Mekhontsev Wedge

Visualization

@
$n=Wedge
$dim=3
a=-1
b=[0,-1,0,1,0,0,0,0,1]
c=[1,0,0,0,0,-1,0,1,0]
h0=1
h1=[1,1,1]
h2=b*a*c^2*[-4,-4,0]
h3=b*c*b^3*a*[0,-4,-4]
h4=b^3*a*c^3*[0,0,-4]
h5=c^2*a*[-4,0,0]
h6=b*[0,-4,0]
h7=b^3*c*[-4,0,-4]
A=2^-1*(h0|h1|h2|h3|h4|h5|h6|h7)*A

Overview

The Mekhontsev Wedge is a convex polyhedron discovered by Dmitry Mekhontsev using IFStile, first published in the Tiling Facebook group in 2026 and subsequently featured by Ed Pegg on Wolfram Community. It is a non-trivial self-similar convex polyhedron: a rep-8-tile in which eight half-scale congruent copies tile the original shape exactly.

The polyhedron has:

• V = 6 vertices
• F = 5 faces (two triangles and three quadrilaterals)
• E = 9 edges

satisfying Euler’s formula $V - E + F = 6 - 9 + 5 = 2$.

It is topologically equivalent to a triangular prism. A cube is a trivial convex rep-8-tile (8 half-scale cubes fill a cube), but the Wedge is the first example with a non-cuboid shape.

Geometry

The six vertices, in a coordinate system where the full shape spans $[0,2] \times [0,2] \times [0,1]$, are:

The shape has a right-angle corner at the origin and tapers toward $v_3 = (1,1,1)$. As a convex polyhedron its boundary consists of five flat polygonal faces, so the boundary has Hausdorff dimension exactly 2.

Self-Similarity

In the AIFS encoding the scalar $a = -1$ is negation, and

are 90° rotations around the $z$- and $x$-axes respectively. The eight maps $h_0, \ldots, h_7$ compose these rotations with translations to orient and place the eight half-scale sub-wedges. The contraction factor is $\tfrac{1}{2}$, so the Hausdorff dimension of the (solid) attractor is:

References

• Mekhontsev, D. (2026). A non-trivial self-similar convex polyhedron. Tiling Facebook Group.
• Pegg, E. (2026). The Mekhontsev wedge: A 3D rep-tile. Wolfram Community.

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