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Patterns in Pascal's Triangle - with a Twist - Cross Products of Cyclic Groups

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

Let us see what happens when we make a PascGalois triangle using the group Z2  x Z2. Remember that the multiplication table for this group is

Now we have to choose which elements to place down the sides of our triangle. Notice that if we chose to place (0,1) down both sides, combining (0,1) with itself gives (0,0) and combining again with (0,1) gives (0,1) again so the element (0,1) cannot generate the whole group. Unlike the case of modular addition, where repeated addition of 1's will generate all of the possible results, in this set there is no single element that generates the whole set. Thus we decide to put the element (1,0) down one side of the triangle and the element (0,1) down the other side. In this triangle, pink is (0,1), yellow is (1,0), purple is (1,1), and teal is (0,0).

Here is the same triangle with a few more rows:

Does this triangle remind you of any of the modular arithmetic triangles? In particular does it remind you of the mod 2 triangle, also known as the Z2 PascGalois triangle? Let's take another look at the Z2  x Z2 triangles with 50 rows. In the following sequence of figures, we begin to change slowly the color corresponding to (1,1) to the color corresponding to (0,0). Then we change the color corresponding to (1,0) until it matches the color corresponding to (0,1). The final triangle is the Z2 triangle redrawn using teal for zero and pink for one. The colors corresponding to the different elements are displayed in order {(0,0),(0,1),(1,0),(1,1)} to the left of each triangle.

 

Z2xZ2 PascGalois triangle with teal identity Z2xZ2 PascGalois triangle with teal identity and (1,1) moving to teal Z2xZ2 PascGalois triangle with teal identity and (1,1)moving to teal
Z2xZ2 PascGalois triangle with teal (1,1) and identity Z2xZ2 PascGalois triangle with teal (1,1) and identity and (1,0) moving to pink Z2xZ2 PascGalois triangle with teal (1,1) and identity and (1,0) moving to pink
Z2xZ2 PascGalois triangle with teal (1,1) and identity and (1,0) moving to pink Z2 PascGalois triangle with teal identity and 1 in pink

The following animation shows the same identification occurring in the 128-row triangles.

Kathleen M. Shannon and Michael J. Bardzell, "Patterns in Pascal's Triangle - with a Twist - Cross Products of Cyclic Groups," Loci (December 2004)

JOMA

Journal of Online Mathematics and its Applications

Dummy View - NOT TO BE DELETED