Despite the scientific importance of semiconductor quantum dots (QDs), little is known about their structure other than the bulk form. Here, we show that one can join segments of tetravalent semiconductors into a seamless icosahedron. Calculations for Si show that pristine icosahedral QDs are more stable than bulklike ones for diameter d<5 nm. Hydrogenated icosahedral quantum dots (IQDs) could also be stable and more atomiclike with larger level spacing and fivefold orbital degeneracy due to the I-h symmetry not possible in the bulk. Experimental feasibility toward synthesizing the IQDs in Si and diamond is also discussed.