Continuously Changing Growth

 

 

1. Simulating real plant growth is quite challenging because plant growth is dependent on many factors that vary, such as rainfall, climate, differences in soil, and so on.
To begin to understand how to find a mathematical model for a growing plant, use as example to model and learn from it. Consider a plant which hypothetically grows in steps which are 0.45 the length of the previous growth and assume that the first measurable stem length is 20 millimeters.
a. Make a table showing the length of the plant’s stem at the end of each of the first 8 steps.
b. Make a graph of the values in the table.
c. Write a formula representing the growth in the simulation based on the pattern of growth from one term to the next.

 

2. It is clear that the equation is not linear since the graph does not represent a straight line.
You know how to determine the equation of a straight line, so the approach is to transform the data in some way to make the plot of the transformed data linear. Use a function on the independent variable, the dependent variable or both in attempt to create a graph which is linear.

 

A Natural Log: Our Innate Sense of Numbers is Logarithmic, Not Linear

People without math training naturally think in terms of ratios

Article from Scientific American.

 

We humans seem to be born with a number line in our head. But a May 30 study 2008 in Science suggests it may look less like an evenly segmented ruler and more like a logarithmic slide rule on which the distance between two numbers represents their ratio (when ­di­vided) rather than their difference (when subtracted).

The mathematical idea of a number line—a line of numbers placed in order at equal intervals—is a simple yet surprisingly powerful tool, useful for everything from taking measure­ments to geometry and calculus.

Previous studies of Westerners showed that people tend to map numbers on a linear scale, with the numerals evenly spaced along the line. But if the numbers are presented as hard-to-count groups of dots, people will logarithmically group the larger numbers closer together on one end of the scale in what researchers call a “compression effect.” Preschoolers also group numbers this way before they begin their formal education in math.

To investigate which number-line concept is innate, neuroscientist Stanislas Dehaene of the College of France in Paris worked with the Mundurukú, an Amazonian culture with little exposure to modern math or measuring devices. The Mundurukú were immediately able to place numbers on a line when asked, but they grouped them logarithmically.

Dehaene says the research suggests that a logarithmic number line might be an intuitive mathematical concept, whereas the idea of a linear number line might have to be learned.

Editor's Note: This story was originally printed with the title "A Natural Log"

 

number scale and use of logarithms.pdf

 

 

Logarithmic spiral

 

This is the spiral for which the radius grows exponentially with the angle. The logarithmic relation between radius and angle leads to the name of logarithmic spiral or logistique (in French).
The distances where a radius from the origin meets the curve are in geometric progression.
                                                              
The curve was the favorite of Jakob (I) Bernoulli (1654-1705). On his request his tombstone, in the Munster church in Basel, was decorated with a logarithmic spiral. The curve, which looks by the way more like an Archimedes' spiral, has the following Latin text accompanied: eadem mutata resurgo. In a free translation: 'although changed, still remaining the same'. This refers to the various operations for which the curve remains intact (see below).
Therefore the curve is also called the Bernoulli spiral.

 

                     

 

The logarithmic spiral is the curve for which the angle between the tangent and the radius (the polar tangent) is a constant.
Suppose that an insect flies in such a way that its orbit makes a constant angle b with the direction to a lamp. Then the poor creature travels in the form of a logarithmic spiral, eventually reaching the lamp ²⁾. It seems that a night-moth flies under a constant angle with respect to the moon. But its orbit resembles much that of a right line, while the orbit of the moth is much smaller that the distance to the moon.
This property gives the spiral the name of equiangular spiral.
For angle b = p/2, the result is a circle.

The logarithmic spiral can be approximated by a series of straight lines as follows: construct a line bundle l_i through O with slope ia / 2p.
Starting with a given point P₁ on l₁, construct point P₂ on l₂ so that the angle between P₁P₂ and OP₁ is b. Then the points P_i approximate a logarithmic spiral with a = cot b.
We call the spiral curve composed of line pieces the  right equiangular spiral.

 

 

 

For b = p / 2, the line series ressembles the spiral of Theodore of Cyrene.
I say "ressembles" because the Theodore of Cyrene's spiral does not define a logarithmic spiral, because not the angles, but the chords are equal (being 1).
Question, what then is the equation of the spiral which the line spiral defines?

 

 

 

 

When dividing a golden rectangle into squares a logarithmic spiral is formed with a = 2 ln f / p (about 0.306)

, where f is the golden ratio. This spiral is called the golden spiral.

 

 

 

 

 

 

 

 

 

 

 

We see the curve in nature, for organisms where growth is proportional to their size. An example is the Nautilus shell, where a kind of octopus hides (showed on this page).
For that proportionality the curve bears the name of the growth spiral: a growth that is proportional to its size. A growth that just adds, is shown by the Archimedes' spiral.
D'Arcy Thompson explains the curve in his book 'Growth and Form'.

Two physical properties related to the spiral are:

• the force that makes a point move in a logarithmic spiral orbit is proportional to 1/r^{3 3)}.
• a charged particle moving in a uniform magnetic field, perpendicular to that field, forms a logarithmic spiral.

Johan Gielis extended the logarithmic spiral to a super spiral.

 

Human vision and logarithmic spirals

Incidentally, logarithmic spirals are, like fractals, self-similar at all scales (f(ka) = e^k f(a)). This may be one of the reasons why they are striking to human vision: Your brain performs early visual computations at several scales (demagnifications of the image) and compares the results. A logarithmic spiral will self-correlate across all scales. In a neural network implementation of the artificial artist's coupled potential fields, this observation was exploited to produce piecewise logarithmic curves as a side-effect of curvature detection.

 

 

 

MATH and NATURE