There is no single country that planes completely avoid flying over. However, flight paths are often altered due to several factors, resulting in planes not flying directly over certain areas. These factors include:
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High Altitude Terrain: Flights may avoid mountainous regions like the Tibetan Plateau due to the increased risk of turbulence and potential engine issues at high altitudes. As noted in a YouTube video discussing Tibet, "[planes] need to quickly descend to 10,000 ft which is almost impossible in Tibet because of its height." The challenging terrain presents significant aviation risks.
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Political Restrictions: Some countries have restricted airspace, prohibiting or limiting overflights for political or security reasons. Historical examples include the former USSR, where flights often rerouted to avoid overflights. Current examples include no-fly zones declared over conflict areas. The FAA, for instance, has imposed restrictions on flights over Haiti below 10,000 feet.
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Safety Concerns: Areas with high levels of air traffic or other safety concerns may see flight paths modified to ensure safety. This isn't a specific country, but rather a consideration across many regions. Certain airspace may be classified as 'Do Not Fly' zones (like Haiti's designated airspace, according to SafeAirspace.net) due to instability.
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Religious Significance: Specific locations with significant religious meaning may have restrictions on overflights out of respect. For example, it's known that passenger aircraft are generally not allowed to fly over Mecca and the Holy Kaaba in Saudi Arabia.
It's crucial to understand that "not flying over" doesn't always mean a complete avoidance of a country's airspace. Flight routes are complex and are constantly adjusted based on a number of factors, resulting in varying degrees of airspace utilization across different countries.