# flies for math addicts

#### studiot

Joined Nov 9, 2007
4,998
Four circus flies stand on the four corners of a one inch square shown blue in the pic.

When the ringmaster blows his whistle each fly begins to walk directly towards the next fly in a clockwise direction, as shown by the green arrows.

As each fly walks is turns so it continues to walk directly towards the next fly, so its direction of motion is always at right angles to the direction of motion of the next fly.
Thus each fly walks a curved path and they meet in the middle as shown in red.

How far does each fly walk?

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#### Mark44

Joined Nov 26, 2007
628
Studiot,
This seems like a tough one that will require some thought. One minor nit: according to the drawing, the flies are walking counterclockwise (or anticlockwise, as our British cousins would say).
As a first guess, the paths look like logarithmic spirals or maybe logarithmic curves.

#### studiot

Joined Nov 9, 2007
4,998
Anticlockwise? Yes you are quite right I meant to say that.
Although if you are under 5 (like me) or watch Mary Poppins or Harry Pottter you say widdershins!

It's quite an interesting excercise in vector analysis, but here's a hint.

There's a hard way and and easy way.

#### thingmaker3

Joined May 16, 2005
5,084
I suspect the hard way involves trigonometry and limits. I'm still working on this mornings first cup of coffee, so I've no clue what the easy way might be.

#### studiot

Joined Nov 9, 2007
4,998
I thought I'd post this one because we have has quite a few questions involving orthogonal vectors lately.

It is often forgotten that two vectors at right angles to each other have no effect on each other.

#### Mark44

Joined Nov 26, 2007
628
It is often forgotten that two vectors at right angles to each other have no effect on each other.
Depends on what operation you perform on them. If you add or subtract them, each has a significant effect on the sum or difference. However, if you multiply them via the dot product, then you get zero. Is that what you're driving at, Studiot?

#### studiot

Joined Nov 9, 2007
4,998
The flies start exactly one inch apart.
One fly starts along the line of the shortest distance (two vectors at right angles) to the next fly.
This fly always proceeds along the line of the shortest distance, adjusting its direction to suit.
Eventually the fly covers the exactly once inch to meet the next fly.
The motion of the next fly is irrelevant because it is at right angles to the motion of the first fly.
The actual path traversed is quite complicated, as has been pointed out.

You can view this problem as a constantly rotating rectangular coordinate sytem.