edit on 6-8-2016 by yampa because: (no reason given)



While browsing the mathematical encyclopedia Wolfram Mathworld, I came across a page on "Euler's Number Triangle". On this page is a plot formed from base-2 binary numbers stacked next to each other. I noticed that these binary numbers formed some quite remarkable patterns.I posted a thread on this about 5 years ago, but most of the images have expired now so I'm going to repost it here.Euler's number triangle is very similar to the famous Pascal's triangle, the only difference is that an extra multiplication takes place when forming the triangle.This animation is showing how Pascal's triangle is formed - Euler's is nearly identical to this:So we take a row from that triangle, convert each number into a binary number, stack them side by side, then we get these pictures:Same images, with black swapped for white:Strange.. Is this pareidolia, or something more important?This is a close up description of how the 13th figure is plotted from the numbers in the triangle. This is a fairly large image, so if you open it in a new window, you should be able to read all the details.A closeup on the first two really interesting rows - row 13 & 14. This is my personal interpretation of what I see - maybe you see something different? Or nothing at all? Is this random noise, or maybe something else?Plotted using mathematical unit cubes:How about row 28? I find this *remarkably* anthropomorphic considering how easy it is to generate these numbers.One section isolated:Some higher rows:A large version of the triangle, showing the algorithm in detail.More details on Euler's number triangle can be found here:The triangle of Eulerian numbers has many deep connections to number theory and combinatorics. So not only do these numbers make cool pictures, they are important and useful in real mathematics!