# The Arrangement of the Osteons and [PERSON]'s Conjecture
## 4 Discussion  
The results presented in this work highlighted the complexity of the Haversian tissue providing reading keys to describe the arrangement of the osteons. The presence of the osteons in the species analyzed confirmed general rules valid in mammals, such as their greater presence in animals most-engaged in biomechanical stress due to a high body mass and aptitude to high speeds and wild animals that generally have more intense locomotor activity than domestic animals. As regards the size of the osteons, among the factors that mainly influence it, the biomechanical stress to which the bones are subjected is the main one [28; 29], whilst the body mass does not seem to matter much [37].  
As is known, osteons can be considered as tridimensional structures organized to better resist mechanical stresses. Indeed, the central canal surrounded by some layers of bone lamellae gives to the osteon excellent properties to oppose the compression, torsion, and tension forces [38; 39; 40]. Osteons with different sizes can be present both in the irregular Haversian tissue and in the H\({}_{1}\) subtype of dense Haversian tissue. In these cases, the points of contact of the cement line with neighboring osteons can be variable in number depending on the closeness of the osteons. In the H\({}_{2}\) and H\({}_{3}\) subtypes, the sizes of the osteons are very similar and the osteonal arrangement is more regular with a geometrical character. As described before, the interosteonal area shrinks from the irregular Haversian tissue to the H\({}_{1}\) subtype of dense Haversian tissue, and so on, to the H\({}_{2}\) subtype and up to the H\({}_{3}\) subtype. This means that, according to the same order, the presence of osteons grows up, and then, the resistance to biomechanical stress increases in the same order (Figure 5). It should come as no surprise that these last two subtypes and, especially, H\({}_{3}\) are found in long bones highly exposed to stress.  
Figure 4: Schematic differences among the subtypes of the Haversian tissue proposed.  
In the case of the H\({}_{3}\) subtype, it is as if the osteons were trying to arrange themselves in the best way to occupy less space. This consideration brings to mind [PERSON]'s conjecture. It is a mathematical theorem about sphere arrangement in a limited space. The problem arose in the 16 th Century when war ships carried cannonballs and needed to transport the largest amount of them in defined spaces. The solution of the problem was sought by mathematicians, astronomers, and other intellectuals of the time until [PERSON] (1571-1630) enunciated his conjecture. In 1611, [PERSON] wrote a short work, _Strena Seu de Nive Sexangula_, where he described for the first time the hexagonal symmetry of snowflakes, extending the discussion on the most-efficient arrangement for packing spheres [41]. [PERSON]'s conjecture, for which, being a conjecture, it has not been possible to demonstrate the falsity and which is, therefore, presumed to be true, states that no arrangement of equally sized spheres filling a space has a greater average density than that of the hexagonal close packing arrangements [42; 43; 44]. The conjecture was enunciated thinking of cannonballs arranged in a space, but finds its analogy in the arrangement of the osteons in the H\({}_{3}\) subtype considering the spheres sectioned by a plane. Just as [PERSON]'s conjecture predicts the maximum packing of the spheres when one sphere is in contact with six other spheres in the same plane, similarly in dense Haversian tissue, the maximum degree of packing occurs when the osteons are arranged in such a way that each is in contact with six other osteons, i.e., in the H\({}_{3}\) subtype of the dense Haversian tissue here described.