chamfering the convex regular

chamfering the cube

We may obtain a chamfered cube, with six squares and twelve equilateral hexagonal faces [1], by uniformly expanding the faces of the cube. Its vertices are also expanded in the direction of the cube’s diagonals. The following images show three steps of this expansion. In the last one, the hexagons have the same edge length as the squares because the distance between the initial and the expanded vertices (the red line segment) equals half the cube’s diagonal (the dashed line segment), as described in [2]:

the faces and vertices of the cube expand until the equilateral version of the chamfered cube

The chamfered cube is also obtainable by truncating the cube with planes parallel to its edges so that hexagonal faces replace the cube’s edges [3]. The image below shows three such plans in red and the intersection of each pair.

Below is a video of a Grasshopper definition I created to prepare a model of the chamfered cube for 3D printing, with a multipipe connecting its edges with an internal structure. The video shows the expansion of the faces and vertices of the cube until its equilateral version.

Some of the models I printed with an internal structure (shown here) broke during printing and were somewhat confusing, so I printed the models below. On the left, a network of pipes replaced the edges, and there is no internal structure. By controlling the pipes’ radius parametrically, I turned them slightly thicker to prevent those between the larger faces from breaking. The model on the right is a simple solid body which, in my opinion, is not very interesting.

[left: circumdiameter 110 mm | pipe radius: 2.0 mm | Total printing time: 11h20; right: circumdiameter 72 mm | Total printing time: 06h40]
  1. Weisstein, Eric W. “Chamfered Cube.” From MathWorld – A Wolfram Web Resource. https://mathworld.wolfram.com/ChamferedCube.html
  2. Viana, V. 2023. Materializing Daniele Barbaro’s Creativity with 3D Printing. Judy Holdener, Eve Torrence, Chamberlain Fong, and Katherine Seaton (Eds.) Proceedings of Bridges 2023: Mathematics, Art, Music, Architecture, Culture. Phoenix, Arizona: Tessellations Publishing. 313-320. https://archive.bridgesmathart.org/2023/bridges2023-313.pdf
  3. Weisstein, Eric W. “Chamfered Polyhedron.” From MathWorld – A Wolfram Web Resource. https://mathworld.wolfram.com/ChamferedPolyhedron.html