We often say somewhere is "in the shadow of" something without it necessarily being true.
» Coventry is in the shadow of Birmingham - technically no.
» Stratford is in the shadow of the Olympic Stadium - not really.
» Crawley is in the shadow of Gatwick Airport - only metaphorically.
The sun does not cast a shadow from the first location to the second, that's not how daylight works, so these places are not "in the shadow of".
Which got me wondering which places genuinely are in the shadow of somewhere else.
Or more to the point, where do you have to be to have a shadow cast upon you?
And I think this question has a brilliant but extraordinary answer, spatially speaking.
Let's start with something familiar in the centre of London - Nelson's Column.
Where could conceivably be "in the shadow of" Nelson's Column?
First important thing - Nelson's Column is 52m tall.
Its shadow isn't going to reach Buckingham Palace no matter what happens.
Second important thing - we need a formula for the length of a shadow.
Length = height ÷ tan α
(where α = angle of elevation of the sun)
So it all depends how high the sun is in the sky.
Don't worry, I'll do the maths.
On the longest day of the year the sun reaches a maximum elevation of 62°.
The formula gives a shadow length of 52 ÷ tan(62°) = 52 ÷ 1.88 = 28m
A shadow of length 28m is pretty small, but that's the minimum it ever reaches on a sunny day.
On the shortest day of the year the sun reaches a maximum elevation of 15°.
The formula gives a shadow length of 52 ÷ tan(15°) = 52 ÷ 0.267 = 195m
That's more like it. A shadow of length 195m is easily long enough to hit the National Gallery.
Of course the sun is often a lot lower than that, especially close to sunrise or sunset.
Technically it reaches 0°, i.e. the horizon, but buildings around Trafalgar Square block sunlight that low down.
Let's say the lowest the sun ever gets in Trafalgar Square is 4°.
The formula gives a shadow length of 52 ÷ tan(4°) = 52 ÷ 0.087 = 600m
That's long enough to hit Piccadilly Circus or Covent Garden but not Tottenham Court Road.
In reality, however, intermediate buildings would stop a 600m shadow from getting that far.
But another important thing to consider is compass direction.
This is the statue of Charles I, the official centre of London.
It's only 40m from Nelson's Column but it's also due south.
The sun never ever shines from north to south, not in the northern hemisphere.
This means the shadow of Nelson's Column is never going to hit the statue, never ever.
Let's summarise.
Shadow pointing west: around sunrise, so shadow quite long. Shadow pointing north: sun at maximum elevation, so shadow quite short. Shadow pointing east: around sunset, so shadow quite long. Shadow pointing south: never happens, so no shadow
Things in the shadow of Nelson's Column: Trafalgar Square, Canada House, National Gallery, National Portrait Gallery, St Martin-in-the-Fields, Charing Cross station Things not in the shadow of Nelson's Column: Charles I statue, Admiralty Arch, Whitehall
If you want to play around and have a go with this yourself, this map is set up to show the shadow of Nelson's Column at noon on 21st December.
Move the yellow circle at the top between sunrise and sunset to see the shadow change length and position.
And I've set up this map to show the shadow of Nelson's Column at noon (GMT) on 21st June.
See how short the shadow gets in the middle of the day.
Zoom out and you can see just how far the shadow stretches just after sunrise or just before sunset.
Ok, let's turn our attention to London's tallest building.
Here's the actual shadow of the Shard, as seen from the observation deck.
I took this photo just after noon at the end of January.
The tip of the Shard has just about reached Monument station.
Due north, this is almost as far as the Shard's shadow ever stretches.
And this is for a 310m tall building!
Again I can let you play around with this on a map.
Let me generalise so I can try to delineate the Shard's potential shadow.
For an extremely long shadow it needs to be sunrise or sunset.
And the direction of sunrise and sunset changes throughout the year.
This diagram shows the direction of sunrise and sunset in London on the 21st day of each month.
In winter, sunrise in London is roughly southeast and sunset is roughly southwest.
In summer, sunrise in London is roughly northeast and sunset is roughly northwest.
Everywhere inbetween these extremes can be the location of a sunrise or sunset.
The red sector shows everywhere that can be hit by a post-sunrise shadow at some point in the year.
The blue sector shows everywhere that can be hit by a pre-sunset shadow at some point in the year.
The red and blue sectors are all "in the shadow of".
But the 'south' sector can never be in the shadow of.
And the 'north' sector is sometimes in the shadow of, but only some of it and only around the middle of the day.
Here's how that looks when you apply it to the Shard.
Everywhere in the west and east sectors can be in the shadow of the Shard.
Nowhere in the southern sector can be in the shadow of the Shard.
And in the northern sector only a boomerang-shaped strip can be in the shadow of the Shard.
Sorry the curve's a bit wonky, my drawing program's not up to drawing hyperbolas.
And to summarise all that, this is the pattern you get if you superimpose every possible shadow over the course of a year.
Everywhere in black is potentially "in the shadow" of the Shard.
A similar pattern applies to smaller buildings, just over a much smaller area.
I have to confess this is not the shape I was expecting.
We often say somewhere is "in the shadow of" something without it necessarily being true.
