It was accepted for publication in the 'Izvestia' of Warsaw University. However, in the following year there was a strike to produce a boycott of Russian Schools in Poland and I did not want to have my first work printed in the Russian language and that is why I had it withdrawn from print in Warsaw's 'Izvestia'.
That is why it was not printed until in the mathematical magazine 'The works of Mathematics and Physics' published by Samuel Dickstein. Each of the students made it a point of honour to have the worst results in that subject. I did not answer a single question I passed all my examinations, then the lector suggested I should take a repeat examination, otherwise I would not be able to obtain the degree of a candidate for mathematical science. I refused him saying that this would be the first case at our University that someone having excellent marks in all subjects, having the dissertation accepted and a gold medal, would not obtain the degree of a candidate for mathematical science, but a lower degree, the degree of a 'real student' strangely that was what the lower degree was called because of one lower mark in the Russian language.
Does this new iteration correlate to the pattern found in Pascal's triangle mod 3? See how this compares to Pascal's Triangle in mod 3! This triangle began by dividing the line segments of the largest triangle into four equal parts. Then, once new triangles were drawn the pattern was repeated.
While this does not correspond to Pascal's triangle mod 4, it is interesting to compare this triangle to the ones above! But what is the connection between this array of numbers and the Sierpinski triangle? If we have enough rows of the Pascal triangle and we highlight only the odd numbers, we discover our old friend: the Sierpinski triangle. Pascal triangle with odds and even numbers highlighted in a different colour. Awesome, right?
Observing a fractal pattern, such as the Sierpinski triangle, within a simple array of odd and even numbers shows that the connections between different mathematical subjects can be found in the most unexpected places.
But is that all? Of course not! Mathematics is about getting a result, and then taking it to its limits. Change something, see what you get and then keep going. The way we obtained the Sierpinski triangle starting from the Pascal triangle was by taking the rows and then dividing the numbers into odds and evens; or, equivalently, we divided each of the numbers by 2 and focused only on the residuals. If this process is continued indefinitely it produces a fractal called the Sierpinski triangle.
Now you are going to work out the dimension of this fractal. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.
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