Lecture 3

Here's a few questions I got about Chapter 3. Think of my comments as the next round of discussion rather than a definitive solution.

On page 33, two paragraphs above (3.6), last sentence of the paragraph: what is the "coherence" referred to?

Don't know.

On page 33, equation (3.6): what is "trunc area"? Is that "trunk area" or "truncated area" and either way, what is it actually measuring? If (3.7) really does "follow" from ( 3.6), shouldn't that mean that "trunc area" is not an area at all, but a length (perhaps trunk diameter)?

I am going to go with "trunc" = "trunk". Based on the last sentence on page 33, I am going to say that "trunc area" means "trunk diameter". Though I may change that guess after I get a copy of Fractal Geometry of Nature.

The paragraph spanning pages 33 and 34 seems to say that the wind resistance of a tree is proportional to its volume (not the volume of tree matter but rather the bounding volume of the tree). Why? Shouldn't it be proportional to the cross-sectional surface area? Or does enough wind pass between the leaves that the drag is more-or-less uniform throughout the volume of the foliage? Even in that case, isn't the foliage predominantly within the outer shell of the volume of the tree and thus still related to the square, not the cube, of tree height? It seems counter-intuitive that the branches supply nearly as much wind resistance as the leaves…

Good forestry question. I don't know much forestry, so I just accepted the statement as is. However, I think what they are saying is that approximately the wind resistance is proportional to the entire surface area. Proportional seems to allow for a constant which reduces the actual wind resistance based on how much wind generally slips between the leaves and branches.

The Poisson disc distribution (p 37) looks, to me, too regular; the simulation results in 8.2 (p 135) seem to agree with me, and the discussion in 8.1 (see paragraph spanning pp. 129-130) implies it is inefficient to compute. Why is it even presented? Is it empirically correct, despite its (to me) over-regular appearance? Is it just the best model we have, knowing it is wrong?

It is pretty regular. As the authors note, its got a low variance. I think what they are trying to do is to say that the Poisson disc distribution is the ideal biological description, but it is too hard to compute so you go with approximations.

I would not infer too much from figure 8.6 for 4 reasons: 1. it is a computer scientists' model and not a forester's model. 2. Figure 8.6(c) shows both the plant and the area spanned by the plant not just a dot at the center of the plant. 3. It may not be the final steady-state growth of the community and more thinning may give a more Poisson-disc-like distribution. 4. It's not clear that the plants drop new seeds.

Objectives for this lecture

After class today, you should be able to…

• compute the fractal dimension of a self-similar object (like a Koch snowflake)
• see fractals in trees
• explain the relationship between phylotaxis, the Golden Section and the Fibbinacci sequence
• recite da Vinci's equation for computing tree trunk widths.
• believe that some biological processes in plants have nice mathematical descriptions.

Topological Dimension

Loosely speaking, if an object has topological dimension $d$ then removing a finite number of slices of dimension $d-1$ creates a break in the object. (I don't have a proof or a reference for this, but it does come up on page 29 of The Science of Fractal Images)

Notes on Digital Design of Nature Chapter 3.

gemoetry versue topology.
Statistics to model probabilistic properties of plants.
almost all botanical terrs are binary

order numbers: assign a number to the branches based on which year they grew.
Strahler order: gowth-indpendent measure

(1)
\begin{align} ord(v_m) = \left\{\begin{array}{ll}max ((ord(v_i),ord(v_j))) & if ord(v_i) \not= org(v_j) \\ ord(v_i)+1 & otherwise \end{array} \right. \end{align}

bifurcation ratios $\beta_k$. The Bifurcation ratio for nodes of order $k$ is the number of nodes with order $k-1$ over the number of nodes with order $k$.

(2)
\begin{align} \beta_k = \frac{\beta_{k-1}}{\beta_k} \end{align}

fractal geometry
consider objects which are built by composing replicas of a smaller object.
But you get to scale, translate or transform the smaller object.
Keep track of two things: the number of copies of the smaller object using to create a larger object and the scaling factor applied to the smaller objects.
Let $N$ be the number of copies.
Let $S$ be the scaling factor.
Let $r(N)$ be, in general, the reciprocal of N.

Also, keep in mind that the smaller object should have a similar shape as the larger object.
That is, they should look alike except one is smaller than the other.

Consider a line split in 1/2. Each 1/2 is similar to the larger line (but differs by a scaling factor).
If you combine the two halves, you get back the whole line.

the relationship

(3)
\begin{align} r(N) = \frac{1}{N^{1/D}} \end{align}

gives the size of the scaling factor as a function of the dimensionality of the final object and the number of copies used to make the final object.

So in 2D in which 4 copies are used to make the final shape we have

(4)
\begin{align} r(N) = \frac{1}{4^{1/2}} = 1/2 \end{align}

which means we have to scale each shape by 1/2 and combine 4 of them to get the final shape.

Now, suppose that we know $S$ and $N$ (because someone told us what they wanted S and N to be).
How do you calculate the dimensionality of the resulting object?
Simple, solve for $D$.

(5)
\begin{align} D = \frac{log N}{log S} = \frac{log N}{log (r(S))} \end{align}

(a geometrically-based derivation of the same result) That was all "Self similarity dimension"

How do you apply to to a tree? or a coastline?

Mandelbrot uses the Hausdorff-Besicovitsch dimension which is roughly the same though not mathematically equivalent. Here's the definition of Hausdorff dimension from the Wikipedia:

To define the Hausdorff dimension for X as non-negative real number (that is a number in the half-closed infinite interval [0, ∞)), we first consider the number N(r) of balls of radius at most r required to cover X completely. Clearly, as r gets smaller N(r) gets larger. Very roughly, if N(r) grows in the same way as $1/r^d$ as r is squeezed down towards zero, then we say X has dimension d.

Phyllotaxis

means the arrangement of outer plant organs such as leaves etc.

the classic relationship between arrangements and math is the Golden Ratio.

Consider a sunflower. Consider the seeds on the sunflower…

to place seed i, in polar coordinates, you start at the origin, go out some radius and rotate some number of degrees.
the radius is $r = c \cdot \sqrt{i}$ to place seed $i$ and the angle is $\alpha = i \cdot \theta$

To get a sunflower, you set $\theta$ equal to a value which depends on a geometrical relationship to the Golden Section. To get a sunflower, you rotate by 137.5 for each seed and come in a little. If you keep doing that, you'll get two things (a) a sunflower and (b) the optimally dense seed packing.

if you pick other angles, you get other less dense packings.

If you number all of the neighbors then take the neighbors who are n numbers apart. If n is a Fibonacci number, then you get a parastichy which is a balanced set of spirals. (To understand this, solve for the recurrence relation which describes Fibbonachi numbers. You'll get a solution which includes the Golden Section).

Duessen calls this kind of model a descriptive model as opposed to functional model which also describes how to build it.

plant populations

Now we move on to descriptions of plant populations.

basically you use a statistical distribution, or a dynamic system or a Markov model or an ecological model depending on how much effort you want to expend.

page revision: 9, last edited: 15 Jan 2008 20:09