### Fun with sub-Nyquist sampling (or Aliasing as Art)

A subwoofer agitates a stream of water at 24 +/- $\epsilon$ Hz, where $\epsilon \sim 1$, while video is recorded at a frame rate of 24  Hz (i.e. sub-Nyquist). Fun ensues!

This effect is known as aliasing, which is also responsible for helicopter blades and car wheels appearing to spin backwards in films. Aliasing is also important in finding planets. We sample the radial velocity variations of stars caused by their planets using instruments such as HIRES at the Keck observatory. If we don't sample with dense enough time coverage (high enough frequency), a sub-sampled radial velocity signal can appear at a shorter or longer period. Here's an example from Wikipedia:

Imagine that the red curve is the true signal and the apparent (measured) signal is blue. You gotta mind your time-sampling! The optimal sampling is less than half the period (twice the frequency), which is known Nyquist samling.

This is what caused planet hunters (including me) to get the orbital period of 55 Cancri e wrong. Bekki Dawson and Dan Fabrycky found the correct signal at a much shorter orbital period than was previously thought. Since the planet was closer to the star, the probability that it would transit increased by a large amount (roughly a factor of 3, if memory serves). This prompted Josh Winn and collaborators to search for transits with a space telescope called MOST. And this is how the brightest transiting planetary system was discovered!

(The fuller story involves a prejudice against the existence of planetary periods less than 1 day, which caused our diagnostic periodogram plots to be plotted starting at 1. This hid the true period near 0.73 days, and drew attention to the aliased signal near 2.8 days. Other more technical details not suited for this blog are covered by Dawson & Fabrycky.)

### On the Height of J.J. Barea

Dallas Mavericks point guard J.J. Barea standing between two very tall people (from: Picassa user photoasisphoto).

Congrats to the Dallas Mavericks, who beat the Miami Heat tonight in game six to win the NBA championship.

Okay, with that out of the way, just how tall is the busy-footed Maverick point guard J.J. Barea? He's listed as 6-foot on NBA.com, but no one, not even the sports casters, believes that he can possibly be that tall. He looks like a super-fast Hobbit out there. But could that just be relative scaling, with him standing next to a bunch of extremely tall people? People on Yahoo! Answers think so---I know because I've been Google searching "J.J. Barea Height" for the past 15 minutes.

So I decided to find a photo and settle the issue once and for all.

I then used the basketball as my metric. Wikipedia states that an NBA basketball is 29.5 inches in circumfe…

### Finding Blissful Clarity by Tuning Out

It's been a minute since I've posted here. My last post was back in April, so it has actually been something like 193,000 minutes, but I like how the kids say "it's been a minute," so I'll stick with that.
As I've said before, I use this space to work out the truths in my life. Writing is a valuable way of taking the non-linear jumble of thoughts in my head and linearizing them by putting them down on the page. In short, writing helps me figure things out. However, logical thinking is not the only way of knowing the world. Another way is to recognize, listen to, and trust one's emotions. Yes, emotions are important for figuring things out.
Back in April, when I last posted here, my emotions were largely characterized by fear, sadness, anger, frustration, confusion and despair. I say largely, because this is what I was feeling on large scales; the world outside of my immediate influence. On smaller scales, where my wife, children and friends reside, I…

### The Force is strong with this one...

Last night we were reviewing multiplication tables with Owen. The family fired off doublets of numbers and Owen confidently multiplied away. In the middle of the review Owen stopped and said, "I noticed something. 2 times 2 is 4. If you subtract 1 it's 3. That's equal to taking 2 and adding 1, and then taking 2 and subtracting 1, and multiplying. So 1 times 3 is 2 times 2 minus 1."

I have to admit, that I didn't quite get it at first. I asked him to repeat with another number and he did with six: "6 times 6 is 36. 36 minus 1 is 35. That's the same as 6-1 times 6+1, which is 35."

Ummmmm....wait. Huh? Lemme see...oh. OH! WOW! Owen figured out

x^2 - 1 = (x - 1) (x +1)

So $6 \times 8 = 7 \times 7 - 1 = (7-1) (7+1) = 48$. That's actually pretty handy!

You can see it in the image above. Look at the elements perpendicular to the diagonal. There's 48 bracketing 49, 35 bracketing 36, etc... After a bit more thought we…