 1 4 ( a 2 c 2 + a 2 d 2 + c 2 d 2 )

( 234 3 ) 2 = ( 1 ) 3 ( 1 ( 3 1 ) ! ) 2 | 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 | = 1 4 | 0 1 1 1 1 0 c 2 d 2 1 b 2 0 d 2 1 b 2 c 2 0 | = 1 4 ( b 2 c 2 + b 2 d 2 + c 2 d 2 )

2.2.3. 例:6个2球正交球心间场为2点间直线的平方

( 12 2 ) 2 = ( 1 ) 2 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 r 2 2 1 r 1 2 0 | = 1 | 0 1 1 1 0 b 2 1 a 2 0 | = a 2 + b 2

( 13 2 ) 2 = ( 1 ) 2 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 r 3 2 1 r 1 2 0 | = 1 | 0 1 1 1 0 c 2 1 a 2 0 | = a 2 + c 2

( 14 2 ) 2 = ( 1 ) 2 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 r 4 2 1 r 1 2 0 | = 1 | 0 1 1 1 0 d 2 1 a 2 0 | = a 2 + d 2

( 23 2 ) 2 = ( 1 ) 2 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 r 2 2 1 r 2 2 0 | = 1 | 0 1 1 1 0 c 2 1 b 2 0 | = b 2 + c 2

( 24 2 ) 2 = ( 1 ) 2 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 r 4 2 1 r 2 2 0 | = 1 | 0 1 1 1 0 d 2 1 b 2 0 | = b 2 + d 2

( 34 2 ) 2 = ( 1 ) 2 ( 1 ( 2 1 ) ! ) 2 | 0 1 1 1 0 r 4 2 1 r 3 2 0 | = 1 | 0 1 1 1 0 d 2 1 c 2 0 | = c 2 + d 2

2.2.4. 例:4个球正交球心为点的平方

( 1 1 ) 2 = ( 1 ) 1 ( 1 ( 1 1 ) ! ) 2 | 0 1 1 0 | = 1 | 0 1 1 0 | = 1

( 2 1 ) 2 = ( 1 ) 1 ( 1 ( 1 1 ) ! ) 2 | 0 1 1 0 | = 1 | 0 1 1 0 | = 1

( 3 1 ) 2 = ( 1 ) 1 ( 1 ( 1 1 ) ! ) 2 | 0 1 1 0 | = 1 | 0 1 1 0 | = 1

( 4 1 ) 2 = ( 1 ) 1 ( 1 ( 1 1 ) ! ) 2 | 0 1 1 0 | = 1 | 0 1 1 0 | = 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

( n ) 2 = n a 2 ( n 1 ) ( n 1 ) ! 2 (3)

例:

( 4 ) 2 = 4 a 2 ( 4 1 ) ( 4 1 ) ! 2 = a 6 9

( 3 ) 2 = 3 a 2 ( 3 1 ) ( 3 1 ) ! 2 = 3 a 4 4

( 2 ) 2 = 2 a 2 ( 2 1 ) ( 2 1 ) ! 2 = 2 a 2

( 1 ) 2 = 1 a 2 ( 1 1 ) ( 1 1 ) ! 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。

文章引用

蔡国伟. 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[J]. 理论数学, 2019, 09(07): 763-770. https://doi.org/10.12677/PM.2019.97100

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