## Wednesday, April 4, 2007

### Learning Journey

The concept of ‘Math in Nature’ is as innate as a person taking their first breath. Most would agree that our conception of math in its basic form has been derived as a means to describe aspects of our environment as an element of a much larger sociological agreement. So to say that “math exists in nature” is as redundant a statement as saying that humans themselves exist in nature. However in researching this topic one can not help but marvel at how well ‘mathematics’ corresponds with the grand scheme of things and ultimately makes one wonder what came first; an issue of the chicken or the egg as it were.
Whatever the case, we can rationalize that math in nature is factual in its tangibility. It is this outstanding quality that makes the use of math in nature a tremendous resource for the classroom. Too often we force mathematical concepts on the basis of blind faith, while examples such as these are quite literally all around us. Demonstrating math in nature is an ideal approach for illustrating what many students will regard as arbitrary information and should be utilized by all teachers as a tool to increase learner interest.

## Tuesday, March 27, 2007

### An Introduction

When math is witnessed in its purest form the realization can be truly amazing. Sometimes the application of mathematics can seem to be separate from the natural world but in actual fact when we take the time, math can be seen all around us. As teachers we will always have to answer the question ‘why’; by providing tangible and authentic examples of math we can empower our students with knowledge and hopefully encourage a love for mathematics that is relevant to their daily lives. But how can we find examples of math in nature? It is as simple as opening our eyes. The majority of our knowledge of mathematics and modern science is strictly based and supported on our observations of our environment. What was once seen as the randomness of nature is now distinguished as the intricate applications of mathematics and illustrates the complexities of our natural world. This web log is dedicated to just a few examples of nature’s mathematic phenomena such as the golden ratio, Fibonacci sequence, fractals and the honeycomb conjecture.

### Fractals

How do you know when you see a fractal in nature? A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each or which is a reduced size copy of the whole. This means that they usually contain little copies of themselves buried deep within the original (Fractal). Fractals are plentiful in nature. A simple example is twigs on trees look like the branches which they grow on, which look like the tree itself. This fits the profile of fractal as the parts of the tree are copies of the larger unit, the whole tree. With this knowledge, it is easier to see how fractals are hiding in your back yard.There are many different examples of fractals in nature that are sure to stimulate students in a classroom. Maybe the most famous fractal is the snowflake. http://library.thinkquest.org/26242/full/fm/fm33.html
This link illustrates how a snowflake is a fractal. It goes through the steps of creation and provides a visual of the smaller parts being similar to the snowflake as a whole. Viewing this site would give students insight to fractals as they apply to nature. Another famous natural fractal is the fern. Fractals are found in many plants but are very obvious in the fern. Students would be able to recognize the properties of fractals easily. The smaller leafs of the shrub are replicas of the plant as a whole. Providing students with concrete examples of fractals makes learning about them much more simplified. Students can carry their knowledge of fractals into their everyday lives and recognize other examples in nature and manmade objects. There are many other fractals in nature that include clouds, mountains, river networks, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels. Fractals are applicable to a variety of fields such as medicine, business, geology, art, and music. They can be taught using nature’s examples as well as games such as Chaos. http://math.bu.edu/DYSYS/applets/chaos-game.html

### Honeycomb Conjecture

There are a variety of places in nature where we can find math. One of the most interesting is the honey bee. When we look at the hexagonal shaped honey combs created by the bee, we wonder if they have been predisposed to geometry. Greek scholars have commented on the unique shape of the honey comb and it has been assumed that they are built this way to minimize the amount of wax used to build the structure. Charles Darwin described the honeycomb as a masterpiece of engineering that is "absolutely perfect in economizing labor and wax." (Peterson, 60-1). The honey bee is a fascinating display of math in nature. Researchers have been curious as to the accuracy of this theory. Mathematician Thomas C. Hales of the University of Michigan has created a proof that supports the theory http://www.math.lsa.umich.edu/~hales/
Students who understand the efficient honey comb conjecture, can apply their knowledge to other circumstances such as the best way to stack oranges and other economic problems.

### Fibonacci Numbers

What are they and what do they show us?

The Fibonacci numbers form a number sequence that is named after Leonardo Fibonacci who was an Italian mathematician and arguably the most talented of the Middle Ages. Fibonacci did not however invent the sequence but did use it as an example in his most prominent work. Fibonacci posed this problem:

“How many pairs of rabbits will be produced in a year,
beginning with a single pair, if in every month each
pair bears a new pair which becomes productive
from he second month on?”

Image courtesy of Dr. Ron Knott

1) After the first month the two rabbits have mated and produced 1 pair.
2) After the second month the female produced another pair, making 2 pairs of rabbits.
3) After the third month the original female produces a second pair, making 3 pairs total.
4) After the fourth month the original female produces yet another pair and the female born two months ago produces, now sexual mature produces her first pair as well, making 5 pairs. total.

The use of breeding rabbits is a simple example to illustrate the existence of this sequence. By starting with 0 and 1 the next number in the sequence can be determined by adding the last two. For example, to determine how many rabbits would exist on the 13th month you would simple add the last two numbers in the sequence 89 &144 to find that 233 is the next Fibonacci number.
The diagram below (Fibonacci squares) represents the proportionality between the 1st month and all sequential months leading up to the 8th month:

This representation may seem arbitrary to its application in the natural world, but when superimposed over the image of a nautilus shell we can finally see the existence of Fibonacci numbers in nature:

### The Golden Ratio

Image courtesy of wikipedia.org

The Golden Ratio is typically represented by the Greek letter Phi and is an irrational number that has found its place in nature. As illustrated in the diagram below, most plants follow a similar when sprouting a new section:

Image courtesy of unitone.org

If we multiply 1 over Phi squared by the 360 degrees in a circle we will get the 137.5 degree of separation that most plant employ. This angle is continued from the last sprouted section as the plant continues to grow upward and in doing so creates a spiral type pattern. This precise pattern produces the lease amount of overlap with respect to the plants leaves and therefore is the most beneficial for the collection of sunlight by the plant.
The human body also exemplifies the fascinating occurrence of Phi in the natural world, for example the bones in our fingers are related to each other by a ratio of Phi:

Image courtesy of unitone.org

The Golden Ratio can be matched to many patterns in nature including the nautilus shell mentioned above. The Fibonacci squares used above can also form Golden Rectangles, which are rectangles that have Phi as the ratio between the length and width. The incredible aspect of a Golden Rectangle is that a square can be cut from the rectangle and a smaller Golden Rectangle will remain. This process can be repeated over and over to create an ever-shrinking Golden Rectangle:

Image courtesy of mathforum.org

The relationship between Fibonacci numbers and the Golden Ratio is simple. The higher you increase in the sequence the closer the difference between the current number and the next numbers is to the Golden Ratio. For example, in the Fibonacci sequence the ratio between 5 and 8 is 1.6, while the ratio between two sequential numbers higher in the scale such as 679891637638612258 and 1100087778366101931 is 1.6180339887, which is much closer to the Golden Ratio.