I’m going to rule out the Aztecs right away because though they built some really awesome pyramids, that’s pretty much about all they did. I believe that they did not reach the mathematical and astronomical advances of the Mayas. Some even say that the pyramids were built by the Olmecs who were there previously.
Why build a pyramid? Note pyramids are the first huge buildings ever built by primitive folks. Start piling rocks, dirt, sand, whatever on a flat surface. After a while, what you do you have? Wa-la! A pyramid. It’s the most simple, basic and reasonable architechtural building. Building those straight up and down things like those 9-11 towers is way harder, plus they have much more tendency to fall over.
So to me it’s down to the Mayas and the Incas. I want to say Incas, with their amazing cities, incredible irrigation systems, and advanced crop breeding. Further, the Incas made it out of the Stone Age 4000 years ago, when they started making stuff of gold, silver and copper and the Mayas stayed in the Stone Age through their entire classic period that ended in 900 or so.
The Mayas are an incredible tale of an extremely advanced Stone Age society! They almost don’t make sense and seem like something out of science fiction. All that great stuff they made was built with stone tools! Why didn’t they work metals? If there were any nearby, maybe they would have, but the nearest iron deposits were 1,500 miles away.
By the time the Spaniards showed up in 1500’s though, the Mayas were out of the Stone Age and were making stuff, most ornaments, out of gold, silver and copper (It’s not true that the Mayan civilization completely collapsed – this is myth).
Some argue that to be really out of the Stone Age, you need to make tools of metal, but I think that’s a bit silly. If you’re working metal at all, you’re beyond Stone Age. Stone Age is rocks, sticks and bone. That’s your working material. True, gold and silver are fairly easy to smelt, but so what? Working metal remains a stupendous cultural advance for any group anywhere, and many never got there. Think the Aborigines.
So it’s down to the Incas and the Mayas. The Mayas had great achievements in mathematics, written language and astronomy and built some pretty amazing structures. They also built roads of stone through the rainforest, yet they had no beasts of burden nor wheels to have the animals pull behind them.
So why the fancy roads? No one knows. Although they never got the wheel, the Mayas did make children’s toys with wheels on them. One could argue, so why not take the next step and make a wagon? A wagon is not much use without beasts of burden to pull it. Humans don’t really want to pull wagons. That’s donkey, camel, cow, and elephant work.
The Incas were the only one to come out of the Stone Age, and they did so long ago. Once again though, they mostly made gaudy stuff and not tools with the metals. In all their achievements, they never figured out how to write down anything about what they were doing, leaving their whole civilization a gigantic question mark. They did not seem to achieve as much in the way of mathematical or astronomical advances as the Maya did.
Roll the dice and see what comes up? Heads it’s Mayas. I take the Mayas barely over the Incas.