O Christmas tree,
O Christmas tree,
One year a fractal made thee!
O Christmas tree,
O Christmas tree,
A heat map can display thee!

From Pascal's matrix we define!
Reflect across, divide by nine.
O Christmas tree,
O Christmas tree,
Self-similar and so divine!

Eventually I will run out of cute ways to use SAS software to create and visualize a Christmas-themed image. But not this year!

My recent article about how to create Pascal's matrix in SAS included a lower triangular matrix that exhibits self-similar patterns. If you take the Pascal matrix modulo 9 and reflect it across the Y axis, you get a triangular array that looks a little bit like a Christmas tree. You can add a "trunk" to the bottom of the tree to improve the resemblance. As shown in my previous post, you can use a heat map to visualize the matrix. In the following SAS/IML program, I use a green palette of colors to visualize the Pascal triangle values modulo 9. The resulting heat map is shown at the top of this article.

```proc iml; start PascalRule(n); m = j(n,n,0); /* initialize with zeros */ m[,1] = 1; /* set first column to 1 */ j = 2:n; /* elements to compute */ do k = 2 to n; /* for kth row, add adjacent elements from previous row */ m[k,j] = m[k-1,j-1] + m[k-1,j]; end; return(m); finish;   /* reflect Pascal's triangle to create left-right symmetry and add a tree trunk to create a Christmas tree image */ start PascalTree(n); m = PascalRule(n); T = m[,n:2] || m[,2:n]; /* reflect; omit column of 1s */ T[ loc(T=0) ] = .; /* replace zeros with missing values */ trunk = j(3,ncol(T),.); /* add a "tree trunk" with value -1 */ midpt = ncol(T)/2; /* note that ncol(T) is even */ halfwidth = ceil(n/10); trunk[,(midpt-halfwidth):(midpt+halfwidth+1)]= -1; return( T // trunk ); finish;   m = PascalTree(25); k = 9; tree = mod(m,k);   ods graphics / width=300px height=380px; title = "Happy Holidays to All SAS Users!"; ramp = palette("greens", k); ramp = "CX26261C" || ramp[,k:1]; /* brown (for trunk) and green palette */ call heatmapdisc(tree) colorramp=ramp displayoutlines=0 title=title;```

It is remarkable that the Pascal matrix has so many amazing mathematical properties. Now you can add "makes a reasonable facsimile of a Christmas tree" to the list!

Happy holidays and a wonderful New Year to all of my readers. You are the reason that I write this blog.

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Distinguished Researcher in Computational Statistics

Rick Wicklin, PhD, is a distinguished researcher in computational statistics at SAS and is a principal developer of SAS/IML software. His areas of expertise include computational statistics, simulation, statistical graphics, and modern methods in statistical data analysis. Rick is author of the books Statistical Programming with SAS/IML Software and Simulating Data with SAS.