## 3 Biological Life
### Biological Fractal Networks
#### 3.1.1 Formation of Main Veins and Skeletons  
In fact, the leaf blade is developed from a cluster of primordia. The tremendous work provided by [PERSON] et al. [38], shows that it grows from an area of about 10 \(\times\) 10 cells (encircled in a dashed blue line in inside a hollow to an inflated \"balloon\". During this process, the leaf vein also develops from a line segment into a network. Obviously, the primary and most important function of the veins is to supply all the cells with nutrients with equal opportunity. Note that water in the mesophyll has a limited diffusion length (R), which could be estimated to be 50-100 \(\upmu\)m, about the length of one to two dozen cells. With the growth of the leaf blade, the older veins must grow both axially and radially and differentiate into secondary veins, due to the limit of R. Based on this simple principle, synchronized duplication of all cells leads to a fractal venation of leaf veins. The main skeleton of leaf venation is formed as early as when the leafage length is around 200 \(\upmu\)m, an ignorable size compared to the adult leaf, much as the animal embryo develops from an ignorable size into the adult body.  
Obviously, the arabidopsis cells divide isotropically, which leads to a round leaf blade. What happens if the cells proliferate anisotropically?  
Figure 2 shows the formation route of various leaf outlines based on the anisotropic division of leaf primordia. Given a cluster of leaf primordia with the typical length \(\sim\)R, the cells on the symmetric axis must differentiate into vein cells and, thus, form the first vein to transport water and minerals to support the division of the germ cells. This vein then develops into the main vein of pinnate and palmate leaves. Since water in the leaf blade is transported via capillary force between the cell walls, the shape of the leaf cells plays a vital role for the leaf shape.  
For fibrous-celled leaves, the veins follow the length direction. If all the fibrous cells originate from the leaf base, the leaf displays a radical division, which leads to radial veins, such as in the palm leaf. If the cells are rooted on the main vein, the leaf would express an axial division, resulting in an oblong leaf, such as a banana leaf. Four steps of division are enough to form the basic venation of leaves, as seen in the four level veins of the banana leaf in the inserted Figure 2i.  
For rectangular or hexagonal celled leaves, division direction also plays an important role. For instance, alternating the sequences of radial and axial divisions leads to palmate leaf venation, such as in plane tree leaves.  
Based on the presumption that water or energy is supplied equally to each cell, but with apical dominance, only the apex cell can divide, while the others grow at the same speed. The apical cell divides only when it grows to the division length (\(L_{d}\)), supposing for the \(n\)th time, then the older cell produced from the \(i\)th division grows to the length \(l_{i}=2^{n-i}L_{d}\), as in the inserted Figure 2 ii for apical dominance. This leads to obovate leaves, such as magnolia denudate leaves. However, if the energy supply to older cells is gradually reduced, it leads to deltoid leaves, such as white poplar leaves.  
The main skeleton of animals could be formed similarly from a few steps of primordial division. Take the formation of the human skeleton, for instance, as shown in Figure 3, which primarily formed within five steps of bone primordial division, although the hand and foot bones needed six more steps.