Known pattern

In general, numbers in pascals triangle produce sierpinski-triangl-like fractals. Some examples can be seen in this answer and are explained in this answer.

If we take the last digit of numbers (binomial coefficients) in the pascals triangle, and color the $\ne0$ digits white, we get the following image shown below; (First $1024$ rows)

enter image description here

Where this process is equivalent to taking $\binom{n}{k}\bmod m$ (where $m=10$ for this case) and coloring nonzero results. This is what was observed in the linked question/answers, for values of $m$ in general. (This is equivalent to taking last digit of binomial coefficients in some number base $m$)

Note that multiple patterns seem to be overlapping in the picture above. This is discussed in the linked answers. Only prime numbers $m$ produce clean patterns, and composite numbers like $10$ above are a combination of multiple prime patterns.

The pattern below does not care if number base (in context of first/last digit) is prime or not. It seems that, that variable in the pattern below, is used as a "zooming" factor to the picture of the pattern. See the gif at the end of the post.

"Reverse" pattern

But what if we take the first digit, instead of the last one, in some number base?

For decimal case, we have the following image, for comparison to the above: ($b=10$)

enter image description here

Where we colored the pixel if the first digit of the binomial coefficient is $"=1"$. (Since $0$ can't be the first digit) Similar pattern holds for any other single digit, in any other number base.

Why does this pattern appear in the pascals triangle, in all number bases?

Can it be somehow mathematically described or linked to any other things?

Like the first patterns links pascal triangle with sierpinski-triangle-like fractals, which can be explained for prime numbers by the Lucas' theorem (see linked answers) ?

You can observe the first $32$ number bases for the first $256$ rows: (10 images per second)

(Starting at base $3$ since in binary we have only one nonzero digit)


When the number bases are increasing, it looks like its "zooming into the tip".

You can also see the number base 3 compared to base 1000 .

Also see, that if we pick another digit $"= d"$ for our $"=1"$ condition, we get a "corrosed" effect on the pattern for bigger digits. See digits $1\dots9$ for $b=10$ example here as a gif .

  • $\begingroup$ Simply beautiful! $\endgroup$ – Mohammad Zuhair Khan Dec 30 '17 at 19:35
  • $\begingroup$ Things like these are why people did maths- there is so much beauty in it! $\endgroup$ – Mohammad Zuhair Khan Dec 30 '17 at 19:36
  • 2
    $\begingroup$ The prominent arcs are almost certainly related to the fact that each entry in the triangle is the sum of the two above it. $\endgroup$ – dbx Dec 30 '17 at 19:57

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