The Maximum Height a Skyscraper can be Built

Nowadays the constructions of skyscrapers have become a widespread phenomenon, we never saw so much new skyscrapers built and under construction all over the world before, skyscrapers are no longer limited in America like in last century, when the majority of tall buildings are located in United States, today even many traditional European cities have their own skyscrapers built, apart from the astonishing amount, the height is getting higher and higher, many cities in China are constructing skyscrapers taller than 500m, and even the height of Burj Khalifa cannot keep it the tallest building of the world longer, there are already two projects of towers with heights of more than a kilometer in progress.

As to the extremity of the height of skyscraper, this article gives a good answer and explanations.
In the article the author proved that a skyscraper could be built to at least as tall as the tallest natural mountain, as long as the base of the structure is large enough like a mountain combined with tapering shape in order to reduce the intensity of pressure on the soil ground, and skyscrapers are far lighter than solid mountains, a skyscraper is mainly composed of air, weighing only around 15 percent of a mountain, it could be 6.67 times taller and still as heavy as a solid mountain, by this way a skyscraper could be built to nearly 59 thousand meters tall, that's 150 times of the height of Empire State Building, how crazy it is.

buildings are largely hollowed
It can be clearly seen before the cladding are installed that buildings are largely hollowed.

However the skyscrapers are not mountain, there are many factors prevent them from having the base as large as what of a mountain.
First, skyscrapers are supposed to accommodate people, which means the natural light is a vital factor to be considered by architects or developers. If a building's base is wide as a mountain, there'll be large amount of floor space lacking of natural lights, nobody wants to live in dark rooms, maybe only suitable for some shopping malls, conference center, in which light illumination are largely used.
Second, this kind of skyscraper will be visually unappealing, many people and social groups will protest against it.

And the most vital factor is the financial difficulties, which is also the lots of skyscraper projects are facing. This factor comes from two aspects, respectively are the scarcity of land and high cost on construction, on average a standard skyscraper occupies an area of 2000 square meters, in some cities like New York such a land requires up to 500 million dollars, and the cost for construction ranges from some 300 million to over a billion, a skyscraper with the base as large as Mount Fuji equals to thousands of standard skyscrapers, which real estate developer in the world has so much money to build it? Today developers are struggling to secure the funds to build those standard size skyscrapers, and in New York City, most of newly built towers are super skinny as they are more easy to build and highly profitable.

Although Mount Fuji is 3776m tall, it has a very large base that as large as many cities combined together.

So taking the mountain as the reference to estimate the maximum height of skyscraper seems very unreliable.
We can't take those normal mountain as the reference, but then how could we estimate this extremity?
Thanks to our amazing nature, there are some bizarre mountains in the world. Some of them appear like normal skyscrapers.
A relatively notable example is the Mount Hallelujah in Hunan, China, see the picture below:
Mount Hallelujah
You may feel familiar when you look at it, yes, it is actually the inspiration for the floating Hallelujah Mountains in the film Avatar, it was formerly called Southern Sky Column and renamed to its current name in 2010 after Avatar became very popular.
The shape of the giant stone pillar is very similar to a skyscraper, it stands 150 meters with no tapering, which measured from its foot to the peak while the elevation surpassing 1000 meters, which cannot be seen as the height of the pillar itself.

150 meters seems too short, let's see another one:

Three giant stone pillars of Mount Jianglang
Three giant stone pillars of Mount Jianglang

These three giant stone pillars are situated on a mountain named Mount Jianglang in Zhejiang Province of China.

These pillars are much taller than Hallelujah mountain, the tallest one is 369m measured from the stone's foot to the peak, the one in the middle not only without tapering but the peak is even wider than the foot.

Three giant stone pillars of Mount Jianglang

Taking the middle pillar as the reference and given the fact that the solid stone is 6.67 times heavier, it can be estimated that the skyscraper could be built to at least 2100 meters by roof height. If the building is designed to be a little tapered (not significantly tapered like a mountain), and utilizing some special stable shape like the Y-shape of Burj Khalifa, or spiral shape of Shanghai Tower, the extremity could be much higher, my guess is more than 3000 meters by roof height.
Therefore theoretically it is still possible to build a ultra-tall skyscraper that exceeds 10,000 feet without using exaggerated tapering shape like Mount Fuji.
The factor of elevators is considered a main reason that stop skyscrapers go higher, the experts say the maximum travel length of elevators is no more than 2000 feet by current technology, if use the newly invented carbon fiber cable, this length could still hardly surpass 1000m.
In fact a 3000m tower itself is a city, people do not need to leave the building frequently everyday, hence this is not a problem, in a few decades we could see a 3000m ultra-tall skyscraper built.

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