Dork Scratchings

Can you always reorient a four legged chair to make it stable on an uneven surface? If we can assume the feet form a perfect square, then: Yes! Without loss of generality let's suppose that the chair starts off in the position shown top left, with its front right foot in the air and the others on the ground.  If we keep the two left feet on the ground while pushing down on the other two, the above ground foot will eventually touch the floor and the remaining foot will sink below it. Now imagine instead rotating the chair by 90$^\circ$, keeping the same 3 feet touching the ground at all times, and ending with the two back feet in the positions previously occupied by the two left feet.  The two back feet and the front left foot occupy three corners of the square described by the feet at the end of the previous paragraph, so the fourth must be below ground.  Since the front right foot starts off in the air and ends up below ground we can infer from t...

 Déjà vu Reality Checkpoint Cambridge ... the feeling of having been somewhere or experienced something before when you know you haven't.  Déjà vu demonstrates that knowing we've been somewhere before and feeling that we have are different things. What makes it so difficult to convince skeptics of the reality of Everett's parallel worlds is that it feels as if time flows through the present moment like water flowing through a hosepipe.  It doesn't feel like the present moment just exists as a droplet in a sea of moments, some of which happen to be a bit future- like , some a bit past- like (but most neither).  Past-like moments invoke a feeling of recognition in us, and we line them up like dominoes and call them the past.  But if you've ever experienced déjà vu you know that such feelings of recognition can't be trusted. So, if we could determine what neural activity or hormone , is associated with déjà vu, then maybe we could develop a drug th...

A few Christmases ago I was playing with a toy xylophone meant for my nephew when I noticed something odd.  I expected going up an octave to halve the bar length, but it didn't.  A quick measurement confirmed my suspicion: to go up one octave you divide by $\sqrt{2}$ instead of by $2$.  I remembered this earlier this week while at my son's school concert, and decided to see if I could work out why! Strings First a bit of background.  Why did I expect halving the length to result in a note one octave higher (i.e. double the frequency)?  The answer is because I'd learned that this was the case with stringed instruments.  Assume that the tension is a constant $T$, and let $z(y,t)$ be the vertical displacement of the string at position $y$ and time $t$.  Then the upward force on a small element of  size $\delta y$ is approximately $$ T \left(\left. \frac{\partial z}{\partial y}\right|_{y+\delta y} - \left.\frac{\partial z}{\partial y}\right|_y...

Image by Goodtiming8871 CC-BY-SA 4.0 Most people learn how AM and FM radio works at some point or another.  These are methods for transmitting and receiving an analogue signal, but nowadays most radio signals are digital.  How is it possible to send bytes from one place to another using analogue electromagnetic radiation?  I thought I'd write a post to try to demystify it.