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.

enter image description here

This is in general an aperiodic tiling of the plane by rhombi that all have the same shape. (Of course for certain half spaces it is periodic.) 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 https://codepen.io/brainjam/full/QrZgLP/)


1 Answer 1


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):

enter image description here

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.

  • $\begingroup$ Funny, I've seen literature on cutting sequences on hyperbolic tessellations (e.g. perso.univ-rennes1.fr/serge.cantat/Documents/…), but the terminology for these Euclidean cases had totally eluded me. Thanks for your very useful answer. $\endgroup$
    – brainjam
    May 18, 2018 at 2:57

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