## Wednesday, April 4, 2007

### Learning Journey

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

### Fractals

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

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

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,*

*b*

*eginning 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:

*Image courtesy of mathacademy.com*

*Image courtesy of mathacademy.com*

### The Golden Ratio

*Image courtesy of wikipedia.org*

*Image courtesy of unitone.org*

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.