The Aztec, Inca, and Mayan represent, collectively, three of the most advanced architectural, structural, and mathematical civilizations in Mesoamerica. To the chagrin of many early civilizations across the ocean, they made these remarkable advances without the use of the wheel.
The Mayan civilization developed a highly accurate calendric system of geared wheels despite the fact that their knowledge was carefully guarded: only taught to the elite (Week 6, lecture 2). This calendar reflects not only that they had an advanced knowledge of the seasons, but also of the cosmos. In fact, they were able to accurately track the other planets along with solar and lunar cycles (Week 6, lecture 2).
The Inca were masters of city architecture. They were able to construct terraces for farming out of the mountain side, and even built the famous city of Machu Picchu close to cloud level (Ghosts of Machu Picchu). The Inca were also able to use their knowledge of architecture to make fountains throughout the city of Machu Picchu from a preexisting mountain stream. These fountains not only gave fresh water to the people, but also filtered down to the terraces, to water the crops of the city (Ghosts of Machu Picchu). What is even more impressive is the fact that these cities, along with their terraces, were able to survive the torrential downpours that could even cause mudslides. This is because the Inca possessed a great knowledge of drainage systems. When a terrace was to be built, for example, the Inca would first use granite waste rock, then sandy gravel, and finally a layer of topsoil to insure water would go into the earth rather than of over and across it (Ghosts of Machu Picchu).
The Aztecs are known as fierce warriors who offered human sacrifices to their gods (Week 6, lecture 2). However, something not many realize at first is that their main city, Tenochtitlan, was built on an island in the middle of a lake (Week 6, lecture 2). Since they had enemies surrounding them, the Aztec civilization was forced to become creative in their agricultural techniques. From this innovation they created structures known as chinampas. Often referred to as “floating gardens,” chinampas were actually artificial islands created by staking out and fencing in shallow areas of the lake bed (http://www.sscnet.ucla.edu/ioa/backdirt/Fallwinter00/farming.html). Once this was complete, the rectangular area was then layered with lake sediment and decaying vegetation, eventually rising above the level of the lake (http://www.sscnet.ucla.edu/ioa/backdirt/Fallwinter00/farming.html). This not only gave the Aztecs more land, but created a very rich soil from which as many as four different crops could be grown each year (http://www.sscnet.ucla.edu/ioa/backdirt/Fallwinter00/farming.html).
Even without the wheel, these three civilizations were able to reign supreme in Mesoamerica due to their mathematical, architectural, and structural achievements.