The asscher diamond cut first saw production in 1902, and its popularity rose in the 1920’s. Up until recently, you would only find these diamonds in antique shops, but since then the shape has resurfaced with cut modifications, which give off more brilliance than the traditional cut.
The modern asscher cut holds many similarities to a square emerald cut, and usually has larger step facets, with higher crown and a smaller table. The combination of all this is what gives more brilliance than the emerald cut. If an asscher is well cut, you will see concentric squares when you look down through the table, which is because of the proper positioning of the pavilion facets under it.
While the asscher cut has cropped corners like the emerald cut, it is primarily square, and so, the whole thing can end up looking like an octagon. If you mount this in a four prong setting, you get a unique shape for the diamond, as well as a square silhouette.