It seems that both the straight dope and ask-a-physicist links describe the effects as something similar to having eight giant mountains on Earth whose summits

It seems that both the straight dope and ask-a-physicist links describe the effects as something similar to having eight giant mountains on Earth whose summits make up the points of a cube. This led me to wonder, what if we consider the case where the Earth is exactly the same (geologically) as it is now, but in the future we put up eight giant towers whose tops describe those summit points? Assuming a supermaterial has been discovered whose properties allow for an extremely light structure with enormous compression strength relative to its weight, like aerogel ( http://bashinginminds.com/2010/01/23/playing-with-nasas-solid-smoke/ http://bashinginminds.com/2010/01/23/playing-with-nasas-soli... ) but even aerogel-ier. What would that be like? Is it then too far-fetched to imagine that some freak process of nature or other might conceivably allow for the creation of a bizarre planet with an immensely dense spherical core and a lighter mantle and crust that take on the shape of a cube externally? Nature is not averse to giving us cubes, after all: http://www.gemstoneslist.com/pyrite.html http://www.gemstoneslist.com/pyrite.html