Flags, striped
shirts and zebras
The Olympic motto is “Citius, altius, fortius” (faster,
higher, stronger).
Our nationalist flag-wavers have decided to emulate this
motto: our flags must go higher, and since the wind there is faster, the flags
must be stronger.
Firstly, the Centre decreed (to inculcate ‘nationalism’) that
every Central University must have a 207 foot high flag.
The proposed flags will be modeled on the one in New Delhi’s Central
Park, which is 207’ high, made of knitted polyester, is 90’x 60’ and weighs 35
kgs.
More recently, the Border Security Force has decided to put
up a flag at the Attari-Wagah border by 2017 which will be 350’ high.
“The current record for the tallest national flag is held by
the one atop a 293-ft-high mast in Jharkhand’s Ranchi. This 66-ft x 99-ft flag
was hoisted in January by Defence Minister Manohar Parrikar — it was,
incidentally, brought down today for repairs after being stuck at half-mast 10
days ago.”
The new flag, to be hoisted in Jan.2017, a top Border Security Force official
said, that at 350 feet, “it can be
seen both from Lahore as well as Amritsar (Amritsar and Lahore are
approximately 18 km away from the international border)”.
“At that height, the flag would have to be proportionately
sized, therefore it would also be the largest Tricolour. Officials say the
material to be used for the flag is also being discussed because at such
heights it may get damaged by rain and high winds.”
The first question is: will the flag be visible in
Amritsar and Lahore. A little simple trigonometry shows that the distance to
the horizon is given by:
S = Ö2hR
Where the height h of the flag is much less than the radius
R of the Earth (assumed as 6,400 kms). If h is not much smaller, an h2
term is to be added within the square root. With h as 106 metres, this distance
is 36.8 kms. So the flag should indeed be visible, as claimed.
The second question is: will the flag be visible to
the unaided eye (no binoculars or telescopes)?
The minimum angle resolved by the human eye is 0.5
milliradian, as opposed to the diffraction limited resolution of 0.2
milliradians under optimum conditions, according to the following website:
At a distance of 18 kms, the smallest visible size would
clearly be: (18x103)(0.5x10-3) = 9 metres.
Even the Ranchi flag is 99’x 66’ wide (30x20 metres) which
would translate to about 3x2 pixels – and the flag planned will be
proportionately bigger.
However, to be a bit more precise, a flag like the American
flag has a pattern of Stars and Stripes. The Indian flag has just 3 stripes
(never mind the Chakra!). So can these stripes be seen?
According to the following website, the appropriate
parameter is the visual acuity of the human eye:
“The ability of the eye to resolve detail is known as
"visual acuity." The normal human eye can distinguish patterns of
alternating black and white lines with a feature size as small as one minute of
an arc (1/60 degree or π/(60*180) = 0.000291 radians). That, incidentally, is
the definition of 20-20 vision. A few exceptional eyes may be able to
distinguish features half this size. But for most of us, a pattern of higher
spatial frequency will appear nearly pure gray”.
This angular resolution is almost the same as that mentioned
by the hyperphysics.phy site, so nothing much changes.
Of course, one might note that the colours in our flag are
not black and white, and that would make the analysis more complicated. And to
be absolutely rigorous one should take into account the modulation transfer
function (MTF) of the eye, because, according to Norman Koren: “The statement
that the eye cannot distinguish features smaller than one minute of an arc is,
of course, oversimplified. The eye has an MTF response, just like any other
optical component.”
The only way I’m gonna do the MTF analysis is:
a) somebody pays me,
or
b) I’m feeling really, really bored and I’ve got nothing to
do…
Anyway, let me just start ending this post by noting that
the world’s highest flag is in Jeddah, Saudi Arabia:
and it is 560 feet (170 m) high. The flag weighs 570
kilograms (1,260 lb) and is 49.5 metres (162 ft) long and 33 metres
(108 ft) wide. The area of the flag is equal to ‘half a football field’.
At 400 ft, the flagpole will be the
tallest the Shebogyan, Wisconsin, the company Acuity has constructed — and the
tallest in North America. The flagpole will support a 60 ft wide by 120 ft long
American flag, which will be weighted at the bottom to prevent it from wrapping
around the pole.
3rd question: can the unaided eye see the Jeddah flag
from outer space, say the International Space Station? The ISS height is about
400 kms above the Earth, so the minimum size needed would be about:
Wmin = (0.5
x10-3)(400 x103) = 200 metres.
I guess it would not be visible – even assuming that the
flag were laid flat on the ground – which would not be countenanced by any
self-respecting nationalist. Maybe if we allow for a satellite just at the edge
of outer space (100 kms altitude), the Jeddah flag may just be visible as
(almost) one pixel.
Can we expect an arms race at the Attari-Wagah border, with
each country trying to push up its flagpole even higher? Anyway, such a
competition is better than war…
Which leads to: ….Q4: what is the limit? With
existing materials, how high can the flagpole go? Victor Weisskopf gave
a derivation for the maximum size of a mountain, which can be tweaked a bit….
Let’s go into that after we touch on striped shirts and zebras – in the
next post (whenever that happens!).
Striped shirt:
If we assume a shirt
with 5 mm wide alternating black and white stripes, and taking the visual
acuity (VA) of a person with 20/20 vision as 0.5 mrad, then the maximum
distance at which the stripes can still be resolved is (0.005)/(0.0005) = 10
metres, under optimum lighting. At a greater distance, the shirt will just look
gray – a very common experience that almost everyone would have had.
The dependence of VA on background illumination, for 100%
contrast between stripes, according to Norman Kopeika in his book, ”A
system engineering approach to imaging” (SPIE Press, 1998) p.389
is:
|
Illumination (foot-lamberts,
fL)
|
VA (mrad)
|
|
1
|
0.36
|
|
10
|
0.28
|
|
100
|
0.20
|
This applies to the photopic system i.e. to the cones in the
human visual system.
1 fL = 10.76391 lux equivalent. Because 10.76391 = (3.2808)2,
the conversion factor from metres to feet.
Zebra
stripes:
There is a paper on the ease with which
predators such as lions, hyenas and humans can perceive zebra stripes – which
are alternating white & black of different widths. The paper is entitled:
“Zebra stripes through the eyes of their predators, zebras and humans” by
Amanda Melin et al (March 2016).
Reference: http://dx.doi.org/10.1371/journal.pone.0145679.
A human with 20/20 vision should in
photopic (daylight) conditions be able to resolve the widest stripes (~3 cms
wide) at about 180 m, while the lion should resolve the stripes at 80 m, and a
hyena at 48 m – while a zebra would resolve the other zebra’s stripes at 140 m.
The authors argue that the ideas that the
B&W stripes serve to ‘break up’ the outline of the zebra and make it
difficult for predators to see them are unlikely to be correct because the
vision of predators is not that good anyway. What I’m interested in is that the
authors use three numbers for the visual acuity of humans:
|
Illumination level
|
Visual acuity
(mrad)
|
|
Photopic (daylight)
|
0.145
|
|
Mesopic (dusk)
|
0.377
|
|
Scotopic (night)
|
6.77
|
The numbers for mesopic and photopic are
close to those for the higher illumination levels (10 and 100 fL) above. As
expected the zebra would have to be a lot closer for one to see the stripes by
moonlight…
The above links to a photo taken of a
zebra at night and the stripes are clearly visible … but the distance is not
stated, and besides the camera has optics…
To conclude: flags, striped shirts and
zebras have something in common: the eye of the beholder.
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