Euler karakteristiği

Matematikte ve daha spesifik olarak cebirsel topoloji ve çokyüzlü kombinatorikte Euler karakteristiği (veya Euler sayısı veya Euler – Poincaré karakteristiği), nasıl olursa olsun topolojik uzayın şeklini veya yapısını tanımlayan bir sayı olan topolojik değişmezdir. Genellikle (Yunanca küçük harf chi) ile gösterilir.

Euler karakteristiği başlangıçta çokyüzlüler için tanımlanmış ve Platonik katıların sınıflandırılması da dahil olmak üzere çeşitli teoremleri kanıtlamak için kullanılmıştır. Platonik katılar için 1537'de Francesco Maurolico tarafından yayınlanmamış bir el yazmasında belirtilmiştir.[1] Konsepte adını veren Leonhard Euler, bunu daha genel olarak dışbükey çokyüzlüler için tanıttı ancak bunun bir değişmez olduğunu kesin şekilde kanıtlayamadı. Modern matematikte, Euler karakteristiği homolojiden ve daha soyut olarak homolojik cebirden kaynaklanır.

Çokyüzlüler

Küpün köşesi, kenarı ve yüzü

Euler özelliği formülüne göre klasik olarak çokyüzlülerin yüzeyleri için tanımlanmıştır.

burada V, E ve F sırasıyla verilen çokyüzlüdeki köşelerin, kenarların ve yüzlerin sayısıdır. Herhangi bir dışbükey çokyüzlünün yüzeyi Euler karakteristiğine sahiptir.

Leonhard Euler tarafından 1758 yılında ifade edilen bu denklem [2] Euler'in polihedron formülü olarak da bilinir.[3] Kürenin Euler karakteristiğine karşılık gelir (yani χ = 2) ve aynı şekilde küresel çokyüzlüler için de geçerlidir. Tüm Platonik çokyüzlülerdeki formülün örnekleri aşağıda verilmiştir.

İsim resim köşeler
v 		Kenarlar
e 		Yüzler
F 		Euler karakteristiği:
V − E + F
Dörtyüzlü 4 6 4 2
Altı yüzlü veya küp 8 12 6 2
Oktahedron 6 12 8 2
Dodekahedron 20 30 12 2
İkosahedron 12 30 20 2

Dışbükey olmayan çokyüzlülerin yüzeyleri çeşitli Euler özelliklerine sahip olabilir:

İsim resim köşeler
V 		Kenarlar
E 		Yüzler
F 		Euler karakteristiği:
V − E + F
Tetrahemiheksahedron 6 12 7 1
oktahemioktahedron 12 24 12 0
Cubohemioctahedron 12 24 10 − 2
Küçük yıldız şeklindeki dodecahedron 12 30 12 − 6
Büyük yıldız şeklinde dodecahedron 20 30 12 2

Düzenli çokyüzlüler için Arthur Cayley, yoğunluk D, tepe şekli yoğunluğu d v ve yüz yoğunluğunu kullanarak Euler formülünün değiştirilmiş bir biçimini türetmiştir.  :

Bu sürüm hem dışbükey çokyüzlüler hem de dışbükey olmayan Kepler-Poinsot çokyüzlüler için geçerlidir.

Projektif çokyüzlülerin tümü, gerçek yansıtmalı düzlem gibi Euler karakteristiği 1'e sahipken, simit gibi toroidal çokyüzlülerin tüm yüzeyleri Euler karakteristiği 0'a sahiptir.

Kaynakça

  1. ^ A History of Folding in Mathematics: Mathematizing the Margins. Science Networks. Historical Studies. 59. Birkhäuser. 2018. s. 71. doi:10.1007/978-3-319-72487-4. ISBN 978-3-319-72486-7.  Yazar |ad1= eksik |soyadı1= (yardım)
  2. ^ "Elementa doctrinae solidorum". Novi Commentarii Academiae Scientiarum Petropolitanae: 109-140. 1 Ocak 1758. 4 Kasım 2022 tarihinde kaynağından arşivlendi. Erişim tarihi: 15 Haziran 2023.  Birden fazla yazar-name-list parameters kullanıldı (yardım); Yazar |ad1= eksik |soyadı1= (yardım)
  3. ^ Richeson 2008

Dış bağlantılar

  • Eric W. Weisstein, Euler characteristic (MathWorld)
  • Eric W. Weisstein, Polyhedral formula (MathWorld)
  • An animated version of a proof of Euler's formula using spherical geometry 3 Nisan 2023 tarihinde Wayback Machine sitesinde arşivlendi..

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