How did the Twin Towers not tip over

Were they designed that way?

All buildings are designed that way.

To make a skyscraper tip, you'd have to gouge a huge chunk out of one side of the building. This didn't happen to the WTC. Instead, the planes put a big hole through the middle, which left most of the supports holding the building up in place.

Eventually, the fires weakened those supports to the point that they couldn't hold up the upper floors. When those supports buckled, the upper floors fell downward. From there, gravity just did its thing. The force of the upper floors falling was enough to collapse each floor below it because the supports are designed to hold the building stationary or wobbling slightly. Add a bunch of downward force to that weight and the supports way as well not even be there.

This is the cause of the reported sound of explosions as the buildings fell. Each floor was crushed in turn, forcing the air on those floors to be blasted out the side of the building.

As mentioned in one of the answers, high rise buildings have very deep foundations (the part that runs into the ground). When the engineers design buildings they also look at the shear (lateral) resistance of these buildings. It is specially important in high rise buildings because of wind and earthquakes. A wind of 50mph at 300ft above ground is more dangerous than a wind of 150mph at ground level.

So, there are mainly three things that provide lateral (side-way) stability to buildings. Columns, Shear Walls, and Core. In the case of TWIN TOWERS the core of the building (which then ran deep into the ground) provided enough stiffness for the buildings not to tip over.

Yes, Each building was designed to have about 3 feet of sway in any one direction. Each of the four outer columns were thin and hollow...think of the towers as a squared donut with a whole in the middle. The outer wall was connected to an inner wall by the floor, which made the 4 out columns shorter...So instead of having 4 very long, long columns, there were essentially hundreds of shorter columns because of the way the floor was connected to each outer wall...this strengthened the structure immensely.