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There are various known 3D Rep-Tiles and Irreptiles. Almost all of them are based on polycubes OR 2D reptiles. What are the 3D rep-tiles and irreptiles not based on polycubes and 2D reptiles?

One infinite set uses $n$ bricks of size $(r^0,r^1,r^2)$ where $r=\sqrt[3]{n}$. The Delian Brick uses two bricks based on the Delian constant.

Delian Brick

The first to use the Delian brick may have been Thickfun and Dale Walton, who expanded this into the Fifth Chair puzzle, a 4-irreptile.

Fifth chair

Of the five space-filling tetrahedra, at least two are 8-reptiles.

tetrahedron reptiles

I've put together code for all of these. Are there any other 3D shapes which can be self-divided into smaller similar shapes, where the underlying shape is not a 2D reptile or a polycube?

Ed Pegg
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    There is a general technique for constructing self-similar tiles in $\mathbb R^n$ using an expansive $n\times n$ integer matrix described in this paper and implemented in Mathematica in the last print edition of The Mathematica Journal. While all the examples in those papers are in $\mathbb R^2$, it certainly works in dimension $n$. – Mark McClure Jul 03 '18 at 17:08
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    In the ArXiv preprint entitled Three-Dimensional Rep-Tiles (2021) by Ryan Blair, Zoe Marley, and Ilianna Richards, all 3D rep-tiles up to homeomorphism are classified. Unfortunately, their approach also relies on polycubes. – Max Lonysa Muller Feb 23 '24 at 20:08

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