The synthesis, structure, and bonding of BaTl4 are described [C2/m, Z=4, a=12.408(3), b=5.351(1), c=10.383(2) angstrom, beta=116.00(3)degrees]. Pairs of edge-sharing Tl pentagons are condensed to generate a network of pentagonal biprisms along b that encapsulate Ba atoms. Alternating levels of prisms along c afford six more bifunctional Tl atoms about the waists of the biprisms, giving Ba a coordination number of 16. Each Tl atom is bonded to five to seven other Tl atoms and to three to five Ba atoms. There is also strong evidence that Hg substitutes preferentially in the shared edges of the Tl biprisms in BaHg0.80Tl3.20 to generate more strongly bound Hg-2 dimers. Cations that are too small relative to the dimensions of the surrounding polyanionic network make this BaTl4 structure (and for SrIn4 and perhaps EuIn4 as well) one stable alternative to tetragonal BaAl4-type structures in which cations are bound in larger hexagon-faced nets, as for BaIn4 and SrGa4. Characteristic condensation and augmentation of cation-centered prismatic units is common among many relatively cation- and electron-poor, polar derivatives of Zintl phases gain stability. At the other extreme, the large family of Frank-Kasper phases in which the elements exhibit larger numbers of bonded neighbors are sometimes referred to as orbitally rich.