Is there a name or a reference for these aperiodic rhomboidal tilings? Fill space with unit cubes and then remove all cubes that are not completely within a given half space. An isometric view of the remaining cubes will look like the following image.  My question is whether these tilings have a name, and whether they are discussed anywhere in the literature. I've seen discussions of this in relation to 'voxelizing' planes (a generalization of Bresenham's Algorithm), but not in relation to generating or analyzing tesselations, even though I think the construction must be well known. (Interactive version of the above picture is at <

These are commonly called _stepped planes_ , _digital planes_ or _stepped surfaces_ (if one considers the cubical faces intersecting a surface rather than a hyperplane). Their one-dimensional counterparts are called _Sturmian sequences_.

They are closely related to _cut-and-project tilings_ , also called _model sets_ , for which Penrose tilings are one particular example (they come from taking particular planar slices in 5 dimensions, rather then 3):

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There is a huge amount of literature on many aspects of their study, from combinatorics and number theory to topological dynamics, ergodic theory and fractal geometry.

Perhaps a good place to start would be a paper of Berthé and Fernique.