2322: ISO Paper Size Golden Spiral

2020-06-19T22:50:17Z

DiegoRapido: /* Explanation */

{{comic
| number = 2322
| date = June 19, 2020
| title = ISO Paper Size Golden Spiral
| image = iso_paper_size_golden_spiral.png
| titletext = The ISO 216 standard ratio is cos(45°), but American letter paper is 8.5x11 because it uses radians, and 11/8.5 = pi/4.
}}

==Explanation==
{{incomplete|Created by a GRAPHICS DESIGNER. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}

This comic strip is about annoying graphics designers and mathematicians, much like [[590: Papyrus]] and [[1015: Kerning]].

The {{w|Golden Spiral}} is a figure made by concatenating squares whose side lengths shrink according to the golden ratio. One can make a similar shape with the {{w|Paper_size#A_series| A Series}} of standard paper sizes, but the figures are rectangles whose side lengths shrink by a factor of the square root of 2, not squares whose side lengths shrink by a factor of the golden ratio. This is meant to parody the numerous questionable examples of the golden ratio in everyday life.

The spiral shown is a logarithmic spiral with a growth factor of sqrt(2), and if the center of the spiral is at the origin, it may be graphed with r = C*2^(θ/π), for any positive constant C.

The title text is a joke, based partly on the fact that the US uses imperial units while much of the rest of the world uses S.I. units. The 11/8.5 ratio is the length/width ratio of US “letter” paper, which is 11 inches by 8.5 inches (not legal, US legal is 14 by 8.5). The value of pi/4 radians is indeed equal to 45 degrees, although Randall takes the cosine in one case and uses the raw angle in the other case in order to get a close coincidence of values. The width/length ratio of A series paper (ISO 216) is exactly cos(45 degrees), which is 1/sqrt(2). As for US letter paper, 11/8.5 is not in fact close to pi/4, but it’s possible that Randall meant to write 8.5/11 instead of 11/8.5. To 4 decimal places, 8.5/11 = 0.7727 and pi/4 = 0.7854.

In reality, the usage of radians vs. degrees is not a geographic or political decision, but generally is delineated by profession. Most engineering and science fields measure angles in degrees or fractions of degrees (arcseconds, or even milliarcseconds in fields like astronomy), while mathematicians and physicists generally use radians. Civil engineers may refer to the slope of a road by its {{w|Grade (slope)|grade}}, which is commonly expressed in terms of the tangent of the angle to the horizontal (either as a percentage or a ratio); for angles up to ~10 degrees, this is close to the value of the angle in radians.

Mathematicians get annoyed by the claim that the golden ratio is everywhere. I love Disney's "Donald in Mathmagic Land" but they make some outrageous claims about the golden ratio's place in art and architecture. BTW, the ISO system of paper sizes is awesome! You can photocopy two A4 pages side-by-side, reduced to fit exactly on a single A4 page.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}

:[Caption inside panel:]
:The golden ratio is everywhere!

:[Picture of the ISO standard paper sizes (i.e. A1, A2, etc.) placed so that they fit together perfectly, overlaid with a spiral resembling that of the golden ratio]

:[Caption below panel:]<br />
:How to annoy both graphic designers and mathematicians

{{comic discussion}}
[[Category:Math]]
[[Category:Comics with color]]

Talk:2322: ISO Paper Size Golden Spiral

2020-06-19T22:47:20Z

DiegoRapido: Mathematicians get annoyed by the claim that the golden ratio is everywhere.

It annoys me that the hover text says 11/8.5 = pi/4, when 8.5/11≈0.77272727272 and pi/4≈0.78539816339. Claiming 8.5/11 equals pi/4 would be a much more beleiveable lie. [[Special:Contributions/162.158.79.37|162.158.79.37]] 15:29, 19 June 2020 (UTC)

The explanation says that the A series "side lengths shrink by a factor of the square root of two" but that's not true. The width of A(n+1) is half the length of A(n) as depicted. The sqrt(2) ratio referenced is between the length and width of any one piece of paper.[[Special:Contributions/172.69.62.124|172.69.62.124]] 15:35, 19 June 2020 (UTC)
:The side lengths do shrink by a factor of sqrt(2): the width of A(n) is sqrt(2) times the width of A(n+1), the length of A(n) is sqrt(2) times the length of A(n+1). Your statement that "the width of A(n+1) is half the length of A(n)" is also true, but it does not contradict that each step in the A-series shrinks the sides by a factor of sqrt(2). [[User:Zmatt|Zmatt]] ([[User talk:Zmatt|talk]]) 16:09, 19 June 2020 (UTC)

Fixed it [[Special:Contributions/162.158.74.61|162.158.74.61]] 15:43, 19 June 2020 (UTC)

Hi ! How come 11/8.5 = Pi/4 ? First one is more thant 1, second one is less than one... Although Pi/4 and 8.5/11 (or the reverse) are pretty similar, as usual in "let's annoy mathematicians" Randall's style...

https://xkcd.com/spiral/ --[[Special:Contributions/188.114.103.233|188.114.103.233]] 17:22, 19 June 2020 (UTC)

I understand why it annoys mathematicians (it's not the golden ratio), but why does it annoy graphics designers? Please add explanation!

It should be noted that the logarithmic spiral this comic implies it is would actually go outside the bounds of the paper. The leftmost point of the spiral would be about 6.4mm to the left of the left edge of the A1 sheet. [[User:Zmatt|Zmatt]] ([[User talk:Zmatt|talk]]) 18:39, 19 June 2020 (UTC)
:This drawing (as opposed to the singular mathematical formula behind the idealised spiral for the partitioning used) basically takes a simple quarter-oval across each distinct sheet size (with, as essentially mentioned elsewhere, the root(2) ratio between sides) alternating x/y and y/x as major and minor axes respectively. Even if it is not obviously discontinuous (x and y inflection transitions occur subtly) any derivative of the curve (as polar, say) would show jumps in gradient at each stage - probably an inclined-stepped/saw-toothy pattern whereas the true logarithmic line would demonstrate itself as a continuous function at any such level of derivation. The true spiral line followed from origin outwards would ''almost'' (not quite, because of the polar gradient) hit the 'outer edge' first in line with the ultimately recursive centre-point then withdraw again to hit the next transition slightly 'inward' of the next level out. The Golden Spiral approximation uses squares for each quarter, which therefore does not switch major and minor axes, but still changes the curve  and thus has the same not-quite-Golden nature. Although it's hard to describe, as you can see from my poor attempt that's probably inadvertently fallen foul of more specialised Pure Mathematics terminology due to the Pedant's Curse... ;) [[Special:Contributions/162.158.155.240|162.158.155.240]] 22:23, 19 June 2020 (UTC)

Mathematicians get annoyed by the claim that the golden ratio is everywhere. I love Disney's "Donald in Mathmagic Land" but they make some outrageous claims about the golden ratio's place in art and architecture. BTW, the ISO system of paper sizes is awesome! You can photocopy two A4 pages side-by-side, reduced to fit exactly on a single A4 page.

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2322: ISO Paper Size Golden Spiral

Talk:2322: ISO Paper Size Golden Spiral