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X-Seed 4000 is the largest building ever fully envisioned and designed. It is 4 km tall (13,123 feet), with a 6km base. X-Seed 4000 was designed by Taisei Construction Corporation, a 130+ years old construction firm based in Japan. The X-Seed 4000 design is meant to be an innovative combination of urban living with a natural environment. Large sections inside the building are open and allow the admittance of light. They could be lined with soil, which would allow the creation of parks and other interior landscapes.
The gargantuan X-Seed 4000 would need to be constructed on water to ensure its stability. The building was designed with 800 floors and is covered in solar paneling for power. Somewhat similar to the Pentagon, X-Seed 4000 would be a self-contained city with 500,000 - 1,000,000 inhabitants, equipped with numerous fast elevators for purposes of firefighting, police and medical dispatches, etc. Regions of the interior could be quarantined with fireproof paneling to ensure that large fires, if they do break out, are confined to relatively small areas and do not threaten the large-scale integrity of the structure.
If built today, X-Seed 4000 would cost about $300-900 billion US Dollars (USD), about half of Japan's nominal GDP. This sounds like a lot of money, but consider that the International Space Station will be estimated to cost $130 billion USD when completed in 2016, and the budget of the US military in 2007 was $530 billion USD. In any case, future advances in robotics and mega-scale construction could bring the cost down enough to actually make it feasible.
Even though it's nothing but a design, X-Seed 4000 is interesting both because of the scope of the vision behind it and the reactions people give when first informed of it. To some, it's the ultimate urban nightmare, despite the fact that the design was originally made to combine together an urban structure with nature, similar to an arcology. The way it packs so many people in a relatively small space by building upwards radically decreases the environmental footprint relative to conventional suburban sprawl. Whether or not structures like X-Seed 4000 are the wave of the future remains to be seen, but for now, they're interesting speculation.