What special considerations do super tall and narrow "pencil skyscrapers" require in design and construction, in order to remain stable?

The challenge with this sort of building is to figure out where to hide the lateral load resisting system. These are the elements of the structure that resist wind and earthquake loads. This can be accomplished with steel cross bracing, perhaps the most famous example of which is the John Hancock building in Chicago:

However, for a number of years, it has been much more economical to use concrete in the form of shear walls, which are thick concrete walls with steel reinforcing bars. Most modern high rise buildings use a series of concrete shear walls to resist all lateral loads. These shear walls are collectively known as the core walls and are often placed near the center of the structure, surrounding elevators and stairwells. The core walls are responsible for resisting virtually all lateral loads in the structure as well supporting a portion of the gravity (vertical) loads of the structure. All remaining columns, whether made of steel or concrete, contribute only minimally to the lateral system of the building, and instead function primarily as gravity load carrying members. 

The biggest architectural advantage of placing the shear walls near the center of the structure is that this allows for unrestricted views, since you have no structural walls at the exterior of the building. The core walls also provide necessary structure for the support and protection of elevators and stairwells (which must be isolated and protected from fires for safety reasons). On the other hand, placing the walls further from the geometric center of the structure gives more structural efficiency by increasing the moment resisting arm between the segments of wall. For this reason, the sizing of the core is a bit of a give and take process. A wall can, for example, at times be increased in thickness to compensate for its less than ideal location. 

An interesting offshoot of this type of construction is that the core walls of a typical high rise can be built in advance of the rest of the gravity structure, while the reverse would be not be true, since the gravity columns, without a core, would be susceptible to collapse from lateral loading.

Here is an image of the concept. It is an in construction image of the Aqua Tower in Chicago. At the center of the image you can see the concrete core walls rising about 3 or 4 stories above the rest of the structure. 

So, how does this all relate to 111 W 57th street and similar buildings? Let us take a look a the typical floor plate of 111 W 57th street:
Where are the concrete core walls? They are the dark grey elements that line the entirety of the longer sides of the building and also wrap around the stairs and elevators. So, to provide enough lateral support for this very slender structure, the designers have effectively placed their entire floor plan inside the structural core, there really is no separate wrap around gravity system to follow after the core is built*. Now, why doesn't every building do this? As previously noted, there is a major trade off, you'll notice this structure can only accommodate tiny, dare I say, prison cell like, windows along the two long faces of the building. You'll notice the rooms with these windows are thoughtfully laid out to be bathrooms and closets, where the impact is minimal. Presumably the structure is oriented on its site so that the favorable site lines for views are at the top and bottom of the above plan, otherwise you'd really be missing out.

Construction wise, I suspect the 111 W 57th street building may actually be built quite rapidly, because the contractor may be able to use a slip form that climbs smoothly upward. This is common practice for the core walls of conventional buildings, and is visible in the above image of Aqua tower, but is not feasible for the main floor plate which is too large, and lacks continuous walls to attach to.   

Now, to further enhance the lateral performance of the building an additional system is being employed at 111 W 57th street. This is a system of tuned massed dampers, placed near the top of the structure. These are heavy weights or liquids that are used to dampen vibrations in the structure by being moved in opposition to the resonant frequencies of the structure via mechanical springs or gravity (in the case of liquid mass dampers). This sort of system is useful in reducing lateral movements in super tall, slender structure, and has been successfully used in buildings such as Taipei 101, which was at one time, the tallest building in the world.  

A few other final comments are in order:

The structure will be 1428 ft tall, but only occupied up to 1134ft. Afterwards the plan area shrinks severely, as it narrows to an almost knife edge at the peak. This is a beautiful aesthetic choice and also provides a nice reduction in lateral loads by reducing the mass and area of the top 300ft of the structure, which would otherwise contribute a disproportionate amount of the lateral load on the structure and base. 

* There are actually four distinct grey columns on the plan, two at the top, and two at the bottom, which may only be gravity elements, though a discussion with the design engineer would be needed to confirm.
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