Tag Archive: Mathematics

Planetary orbital foci

May 24, 2012 12:59 pm Published by Leave your thoughts

Planetary orbits are not perfectly circular; in fact, they are ellipses. An ellipse is a mathematical shape approximately equivalent to what is typically called an oval. An ellipse, though, meets some very specific criteria. One is that, unlike a circle, it has two foci instead of a single center. Where a circle is defined as the set of points whose distance to the center are some constant distance (the radius), an ellipse is the set of points whose distances to the two foci add to a constant. This allows you to construct an ellipse with some pins, some string, and a pencil.

So what are the two foci of planetary orbits? Well, one is the Sun. The other one? Just some random spot in space. And because each planet has a different size orbit with a different eccentricity (a measurement of how non-circular the orbit is), each planet has a non-Sun focus in a different place. Here is a Google doc spreadsheet with information on each planet’s orbit. A visualization after the jump.

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Choosing random keys

May 4, 2012 10:45 am Published by Leave your thoughts

Say you had code that generated random keys. These random keys were 6 letters long, all caps, with no duplicates. Here’s a few, as an example:

NTCYAR DHIEWM INBVTX IOELUC
RKNBJX GKRANB DRYVQU YIFKTS
VAUPSG ALWPOS CERSUY WAHJVM
MTXJSZ RNLFXZ VFIEXT VEOKIH

What are the chances that the key that you generate will be in alphabetical order? For instance, above, there’s only one in alphabetical order (CERSUY, in bold) And then, if you think you have that, generalize: For any string of length k distinct characters chosen from a set of n, what are the chances that they will be in order? My answer after the break.

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Idea: OEIS blog

April 15, 2010 2:54 pm Published by 1 Comment

Someone should make a blog based on the Online Encyclopedia of Integer Sequences. Each week (or twice a week, or daily, or something), there’d be a post which would discuss a single sequence. It’d give some glance at the theory and some history behind the concepts, be they Turing machines, prime numbers, set theory, or bi-directional graphs.

I’m adding this to my “to do when I retire” list.

The Feynman point

March 14, 2007 7:30 am Published by 1 Comment

Famed Physicist Richard Feynman once joked that he wanted to memorize π up to the 762nd digit. Why? Because that’s where pareidolia kicks in, and the digits appear to briefly coalesce into rationality:

3.1415926535 8979323846 … [727 digits] … 9605187072 1134999999 …

He would end a hypothetical recitation at that point, the implication being that from there on the decimal repeats. Virtually everyone, obviously including Feynman, knows π is both irrational and transcendental, and thus such a fact is impossible. The so-called “Feynman Point” is the first incidence of six consecutive identical digits in π and it also happens to be the first incidence of four and five consecutive digits.

Happy Pi Day.

Melting down pennies

July 28, 2006 9:07 am Published by 4 Comments

Not many people know that there is very little copper in modern pennies. In 1982, they were changed from 95% copper to simply copper-plated zinc. If you cut open a penny made after that date, you’d see the grey-colored metal on the inside. In fact, in 1981, pennies of both recipes were made, and collectors have to listen to the sound they make when they bounce to tell the difference. Do it yourself with a penny from the 70s and one from the 90s; the difference will be obvious.

The price of copper closed last night at $3.48 a pound. That means that a 3.1 gram penny has about 2.26 cents of copper in it. Melt down a big wad of those, and you can make yourself a tidy profit. The price of zinc has been escalating in the last few months, too, so it might not be long until it’s worthwhile to melt down new pennies, too.