Why do so many powers of 2 start with “1”?

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384 …

If you liked math in middle school, odds are that maybe you memorized the powers of two. But did you ever think about the fact that so many of them start with the digit “1”? Is there a reason for it? How would you go about stating the problem in more formal mathematical terms?

Dmitry Kleinbock from the Brandeis Math department explains in this Numberphile video:

Be sure to watch the extra content (below) for a slightly more technical, but still completely approachable, additional explanation, where the problem reduces to the so-called equidistribution property of irrational rotations of the unit circle.

Are there other numbers more likely to start with the digit “1”?  It’s pretty easy to convince yourself that there are.

3-D Turing pattern formation in a chemical reaction system

In a report in this week’s issue of Science, Brandeis professor Irving Epstein, senior research associate Vladimir Vanag and postdoc Tamas Bansagi use tomographic methods, like those employed in a medical CAT scan, but using visible light in this case, to obtain the first three-dimensional images of Turing patterns. These patterns have been proposed as a mechanism for morphogenesis in living systems, perhaps offering an explanation for phenomena like “how the leopard gets its spots” or skeletal structure in developing limbs. .

Commentary: Wired Science

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