Euler characteristic

Japanese: オイラー標数 - おいらーひょうすう
Euler characteristic

...The term analysis situs was in use for a long time until the early 20th century, but the term is no longer used today.
[Topological invariant - Euler characteristic]
There are five regular polyhedrons, all of which are homeomorphic to one another. For any regular polyhedron, if the number of vertices is a , the number of edges is b , and the number of faces is c , then a - b + c = 2 holds true.

From [Euler's Polyhedron Theorem]

...A polyhedron is called a convex polyhedron if the planes containing its faces do not intersect with the polyhedron at any other point. If the number of vertices of a polyhedron is α 0 , the number of edges is α 1 , and the number of faces is α 2 , then α 0 - α 1 + α 2 is called the Euler characteristic (genus) of the polyhedron. The Euler characteristic of a convex polyhedron is always 2 (Figure 1). ...

From [Number of species]

...For example, the genus of a Klein bottle is 2 (Figure 3). If the numbers of vertices, edges, and faces of a polyhedron are a 0 , a 1 , and a 2 , respectively, a 0 - a 1 + a 2 is called the Euler characteristic. This number is always 2-2 n when the polyhedron is an orientable closed surface of genus n , and always 2- n when the polyhedron is an unorientable closed surface of genus n . ...

*Some of the terminology explanations that mention "Euler characteristic" are listed below.

Source | Heibonsha World Encyclopedia 2nd Edition | Information

Japanese:

…analysis situsという言葉は，20世紀初期まで長く通用していたが，今日この言葉は用いられない。
［位相不変量――オイラー標数］
　正多面体は5種類あり，すべて互いに同相である。どの正多面体についても，その頂点の数をa，辺の数をb，面の数をcとすると，a－b＋c＝2が成り立つ。…

【オイラーの多面体定理】より

…とくに，多面体の面を含む平面がこの面以外では多面体と交わらないようなものを凸多面体という。一つの多面体において，その頂点の個数をα0，辺の個数をα1，面の個数をα2とするとき，α0－α1＋α2をその多面体のオイラー標数(種数)という。凸多面体ではオイラー標数はつねに2になる(図1)。…

【種数】より

…例えば，クラインの瓶の種数は2である(図3)。多面体の頂点，辺，面の個数をa0,a1,a2とするとき，a0－a1＋a2をオイラー標数という。この数は，多面体が種数nの向きがつけられる閉曲面のときはつねに2－2nとなり，種数nの向きがつけられない閉曲面のときはつねに2－nとなる。…

※「オイラー標数」について言及している用語解説の一部を掲載しています。

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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