![\"Binomial\"](\"http:\/\/farm9.staticflickr.com\/8171\/7954507004_e2c202a26c_z.jpg\")
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Above is a discrete fractal generated by creating a table of zeros and ones by deciding if the binomial coefficients<\/a> are even or odd. The “key” here is paint black if odd, otherwise leave light blue. <\/p>\n The pattern is closely related to Pascal’s triangle. <\/p>\n The pattern clearly shows self-similarity as all fractals do.<\/p>\n As far as I know, this pattern was first noticed in [1]. Also note that we have a structure very similar to the Sierpinski Sieve<\/a>. In the limit of infinite rows we recover the Sierpinski Sieve, up to a shift in the positions of the zeros and ones.<\/p>\n A slight variant<\/strong><\/p>\n Just for fun I used the same algorithm to study the pattern associated with modified binomial coefficients of the form<\/p>\n \\(\\left( \\begin{array}{c} (-1)^{k}n\\\\ k \\end{array} \\right)\\) <\/p>\n Again the pattern shows lots of self-similarity. <\/p>\n References<\/strong><\/p>\n [1] S. Wolfram: American Mathematical Monthly<\/em>, 91<\/strong> (November 1984) 566-571 <\/p>\n","protected":false},"excerpt":{"rendered":" Above is a discrete fractal generated by creating a table of zeros and ones by deciding if the binomial coefficients are even or odd. The “key” here is paint black if odd, otherwise leave light blue. The pattern is closely related to Pascal’s triangle. The pattern clearly shows self-similarity as all fractals do. As far … Continue reading Fractal from Binomial Coefficients<\/span> ![\"Binomial2\"](\"http:\/\/farm9.staticflickr.com\/8436\/7954509304_d9be849f4e_z.jpg\")
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