You are here

Patterns in Pascal's Triangle - with a Twist - Zooming In: Self-Similarity

Author(s): 
Kathleen M. Shannon and Michael J. Bardzell

Faculty Notes

One thing you can notice when you study our triangles is that sometimes when you "zoom out" or increase the number of rows you get triangles that look very similar. The following animations show what happens when you increase or decrease the number of rows in certain PascGalois Triangles. In each case, notice how the general structure or appearance of the triangle is preserved as you "zoom out".

Pascal's Triangles mod 3: Each triangle has either three
times as many or a third as many rows as the previous one.
PascGalois Triangles for Z3 tripling number of rows
Pascal's Triangles mod 5: Each triangle has either five
times as many or a fifth as many rows as the previous one.
PascGalois Triangles for Z5 quintupling number of rows
PascGalois Trianglesfor D4:
Each triangle has either twice as many or half as many
rows as the previous one. PascGalois Triangles for D4 doubling number of rows
PascGalois Trianglesfor D3:
Each triangle has either twice as many or half as many
rows as the previous one. PascGalois Triangles for D3 doubling number of rows

If something looks the same when viewed at all different scales, it is said to be self-similar. For example, imagine looking at a satellite image of a jagged coastline. Now consider looking at an image taken from an airplane of a smaller section of the same coastline. Finally, think of an image of the coastline taken from a tower nearby of a yet smaller section of the coast. Assuming the coast is natural and undeveloped, all three images would likely look pretty much the same. By changing the scale, you do not change the essential nature of the "curve" that would outline the coast.

You can perform a similar exercise by looking at snapshots taken at different distances for some other image. For example, think of the patterns in the growth of a fern or a tree. A single branch of a tree looks pretty much like a tree itself.

These are all examples of the self-similarity property found in geometric objects known as fractals. See Burger and Starbird (2000, pp. 398-511), Field and Golubitsky (1992), or Gleick (1987) for more information about fractals and fractal geometry. Fractals arise in an area of mathematics called Dynamical Systems or Chaos Theory that became very popular in 1980's. It continues to be a rich area of mathematics because these fractal patterns frequently correspond better to many things we see in nature than do the shapes (circles, triangles, rectangles, general polyhedra, etc.) that had previously been used to describe them.

PascGalois Triangles exhibit their own kind of self-similarity, and the tools that have been developed to study the self-similar structure of fractals, such as the notion of fractal dimension, can be modified to study them as well. This is one avenue of further study related to these images. See Bardzell and Shannon (preprint) and Wolfram (1984) for more information.

JOMA

Journal of Online Mathematics and its Applications

Dummy View - NOT TO BE DELETED