Dual - Sotsui (English spelling) dual

Japanese: 双対 - そうつい（英語表記）dual

In mathematical theory, when two or more concepts are matched, a pair of theorems sometimes occurs, each of which has the same structure. This phenomenon is called duality, and the corresponding concepts or theorems are called duals of each other. For example, in plane projective geometry, duality can be seen when the concepts of "point" and "straight line" correspond to each other, and the concepts of "contain" and "contained by" correspond to each other. To give a specific example, the dual of Pappus's theorem, which states, "When three points A, B, and C are contained in line l , and three points A', B', and C' are contained in line l ', if P is the point contained in both the line containing B and C' and the line containing B' and C, Q is the point contained in both the line containing C and A' and the line containing C' and A, and R is the point contained in both the line containing A and B' and the line containing A' and B, then points P, Q, and R are contained in a common line" (Figure 1), is expressed as follows: "When three lines a , b , and c contain point L, and three lines a ', b ' , and c ' contain point L', then p is the line containing both the point contained in b and c ' and the point contained in b' and c , q is the line containing both the point contained in c and a ' and the point contained in c' and a , and r is the line containing both the point contained in a and b ' and the point contained in a' and b, then lines p , q , This results in the theorem that r contains one common point (Figure 2).

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

数学の理論において，いくつかの概念を二つずつ対応させるとき，定理が1対となって，そのおのおのは同じ構造をもつことがときどき起こる。この現象を双対性といい，対応する概念や定理を互いに他の双対という。例えば，平面射影幾何学で，“点”という概念と“直線”という概念を，“含む”という概念と“含まれる”という概念を対応させるとき，双対性がみられる。具体例をあげれば，〈3点A,B,Cが直線lに含まれ，3点A′，B′，C′が直線l′に含まれているとき，BとC′を含む直線およびB′とCを含む直線の両方に含まれる点をP,CとA′を含む直線およびC′とAを含む直線の両方に含まれる点をQ,AとB′を含む直線およびA′とBを含む直線の両方に含まれる点をRとすれば，点P,Q,Rは共通の1直線に含まれる〉(図1)というパップスの定理の双対は，〈3直線a,b,cが点Lを含み，3直線a′，b′，c′が点L′を含んでいるとき，bとc′に含まれる点およびb′とcに含まれる点の両方を含む直線をp,cとa′に含まれる点およびc′とaに含まれる点の両方を含む直線をq,aとb′に含まれる点およびa′とbに含まれる点の両方を含む直線をrとすれば，直線p,q,rは共通の1点を含む〉(図2)という定理となる。

出典　株式会社平凡社世界大百科事典第２版について　情報

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