The geodesic dome was invented in the late 1940's

The 'big ball' at Epcot in Orlando Florida, is a Geodesic sphere. The old dome-like playground equipment is based on the same structure as a geodesic dome. I've posted a couple of links about geodesic domes with some pictures.

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No they did not. The Romans made domes either with unreinforced concrete of in bricks. Geodesic domes are a modern invention.

A geodesic dome is a spherical shape generally created using small triangles strategically places to make a round shape. One example of a geodesic dome is the Epcot Center in Disney World. They also have play grounds for children that are half spheres made from the triangles mentioned above that would technically qualify as a geodesic dome.