Personal Blogs

The local choir was going to perform Carmina Burana but they’ve decided to Carl it Orff.

Here's something I wrote a few years back and had forgotten about. It's an experiment you can try anywhere, when you have a few spare moments.

Equipment

Your imagination.

A stopwatch (or phone app or you can just count seconds: one hundred, one hundred-and-one, one hundred-and-two, one hundred-and-three… or get someone else to do it for you).

Background

There is no background, you must imagine it.

Preparation

Have your stopwatch (or phone app, or mental counter or helper) handy to start and stop without needing to look at it (you’ll see why shortly). Then:

READ THE INSTRUCTIONS BELOW before starting the experiment.

Because for most of the time your eyes need to be closed tight shut.

I had to rehearse it myself. It’s not that hard but a dry run helps with starting and stopping the timer without breaking concentration.

Instructions

Shut your eyes and think of an elephant.

Your elephant is facing directly toward you. Picture its eyes, tusks, greyness, sturdy legs, big flapping ears.

Are you ready? Do you hold a solid four-ton elephant in your imagination?

Rehearsal

Now you must rotate your elephant.

Your mind alone must do the heavy lifting (or in this case heavy rotating). Think of yourself as using the power of thought to act on the imaginary elephant, which you have to heft round so it is now facing away from you, so you see its bum and tail.

1. Ready steady go! Mentally rotate elephant.

2. When elephant in correct orientation stop timer.

3. Open eyes and write down time.

Experiment

Now that you have conducted the rehearsal:

1. Repeat the same procedure, that is record how long you take to turn the elephant so it goes from facing you to facing away from you.

2. Do it again but with this variation: now you have to mentally rotate the elephant full circle so that at the end it looks toward you again, and note the time as before.

Results

If you are like me you find that rotating the imaginary elephant takes time! Something is happening inside your head that is like physically manipulating an object, even though there is no real object. And a full rotation of my elephant takes about – guess what – twice as long as a half-rotation. It varies but usually I take about 6 s for a half rotation, 12 for a whole turn.

Conclusion

This is not of course a rigorous scientific experiment. But it does suggest that when we move mental objects we base the imaginary movement on experience of shifting real things. There is a considerable body of research on this, involving much better experimental designs that leaves little room for doubt. Shepard and Metzler were the pioneers (scroll down to find their article).

And did you rotate your elephant to the left or right? I didn’t say which to do. Are you are left or right handed? I’d be interested if you could post a comment to tell me the two things, so we can see if there is a connection.

And does the size of an imaginary animal matter? What if you try to mentally rotate a blue whale, about 50 times bigger by weight than an elephant? Or a human, about 50 times smaller?

"Quick, try to grab the toy bear's attention!", said Tom distractedly.

This is by Lord Byron and very evocative. It may be his best-known poem today. Manfred and Don Juan made a huge stir 200 years ago. I've not read Manfred but I have read Don Juan, and probably will again. But it is not a moving poem; it is funny and racy and cleverly-versified; and the references were topical at the the time it was written. I like it, but I don't think it is very deep.

Lord Byron was an Hellenophile and famously went out to support the fight for Greek independence. He died at Missolonghi in 1824, during the military campaign, it seems of malaria, and becames a Greek national hero. When I first visited Greece in the 1970s, small bars and restaurants often had a picture of Lord Byron on the wall, 150 years after his death.

I'm determined to beat procrastination. Starting tomorrow.

I'm a big doodler. Last night I drew some dots in a rough octagon and started drawing lines joining pairs of dots.

Then I wondered how many lines I could draw before I formed a triangle. I don't mean a triangle formed by three lines (although that's an interesting question too), but one with dots at its three corners.

With any problem where you have a series it's always a good idea to look at the small cases. Sometimes it can be misleading though. Still I drew some little sketches

It's easy to be sure the count is right for n = 2, 3 and 4, but even by 5 it gets less obvious. Is there a general formula? I wondered if this was a novel puzzle (and hoped it was); sadly it turns out no, it was investigated by Mantel in 1907. There are various proofs of the result Mantel found, but they are more or less technical and not intuitive. So I wondered if I could present the basic idea of one proof in a more accessible way.

With 5 dots consider a pair x and y that are joined by a line. That leaves 3 others and if we think about any one of these, let's call it z, we see x and y cannot both be joined to z. Otherwise we would have a triangle! So the maximum of lines from either x and y to the other dots is 3. Finally focusing on the 3 other dots, the maxiumum number of lines possible amongst these is 2, otherwise we would have a triangle.

Adding this all up as shown the greatest possible number of lines is 1 + 3 + 2 = 6.

Let's see if we can generalise this idea. Suppose we have n dots and the maximum number of lines that can be drawn without forming a triangle is f(n).

Following the same argument as before, if we are to avoid a triangle, we can have one line joining x to y, a maximum of n - 2 lines joining either x or y to one of the remaining n - 2 dots, and f(n - 2) lines amongst tos n - 2 others. So

f(n) = 1 + n - 2 + f(n - 2) = n - 1 + f(n - 2)

So we can immediate calculate further values

f(6) = 6 - 1 + f(6 - 2) = 5 + f(4) = 5 + 4 = 9

f(7) = 7 - 1 + f(7 - 2) = 6 + f(5) = 6 + 6 = 12

f(8) = 8 - 1 + f(8 - 2) = 7 + f(6) =  7 + 9 = 16

So now we have (for completeness we include n = 1 with 0 lines)

0, 1, 2, 4, 6, 9, 12, 16

Can you spot a pattern?

I’ve got a new job; vaccinating people. Can’t wait to get stuck in.

Repeat five times after me

"On an anonymous ominous omnibus."

I worked with a men’s neckwear company for many years. But it’s all over now, and we've severed our ties.

The police caught me in their dragnet. First they threw an edible water plant over me. Then they shouted, “You’re under a cress!”