  In 1935, Pauling estimated the residual entropy of water ice with remarkable
accuracy by considering the degeneracy of the ice rule {\it solely at the
vertex level}. Indeed, his estimate works well for both the three-dimensional
pyrochlore lattice and the two-dimensional six-vertex model, solved by Lieb in
1967. The case of honeycomb artificial spin ice is similar: its pseudo-ice
rule, like the ice rule in Pauling and Lieb's systems, simply extends a
degeneracy which is already present in the vertices to the global ground state.
The anisotropy of the magnetic interaction limits the design of inherently
degenerate vertices in artificial spin ice, and the honeycomb is the only
degenerate array produced so far. In this paper we show how to engineer
artificial spin ice in a virtually infinite variety of degenerate geometries
built out of non-degenerate vertices. In this new class of vertex models, the
residual entropy follows not from a freedom of choice at the vertex level, but
from the nontrivial relative arrangement of the vertices themselves. In such
arrays, loops exist along which not all of the vertices can be chosen in their
lowest energy configuration: these loops are therefore vertex-frustrated since
they contain unhappy vertices. Residual entropy emerges in these lattices as
configurational freedom in allocating the unhappy vertices of the ground state.
These new geometries will finally allow for the fabrication of many novel
extensively degenerate artificial spin ice.
