This remarkable 3D geometric object tiles space! It is related to a theory of “spacial networks” extensively studied by Michael Burt and a few of his students. The network associated to this object is described in the picture below.

And below you can see the first step in tiling the whole space: How two tiles fit together.

Michael Burt started finding and classifying related objects in his 1967 Master thesis. My academic grandfather Branko Grunbaum (below left) helped him in his early works. The thesis was so impressive that Michael (below, right) was awarded a doctorate for it.

You can find more about it in Michael’s site. The site includes slides for various lectures like the one on UNIFORM NETWORKS, SPONGE SURFACES AND UNIFORM SPONGE POLYHEDRA IN 3-D SPACE.

Below are some more pictures from Michel Burt’s home. In one of them you can see Michael, his wife Tamara, and my friend since university days Yoav Moriah who initiated the visit.

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David Eppstein commented on some well known structures of the kind that Burt is studying: —the diamond cubic (https://en.wikipedia.org/wiki/Diamond_cubic) and Laves graph (https://en.wikipedia.org/wiki/Laves_graph) particularly. The Laves graph is also closely related to the gyroid surface (https://en.wikipedia.org/wiki/Gyroid) but I know less about that.

Let me also mention the 1974 book “Infinite Polyhedra” by A. Wachman, M. Burt, and M. Kleinmann (2nd edition, 2005). Where a lot of infinite polyhedra in space are described and modeled.

Here is a typical page from the book:

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Let me mention that the journal “structural topology” which was devoted to framewrk rigidity and related connections between math and engineering is available on line http://www-iri.upc.es/people/ros/StructuralTopology/