FRACTALS + COASTLINES = PANGAEA
Ashley Sheahan

Abstract
The purpose of this experiment is to provide evidence to Pangaea by applying fractal mathematics and determining the fractal complexity. Shorelines connected in Pangaea will be tested and their fractal complexity compared against each other. The hypothesis states that a mathematical fractal pattern will exist in the shoreline relationships associated with Pangaea. The null hypothesis states that a mathematical fractal pattern will not exist in the shoreline relationship associated with Pangaea. Box counts of continent coastlines theorized to correspond prior to Pangaea were determined. Fractal complexity was determined by taking the log of the box count and the log of the scale and in calculating the linear regression. The TI-83 + was used to determine the complexity of each fractal dimension. Coastlines selected include the following pairs: Africa and South America; India and Madagascar; North America and regions of Africa; Australia and Antarctica; Canada and Greenland; Madagascar and Africa. Finally, the coastlines of non-associated sides of Africa and South America were analyzed. Fractal complexities indicate similarities between associated coastlines providing evidence of Pangaea. The hypothesis was accepted and the null hypothesis was rejected.

Introduction
If South America and Africa were puzzle pieces would they fit together? Nine decades ago Alfred Wegener thought so. He proposed a revolutionary theory for this jigsaw puzzle of the arrangement of continents using plate tectonics and the Continental Drift Theory. Wegener’s theory to explain Earth’s development has long been under scientific scrutiny. This project will evaluate the theory of Pangaea by applying modern fractal mathematics.
When Pangaea, one landmass, existed it was surrounded by one ocean called Panthalassa. Obviously, it was a different world. The question is; how did one landmass become seven continents? This natural occurrence can be explained by plate tectonics. The Earth is made up of plates that move over magma in the Earth’s core. Scientists theorize that about 130 million years ago Pangaea broke up into two continents, Laurasia and Gondwana, then later into the continents today (1). Research indicates that theses pieces or plates are still moving. It is predicted in 150 million years; California will collide with Alaska (2).
Evidence to support the theory of Pangaea is limited. The most noticeable evidence is the visual observation that the continents appear to fit together. Second, similar fossils can be found in continents that are now an ocean apart. Third, the rock layers in Africa and South America appear to be the same age. Also, the presence of coal deposits in Antarctica raises an interesting question. Coal can only form in warm and wet conditions while Antarctica’s position today is cold and dry. If Antarctica took part in Pangaea, it would have had the conditions to make coal (3)
Natural phenomena have been studied for their fractal patterns for many years. Fractal complexity has been found for Great Britain’s shoreline (4). The slope of the line the data produces determines fractal complexity. The more complex shoreline has a larger slope. A previous study of fractal mathematics was conducted in an experiment “Take a Walk on the Shoreline Fractal”. This project studied the fractal patterns of Oklahoma lakes. The lakes did show a mathematical relationship, which introduced the idea of fractal patterns on Pangaea’s shorelines. Since fractal patterns in shorelines exist, fractal mathematics could be used to provide further evidence to support Pangaea.
What is fractal mathematics? The Latin word for fractal is frangere, which means ‘to break’, and the Latin adjective fractus means ‘broken and irregular’ (7). Being scale-independent the very small or very large object looks the same whether you are close to it or far from it (5). The fractal is a geometrical figure consisting of a pattern that is repeated in finer and finer scales (6). It is predicted that even if the shorelines have been weathered the fractal will still reveal itself. This thought comes from the meaning of a fractal, breaking down into smaller scales, and an accurate calculation can be made still after millions of years.
The purpose of this experiment is to provide evidence to Pangaea by applying fractal mathematics and determining the fractal complexity. Shorelines connected in Pangaea will be tested and their fractal complexity compared against each other. The hypothesis states that a mathematical fractal pattern will exist in the shoreline relationships associated with Pangaea. The null hypothesis states that a mathematical fractal pattern will not exist in the shoreline relationship associated with Pangaea.
MATERIALS

• Maps of continents Tape
  
• Transparencies of grids: _ Erasable marker
  

RESULTS
Data Table I Comparison of South America and Africa Test I

Data Table IV Comparison of Africa and North America Test I

Data Table V Comparison of Africa and North America Test II

Data Table VIII Comparison of Madagascar and Africa Test III

Data Table X Comparison of Australia and Antarctica

This Section includes Sample Graphs for the data collected. For a complete listing of all graphs see the appendix
Graph I-A and I-B show the linear regression and fractal complexity of coastlines that were thought to be connected by Pangaea. Both sets of data show a good correlation for the data series. The similar fractal complexities provide support to the relatedness of the coastlines and Pangaea

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Graph XIIA and B show the linear regression and fractal complexity of the two coastlines of South America and Africa that are not associated with Pangaea connections (far sides or “outsides” of the continents. The data series show a good correlation within each set of data, however the fractal complexity differences indicate the coastlines on the “outside” of each continent do not show a Pangaea relationship (as predicted)

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Graphs VIIIA and VIIIB indicate a good correlation between both data sets. The similar fractal complexities indicates possible evidence that these two coastlines were once connected by Pangaea

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This investigation provides mathematical evidence to support the idea that fractal patterns could be found on continental coastlines. The hypothesis of this investigation states that a mathematical fractal pattern will exist in the shoreline relationships associated with Pangaea. The null hypothesis states that a mathematical fractal pattern will not exist in the shoreline relationships associated with Pangaea. The following sets of data indicate a positive result. They show similar fractal complexity and a high (approaching one) linear regression: South America and Africa corresponding coastlines test I, South America and Africa corresponding coastlines test II, Africa and Saudi Arabia corresponding coastlines, North America and Africa corresponding coastlines test I, North America and Africa corresponding coastlines test II, Madagascar and Africa corresponding coastlines test III, India and Madagascar corresponding coastlines. Two tests were conducted on South America and Africa due to finding a more detailed map. Both tests were positive and showed similar complexity. One test seems more positive than the other because it was conducted on a map with better shoreline definition. These test support the hypothesis. The following sets of data indicated a negative result. They do not have a similar fractal complexity to each other and the linear regression varies: Madagascar and Africa corresponding coastlines test I, Madagascar and Africa corresponding coastlines test II, Canada and Greenland corresponding coastlines, Antarctica and Australia corresponding coastlines. The negative result of Canada and Greenland demonstrate that some coastlines initially thought to be connected were not. A different model of Pangaea was found that supported the negative test. Three tests were conducted on different areas of Madagascar and Africa. The first two were negative tests. The model of Pangaea showed that they should have fit together with a positive result. A final test was performed and indicated a positive correlation because the correct region of coastline was compared. The combination of positive and negative tests prove the hypothesis in that a correct coastline match will have similar complexity and an incorrect coastline match will not show similarity. Antarctica and Australia tested negative because the Australia map shows the shape of the ice not the bedrock. The following data show that non-associated coastlines have their own fractal complexity but do not correlate with any other coastline: Non-associated South America coastline, Non-associated Africa coastline. In this experiment the hypothesis was accepted and the null hypothesis was rejected. The data of associated shorelines showed similar fractal complexities. Where as shorelines not associated with each other did not show similar fractal complexities. Future studies might be to further extend this investigation with more detailed maps and include Antarctica bedrock.
LITERATURE CITED

1. Author unknown. Pangaea Theory _Available online_ www.library.thinkquest.org July, 2002
   

Fractal Reference Film
Fractals-An Animated Discussion with Edward Lorenz and Benoit Mandlebrot Production and Copyright Spektrum der Wissenschaft in cooperation with Graphics Lab Dynamical Systems University of Bremen, 1990.

ACKNOWLEDGEMENTS
I would like to thank the following people for helping in the completion of this science experiment:
Mrs. Fenska for advice and guidance in experimentation
Mrs. Drake for mathematical and calculation help
Mrs. Benne for mathematical help, guidance, and advice
Mrs. Oexman for help in English matters
MHS computer lab and printer
Family and friends for encouragement and support