We argue that the prevalence of continuous flat floor surfaces in architecture is comprehensible but fallacious, and that this chimaera can be overcome through studying and employing the sculptural potential of algebraic surfaces which suggest spatial possibilities that enrich designersí vocabulary enormously. We continue, deepen and extend research the basics and early results of which were presented at the last two eCAADe conferences in Istanbul and Z¸rich. We present and discuss a university-based experimental design and research project that demonstrates how Algebraic Surfaces can drastically amplify the so far only tentative exploration of the possibilities of non-fl at fl oor surfaces in Architecture.