# Romanesco Broccoli

(Difference between revisions)
 Revision as of 17:07, 17 June 2013 (edit)← Previous diff Current revision (11:35, 18 June 2013) (edit) (undo) Line 6: Line 6: |ImageDesc=Proof that the Romanesco Broccoli is a natural example of the Fibonacci sequence: |ImageDesc=Proof that the Romanesco Broccoli is a natural example of the Fibonacci sequence: - It looks as if the Romanesco Broccoli is a natural example of the Fibonacci sequence. However appearances can deceive. A mathematical and scientific proof does not. I took the main image of the broccoli and I put points on each of the vertices of each fractal iteration. Then I connected each point and made line segments between them. + It looks as if the Romanesco Broccoli is a natural example of the Fibonacci sequence. However appearances can deceive. A mathematical and scientific proof does not. The main image of the broccoli was taken and points were placed on each iteration of the fractal points. Then the points were conncted to form line segments. [[image:Ratios_Fibonacci_2.png]] [[image:Ratios_Fibonacci_2.png]] - I then measured the line segments, and going from largest to smallest, I made ratios between the largest line segment and the next largest until the last line segment was reached.The image I used for this was a side view of the broccoli. + The line segments were measured from largest to smallest. Ratios were made between the largest line segment and the next largest until the last line segment was reached.The image used for this was a side view of the broccoli. - I repeated the process with an image of the broccoli from above. + The process was repeated with an image of the broccoli from above. [[image:Ratios_Fibonacci_1.png]] [[image:Ratios_Fibonacci_1.png]] - I then took the first few numbers of the Fibonacci Sequence and created ratios between them from largest to smallest, just like I did with the line segments on the broccoli. I created a chart with all these ratios. + For the first few numbers of the Fibonacci Sequence, ratios were created from the largest to the smallest numbers. This was to ensure a point of comparison to the line segments on the broccoli. Then a chart was created with all these ratios. [[image:Table_of_Ratios.PNG]] [[image:Table_of_Ratios.PNG]] - I took the chart and made a line graph with the ratios. I noticed that the line made by the ratios of the Fibonacci sequence was very similar to the line made by the third set of broccoli ratios. + The chart was used to create a line graph with the ratios. Notably, the line made by the ratios of the Fibonacci sequence was very similar to the line made by the third set of broccoli ratios. They both followed the same upward growth, but they had different intercepts, which does not change the slopes of the lines. The Fibonacci Sequence has a higher bound of growth, limited at 2, while the broccoli was limited at 1. [[image:Original_Ratio_Graph.PNG]] [[image:Original_Ratio_Graph.PNG]] - I decided to delete the other two lines and only keep the Fibonacci line and the line from the third set of broccoli ratios. I made a trend line for both and I displayed the equation. The y intercepts were different, however the slopes were nearly the same. They was only a 0.0116 difference between them. + For a more clear comparison between the broccoli and the Fibonacci Sequence, only the Fibonacci line and the line from the third set of broccoli ratios were kept. A trend line was created for both with their equations displayed. The y intercepts for the equations were different, however the slopes were nearly the same. There was only a 0.0116 difference between them. The following is the graph and proof the Romanesco Broccoli grows in accordance to the Fibonacci Sequence: The following is the graph and proof the Romanesco Broccoli grows in accordance to the Fibonacci Sequence:

## Current revision

Romanesco Broccoli
Field: Geometry
Image Created By: KatoAndLali
Website: [1]

Romanesco Broccoli

This is the Romanesco Broccoli, which is a natural vegetable that grows in accordance to the Fibonacci Sequence, is a fractal, and is three dimensional.

# Basic Description

Although the broccoli looks like it grows in accordance to the Fibonacci sequence, does it really? By making ratios between the distances of the vertices of each iteration of the fractals, and then making ratios between the numbers in the fibonacci sequence, then plotting them, the growth of the broccoli in accordance to the sequence was proved.

# A More Mathematical Explanation

Proof that the Romanesco Broccoli is a natural example of the Fibonacci sequence:

It looks as if the [...]

Proof that the Romanesco Broccoli is a natural example of the Fibonacci sequence:

It looks as if the Romanesco Broccoli is a natural example of the Fibonacci sequence. However appearances can deceive. A mathematical and scientific proof does not. The main image of the broccoli was taken and points were placed on each iteration of the fractal points. Then the points were conncted to form line segments.

The line segments were measured from largest to smallest. Ratios were made between the largest line segment and the next largest until the last line segment was reached.The image used for this was a side view of the broccoli.

The process was repeated with an image of the broccoli from above.

For the first few numbers of the Fibonacci Sequence, ratios were created from the largest to the smallest numbers. This was to ensure a point of comparison to the line segments on the broccoli. Then a chart was created with all these ratios.

The chart was used to create a line graph with the ratios. Notably, the line made by the ratios of the Fibonacci sequence was very similar to the line made by the third set of broccoli ratios. They both followed the same upward growth, but they had different intercepts, which does not change the slopes of the lines. The Fibonacci Sequence has a higher bound of growth, limited at 2, while the broccoli was limited at 1.

For a more clear comparison between the broccoli and the Fibonacci Sequence, only the Fibonacci line and the line from the third set of broccoli ratios were kept. A trend line was created for both with their equations displayed. The y intercepts for the equations were different, however the slopes were nearly the same. There was only a 0.0116 difference between them.

The following is the graph and proof the Romanesco Broccoli grows in accordance to the Fibonacci Sequence:

# Why It's Interesting

This is interesting because it is proof that the Fibonacci sequence occurs naturally, and the numbers are something derived from nature. Nature has math in it.