The Arrangement of the Osteons and Kepler’s Conjecture

Abstract

The studies of bone tissue have mainly highlighted the morphometrical characteristics of the osteons, rather than their spatial distribution. This work aimed to verify if the topographical distribution of the osteons responds to geometrical order. From an analysis of hundreds of bone sections of domestic and wild mammals collected over 60 years, it is evident that the spatial distribution of osteons varies from a random arrangement in the irregular Haversian tissue to an ordered geometric arrangement in the dense Haversian tissue. In this work, a new method of classification of Haversian bone tissue was introduced based on the number of points of contact that the perimeter of an osteon has with neighboring osteons. When the functional commitment of the bone is maximum to resist biomechanical stresses, the osteons are smaller and crammed adjacent to each other as if to occupy less space. Their spatial arrangement, in this case, reminds us of Kepler’s conjecture, which predicts the ideal arrangement that spheres must have to occupy as little space as possible. The conjecture was elaborated by Kepler in the Seventeenth Century to solve the practical problem linked to the need to transport the largest number of cannonballs in warships.

Keywords

• bone tissue
• osteon
• Haversian tissue
• interosteonal area
• Kepler’s conjecture