Not the golden rectangle

September 27, 2005 11:12 PM

Is there a specific name for a rectangle which, if you divide it in half, the two halves have the same proportions as the original rectangle?

In other words, take a rectangle with long side A and short side B. Divide it at a point half-way along side A. Divide one of the resulting rectangles again, this time half-way along side B, and so on.... What would be the ratio so that A/B = B/0.5*A = 0.5*A/0.5*B ... to infinity? Using simple math I can easily figure it out to somewhere between 1.4117 and 1.4167 (or 24/17 and 17/12).

Whatever the ratio is, does it have a special name, and is it particularly significant in any way? For some reason, this question has stuck in my mind for over twenty years, and I'd now like closure. And no, it's not the golden section or golden rectangle.

posted by e-man to Science & Nature (12 answers total)

(A/B)=(B/.5A)

(1/2)A^2=B^2

B^2/A^2=1/2

A/B=sqrt(2)

sqrt(2)=1.4142...
posted by vacapinta at 11:26 PM on September 27, 2005

It does apparently have a name.

The silver rectangle. The ratios are known as the Lichtenberg ratio.
posted by vacapinta at 11:34 PM on September 27, 2005

It's worth pointing out that this is true for the ISO 216 paper sizes (e.g. A4) used outside the U.S.
posted by grouse at 11:41 PM on September 27, 2005

It's worth pointing out that this is true for the ISO 216 paper sizes (e.g. A4) used outside the U.S..

Nonsense, that would mean that the sides of the paper have irrational lengths, but you can never measure something to be irrational.

Ok, done being an ass.
posted by dsword at 11:59 PM on September 27, 2005

Yes, the paper the rest of the world uses largely conforms to this ratio. A4 folded in half lengthwise is A5.

It makes it worth it to buy A4 or A5 paper, especially if you are making booklets or are drawing at double the printed sixe.

That ratio being x : sqrt(2)*x.

It's related to the 30-60-90 triangle.
posted by blasdelf at 12:45 AM on September 28, 2005

but you can never measure something to be irrational.

Select a different basis.
posted by j.edwards at 1:13 AM on September 28, 2005

grouse: "It's worth pointing out that this is true for the ISO 216 paper sizes (e.g. A4) used outside the U.S."

dsword: "Nonsense, that would mean that the sides of the paper have irrational lengths, but you can never measure something to be irrational."

Technically true, although pedantic. The ISO 216 paper sizes are rounded down to the nearest millimeter.
posted by Plutor at 5:01 AM on September 28, 2005

blasdef, you might be thinking of the isosceles (45-45-90) right triangle, the ratio of whose legs to its hypotenuse is 1:sqrt(2). The 30-60-90 triangle has sides in ratio 1:sqrt(3):2.

dsword, you can make a physical object conform to an irrational measurement just as precisely as you can make it conform to a rational one; how would you make a piece of paper that was exactly half as wide as it was long?
posted by escabeche at 8:46 AM on September 28, 2005

dsword, you can make a physical object conform to an irrational measurement just as precisely as you can make it conform to a rational one

My comment was a bit of a dumb joke. Yes, you can clearly construct a piece of paper that fits the criteria. But my point remains valid. You will never measure anything to have an irrational length. You may calculate it to be so, but any honest measurement will turn up a rational multiple of your unit plus or minus an uncertainty. Now, I can make a right isosceles triangle with the two similar sides having length 1 inch and then use the hypotenuse to mark sqrt[2] inches on my ruler (this is NOT a measurement, it is a calculation). Then I can go use that mark and measure something else to be sqrt[2] inches. If I publish this as data, however, I will be laughed at.

Plutor's comment was exactly right: my comment was as pedantic as they come. We can definitely make paper in useful sizes.
posted by dsword at 4:24 PM on September 28, 2005

I thought it was funny, dsword.

We use ISO paper in Australia. Almost nobody, however, knows that A5 is half of A4 is half of A3 is half of A2 and so on. My coworkers were amazed when I explained how big our new A2 whiteboard would be by laying out a couple of sheets of A3 paper from the photocopier.
posted by obiwanwasabi at 6:36 PM on September 28, 2005

Response by poster: Thanks! A deceptively simple solution. Now I can rest easy ;-)
posted by e-man at 9:36 PM on September 28, 2005

My comment was a bit of a dumb joke. Yes, you can clearly construct a piece of paper that fits the criteria. But my point remains valid. You will never measure anything to have an irrational length.

You will never mesure anything to have a rational length, either. Think about that!
posted by delmoi at 1:41 PM on October 1, 2005