Book aspect-ratios

[RogerOwens]

[Edited a few minutes after posting]

Below, where I call a number an "outlier", and calculate results that don't include it, I do that only to look more at typical books.

Of course in many statistical samplings, an outlier is omitted because it might be felt to be a likely error. I didn't mean "outlier" in that way here.


Book aspect-ratios:

I measured the aspect-ratios of 8 books. They were paperbacks, and their covers have the same dimensions as their pages. That's why I chose paperbacks, for easier measurement.

It was a small sample, but the distribution was very symmetrical--The arithmetic mean and geometric mean were the same as eachother (to the 2nd or 3rd digit after the decimal-place) for the sample and eachof the sample-subsets for which they were determined. So I just list "mean" below, referring to both means.

...And the medians were very close to the means.

Here's some information about the book aspect-ratios:

Maximum: 1.636
Minimum: 1.305
Max/Min: 1.25

But 1.305 was an outlier .

If that outlier is left out, the minimum was 1.4917

...and Max/Min is then only 1.097

Here are the aspect-ratios:

1.636, 1.628, and 1.6106, 1.5714, 1.549, 1.519, and 1.4917

The maximum differs from the Golden-Ratio (about 1.618) by a factor of only 1.0111

As I mentioned, all of the geometric means that were determined were the same as the arithmetic means, to the 2nd or 3rd digit after the decimal point.

Here are some means:

All: 1.54
All but outlier: 1.57

Here are some medians:

All: 1.56
All but outlier: 1.57

Michael Ossipoff

[RogerOwens]

As of 4:02 p.m. Eastern DST,I've edited the initial post to this thread, to delete some errors.

Michael Ossipoff

[RogerOwens]

Three Atlases:

I measured the aspect-ratios of 3 world atlases.

The 3 values were: 1.349, 1.4234, 1.258

Mean: 1.34555
Median: 1.34924

Note that the mean and median are nearly identical, about 1.35.

According to Wikipedia, even with a sample-size as small as 3, a decent estimate for the most likely standard deviation of the population can be gotten by dividing the sum of the squared deviations by N-1.5, instead of by N-1

(and then taking the square-root).

The division by N-1 instead of N reduces the bias of sample standard deviation, and, according to Wikipedia, division by N-1.5 reduces bias more satisfactorily in the very smallest samples, like a sample-size of 3.

By that measure, the sample standard deviation for that sample was:

.09566

That sample standard-deviation is .0712 times the mean.

Max/Min = 1.1315


Larger Book-Sample:

My initial small sample consisted of the 8 books that were within easy reach from where I was sitting, and not under too many other things. Later I measured some more books, for a total of 31 books.

Mean aspect-ratio: 1.459

Sample standard-deviation: .13199

That sample standard deviation is .09 times the mean.

Max/Min = 1.33

Range: 1.2533 to 1.667

The maximum value differs from the Golden-Ratio by a factor of 1.03

In other words, it differs from the Golden-Ratio by 3%.

Michael Ossipoff

[RogerOwens]

When I said that the median of the aspect-ratios of the 3 atlases was 1.4234, I copied the wrong number.

The median aspect-ratio of those 3 atlases was 1.34924

(Of course I don't really claim that the numbers are accurate to that many digits)

Note that the mean and the median are very nearly the same, about 1.35

And I'll mention that thebook-page aspect ratio that is just right for the best display of Equal-Area PF8.32 or Behrmann is also nearly the same as that 1.35 mean aspect-ratio for the atlases.

...only about 0.2 standard deviations from the mean.

(based the rough guesses for the population's mean and standard deviation gotten from measurement of 3 atlases)

The fact that that sample's (sample standard deviation)/(mean) is very close to that of the 31-book sample suggests that the estimates based on the 3 atlases are fairly good.

And that ideal page aspect-ratio for PF8.32 and Behrmann is only 1.1 standard deviations from the mean of the aspect-ratios of the books in the 31-book sample.

The book aspect-ratio that is just right for the best display of Behrmann is 1.36

Obviously you get a bigger map if you use a page-spread, instead of a page.

If the east an west halves of the map, printed on 2 facing-pages, are oriented with their equator parallel to the page's long axis, then the page-spread's unusable part isn't in the middle of the map, occupying all of the map's central-meridian. That unusable part is only at and very near to the poles.

...a big improvement over the usual way of using a page-spread.

...and which, for the best display of PF8.32 and Behrmann, calls for a page aspect-ratio that is very close to the means of the aspect-ratio distributions for books and for atlases, where the distance from the mean is expressed in standard-deviations.

Behrmann has an aspect-ratio of about 2.72 (unless I made an error). I haven't determined the aspect-ratio of Equal-Area PF8.32, but, because that map is nearly cylindrical up to the Arctic, it, with the same standard parallel, is very similar to Behrman for most latitudes, and one would expect is aspect-ratio to be similar.

For a map with an aspect-ratio of 2.72, you want a book with an aspect ratio of 1.36

I'd have to find its formulas to find out, but the aspect-ratios of Equal-Area PF8.32 and Behrmann are surely very similar.

Michael Ossipoff



...

[Atarimaster]

Behrmann has an aspect-ratio of about 2.72 (unless I made an error).

You did.
Behrmann’s aspect ratio is actually ~2.36 – according to Wikipedia, and to make sure that this isn’t a typo, I tried the results from both Geocart and G.Projector.

2.72 is somewhere around standard parallels at 21°, shown in the image below (which is actually closer to 2.37).

[RogerOwens]

(The website insists on adding an additional unquote after my quoted statement, messing up the display of quotes. After repeatedly fixing it in edits, only to have the extra unquote re-appear, I finally gave up trying to fix it. ...hence the mis-displayed quote of your comment.)

Behrmann has an aspect-ratio of about 2.72 (unless I made an error)

You did.
Behrmann’s aspect ratio is actually ~2.36 – according to [url=https://en.wikipedia.org
{/quote]

Yes, I made the same error when I was posting the formulas for the equal-area cylindricals that I was proposing, one of which is conformal at lat 30, and the other of which give equal and opposite NS/EW scale-disproportion to (lat 45, lon 0) and to places at the equator.

Afterwards, I noticed the error, and corrected the formulas. So, my formula for Behrmann (which was one of the CEA versions that I was and am proposing) was correct, because I corrected it soon after first posting it.

Here's what the error was:

In Cylindrical Equidistant (CE), The NS scale is the same everywhere, and the EW scale at lat 30 is 1/cos 30 times what it is at the equator...because the parallels are the same length everywhere, though the parallel at lat 30 should be cos 30 times shorter than the equator.



So, if the map is conformal at the equator, then, with CE, places at lat 30 are therefore disproportioned by a factor of cos 30.

Considering only the horizontal widening at lat 30, due to all places having the same parallel-length, and forgetting the variation of CEA's vertical scale, and with conformality at the equator, I considered places at lat 30 to be disporportioned by a factor of cos 30. ...for the reason given above, for CE.

But, in order to make CEA equal-area, if a place at lat 30 is EW expanded (due to every latitude having the same parallel-length) by a factor of 1/cos 30, then, to achieve equal-area, places on the 30th parallel must be flattened vertically by a factor of cos 30.

Therefore, with those two equal and opposite scale-distortions in the two dimensions at lat 30, places at lat 30 must be disproportioned by a factor equal to the square of cos 30.

That was the error that I made, and corrected right away, when I recently posted the formula for Behrmann.

And, when posting (in the post that you're replying to) about the aspect-ratio of Behrmann, I made the same error again, leading me to give 2.72 as the aspect-ratio of Behrmann.

Behrmann's aspect-ratio is cos-squared (30) times pi, or .75 pi, or about 2.356

Michael Ossipoff