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Геометрія гілок у дерев

товщина гілок у дерева

Великий стовбур дерева розділяється на гілки, по-перше нечисленні й потужні, а ті , у свою чергу - на все більше тонкі. Це так прекрасно й так природно, що навряд чи хто-небудь із нас звертав увагу на просту закономірність, помічену, однак, великим Леонардо.

 Справа в тому, що загальна товщина гілок на певній висоті завжди дорівнює товщині стовбура. Дотепер ніхто не міг пояснити, для чого дерева строго слідують цій закономірності.

На малюнку -  створені Кристофером Елоєм моделі, використаної для доказу його гіпотези. Праворуч - фрактальний «кістяк» моделі дерева до того, як алгоритм почав міняти діаметри його стовбура й гілок, підбираючи оптимальні розміри

«Правило Леонардо» справедливо практично для всіх відомих видів дерев. Про нього інформовані й творці комп'ютерних ігор, що створюють реалістичні тривимірні моделі дерев. Більш точно, правило це встановлює, що в місці, де стовбур або гілка роздвоюється, сума перетинів роздвоєних гілок буде дорівнює перетину вихідної гілки. Коли потім і ця гілка роздвоїться, сума перетинів уже чотирьох її відгалужень буде як і раніше дорівнювати перетину вихідного стовбура. І так далі.

Ще красивіше це правило записується математично. Якщо стовбур діаметром D розділяється на довільне число галузей n з діаметрами d1, d2 і так далі, сума їхніх діаметрів, зведених у квадрат, буде дорівнює квадрату діаметра стовбура. По формулі: D2 = ?di2, де i = 1, 2, ... n. У реальному житті ступінь не завжди строго дорівнює двом і може варіювати в межах 1,8-2,3, залежно від особливостей геометрії того або іншого дерева, але в цілому залежність строго виконується.

Щоб пояснити цей феномен, ботаніки припустили, що подібне відношення оптимально для роботи системи трубок, по яких вода піднімається від кореня до листя. Ідея виглядає цілком обґрунтованою хоча б тому, що від квадрата радіуса прямо залежить площа перетину, що визначає пропускну здатність труби. Однак французький фізик Кристоф Елой (Christophe Eloy) із цим не згодний - на його думку, зв'язана така закономірність не з водою, а з повітрям.

Для обґрунтування своєї версії вчений створив математичну модель, що зв'язує площу листя дерева з діючою на злам силою вітру. Дерево в ній описувалося, як закріплене лише в одній точці (місці умовного входу стовбура під землю), і являє собою розгалужену фрактальну структуру (тобто таку, у якій кожний менший елемент являє собою більш-менш точну копію старшого).

Додавши до цієї моделі тиск вітру, Еллою ввів певний постійний показник його граничної величини, після якої галузі починають ламатися. Виходячи із цього, він зробив розрахунки, які показали б оптимальну товщину гілок, які розгалужуються, таку, при якій опір силі вітру був би найкращим. І що ж - він прийшов рівно до тієї ж залежності, причому ідеальне значення тієї ж величини лежало між 1,8 і 2,3.

Простоту й елегантність ідеї і її доказу вже оцінили фахівці. Так, месачусетський інженер Педро Рейс (Pedro Reis) коментує: «Дослідження ставить дерева на висоту штучних споруджень, спеціально перелічених для опору вітру - кращим прикладом яких є Ейфелева вежа». Залишилося дочекатися, що скажуть на це ботаніки.

 
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