The elliptical trainer

December 5th, 2011

An “elliptical trainer” is an exercise machine, intended to give its user the same physical experience that he could get from climbing up stairs or just going for a walk. Here’s my “artist’s impression” of one of these things:

The basic idea is that the athlete (blue in this picture) steps up and down on two boards (only one shown here) which are attached to a wheel behind him (that can be slowed down with a brake to make him exert himself more) and to vertical bars that he holds on to, which swivel fore and back around a pivot marked “T” in this sketch.

All this time, the athlete’s foot remains on a spot “F” on the moving board, and apparently this point F moves along an elliptical curve (green in the sketch above) — hence the name of this machine.

This contraption was recently the subject of a limerick in the OEDILF, and in a private discussion with my fellow OEDILF author PGS (Holger Martin), the question arose whether the green curve in this sketch really is an ellipse, or only an egg-shaped non-circular shape. In other words: is the elliptical trainer really “elliptical”?

Obviously, this is just a simple question about planar geometry involving a circle, a circular arc and some straight lines, so I thought I’d be able to determine the green shape quite easily. It’s turned out to be a real pest though. I hope I can get it worked out later this week.

Correlation

December 1st, 2011

I’ve been a member of the OEDILF, the Omnificent English Dictionary In Limerick Form, since February 2007, an online project attempting to define every word in the English language by a suitable limerick. When a limerick alone isn’t enough, one may add an “author’s note” to provide additional information. Over the last years, I’ve written a number of limericks defining mathematical terms — a subject shunned by most and feared by some other OEDILF authors, and I’ll be presenting some of them here. Here’s an old one from February 2008:

Correlation

Recent tests show that so-called “IQs”
Bear no strong correlation to who’s
Rich or poor, black or white,
Male or female, but might
Correspond with the size of your shoes.

Statistics do, in fact, prove a strong correlation between IQ-test results and shoe sizes. Young children tend to have very small feet, and their test results, given the same tests, hardly ever beat those of adults (whose feet are larger). Such cases show that correlation between two seemingly unrelated numbers (test results and shoe sizes) may be due to a hidden factor (age).

Many statistics of this kind seem profound and meaningful until the hidden connexion is revealed, and then they look trivial. Say someone claimed that a study found a large majority of all traffic accidents in Israel were caused by Jews. That might lead you to think that Jews are bad drivers — until you start asking who lives in Israel and might own a car. So don’t trust statistics alone; use your brain!

The moon is tumbling down

November 30th, 2011

Our balcony faces due south, and since we don’t smoke in the flat, I’ve been watching the moon quite often over the last weeks, even months. And I noticed something that appeared rather strange at first. Each evening/night the moon wanders from the left (east) side of the sky to the right (west) side — which is to be expected, if you remember that this apparent motion is the same as the motion of the sun (both are due to the earth’s rotation). But when the moon is only a half moon, it seems to “tumble” while it moves from left to right. Here’s what it looks like, roughly, at three times during one evening:

The moon is tumbling down

At first I wondered why this should happen, till I connected the three moon shapes in this picture with a circular arc and realised that the moon isn’t tumbling at all; it’s just following a circular (or near-circular) path along the sky and keeping the angle of its “crescent” perpendicular to this path all the time.

Problem solved, but then I ran straight into the next one, which I couldn’t solve just with pencil and paper (and a graphics program) — what would the motion in the picture above look like if I’d observed it from the southern hemisphere, say from Australia?

Since I have some online friends in Oz and NZ, I wondered whether I should ask them whether it’s the same there. But then I worked it out on my own. Can you?

What’s this all about?

November 15th, 2011

For quite some time I’ve been meaning to write a diary, in the way that great people did in the past, jotting down ideas about world politics or sketches for machines that I dream up. Unlike my predecessors (I’m thinking of men like Newton or Darwin) I have no shelf space for leather-bound tomes filled with ink, so I’m resorting to a modern medium.

Like many others, I think of many weird things every day, sometimes leading me to a brilliant new insight about how the world “works” (or doesn’t), and sometimes leading me nowhere. And sometimes I forget the whole thing after a few hours. That’s another reason why I’d like to record such thoughts in this space.

I hope at least some of my visitors wil enjoy at least some of my ramblings.