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After my observation today at the Skytree in Tokyo, I am wondering if this building could be used as a giant sundial?

As an aside, it would be interesting to consider how the Skytree might be used to display the time.

Tokyo Skytree

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    $\begingroup$ Anything that casts a shadow could be used as a sundial. Just paint big numbers (10, 11, 12, etc) on the roofs of the buildings to know the time. :-) $\endgroup$ – JohnHoltz Nov 12 '19 at 15:10
  • $\begingroup$ In theory, you can work this problem in reverse: use something like CSPICE to calculate Skytree's shadow's tip position for various dates and times and paint the buildings (or display on OSM (or Google) Maps if the Japanese object to having their buildings painted). This might be project worthy. $\endgroup$ – user21 Nov 13 '19 at 15:49
  • $\begingroup$ astronomy.stackexchange.com/questions/23165/… is a similar question $\endgroup$ – user21 Nov 13 '19 at 16:47
  • $\begingroup$ shadowcalculator.eu/#/lat/35.70992225357101/lng/… (zoom out slightly seems to do this, but the shadows are fixed unless you draw your own polygons. pintsinthesun.co.uk/#51.52491267801333/-0.07539689540863037 might do this for a limited area, and youtu.be/kGh2lpTHjDQ?t=86 suggests Google Earth can do this. $\endgroup$ – user21 Nov 13 '19 at 17:06
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In most sundials, the gnomon is aligned with the Earth's polar axis so that the shadow indicates the Sun's hour angle. A vertical tower instead casts a shadow indicating the Sun's azimuth. The time of day can be computed from this, but simple hour marks won't quite do it.

Let's try it with your photo. The Skytree is at 35.7°N, 139.8°E, and the antenna tip is 634 m above ground. The buildings near the shadow tip appear to match the satellite image about 830m away and 1° west of due north. So the Sun's azimuth at that time is 179° and its altitude is atan(634/830) = 37°. Assuming a date of 2019-11-12, and jiggling the inputs of the NOAA Solar Calculator, I find a matching azimuth at 11:22 AM JST (UTC+9).

Alternatively we can solve the coordinate conversion formulas for hour angle:

ha = atan2(-sin(az), -cos(az) * sin(lat) + tan(alt) * cos(lat))

which works out to -0.8° or about 11:57 apparent solar time. Subtracting 19 minutes for Tokyo's longitude east of the JST central meridian, and 16 minutes for the equation of time on Nov 12, we also get 11:22 AM.

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  • $\begingroup$ It's a lot easier to just check the time of year and the sun's position in the sky. Once you get NSEW sorted, and make a habit of looking, you can get within half an hour easily; no shadows involved. $\endgroup$ – Wayfaring Stranger Nov 14 '19 at 16:12
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Anything which casts a shadow can become a sundial, as JH's comment points out.

The other thing you need to know is the latitude of the building, as that affects the change in shadow direction per hour of elapsed time. Worse, due to the eccentricity of Earth's orbit (and rotation axis), there's an offset between clock time and sundial time which varies over the year. You can get most of the formulas at the Wikipedia page .

There's a handy calculator at blocklayer.com

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