Формула Шлефли

Формула Шлефли — соотношение на производные двугранных углов и длины рёбер семейства многогранников. Предложена Людвигом Шлефли^[1].

Формула

Пусть $P(t)$ есть гладкое однопараметрическое семейство многогранников в евклидовом пространстве. Обозначим через $\theta _{i}=\theta _{i}(t)$ и $\ell _{i}=\ell _{i}(t)$ двугранные углы и длины рёбер $P(t)$ . Тогда

\sum _{i}\ell _{i}{\frac {d\theta _{i}}{dt}}=0

Вариации и обобщения

Формула имеет естественные обобщения на случай многомерных евклидовых пространств^[2] и пространств постоянной кривизны.

См. также

Символ Шлефли

Ссылки

↑ L. Schläfli, Quart. J. Pure Appl. Math. 2 (1858); ibid 3 (1860)
↑ R. Alexander, Lipschitzian mappings and total mean curvature of polyhedral surfaces. I, Trans. Amer. Math. Soc. 1985. Vol. 288, no. 2, 661—678.

Многогранники

Правильные

Трёхмерные (Платоновы тела)	Правильный тетраэдр Куб Октаэдр Додекаэдр Икосаэдр
Четырёхмерные	Пятиячейник Тессеракт Шестнадцатиячейник Двадцатичетырёхъячейник Стодвадцатиячейник Шестисотячейник
Большей размерности (указана в скобках)	Правильный симплекс Гиперкуб (Пентеракт (5) Гексеракт (6) Гептеракт (7) Октеракт (8) Эннеракт (9) Декеракт (10)) Гипероктаэдр

Правильные
невыпуклые

Трёхмерные по количеству
граней (указано в скобках)

Многогранники Джонсона

Формулы,
теоремы,
теории

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