Sunday, August 15, 2010

Cubic trisections

A few days ago I explored many novel ways to trisect a cube into three identical parts. Could they be adapted into a puzzle? To find out I printed three samples, each using different parameters.

Each cube is made of 3 identical parts, which slide together (or apart) with a novel twisting action.

Here are the individual pieces of one trisected cube.

Because the model included some spots as thin as 0.3 mm I chose the build orientation carefully to exploit the SD300's ability to build thin layers in the Z axis.

Light reflects and refracts spectacularly inside each model. I don't regard it as a finished puzzle, but the SD300 encouraged me to try out an idea I might have ignored.


  1. Very cool! I like the effects from the transparency. Why don't you regard it as a finished puzzle?

    George Hart has done some similar cube dissections by a helical cut.

  2. I hadn't seen Prof. Hart's cubes before; thanks for calling them to my attention! He has an excelling illustration of them at

    My trisections would undoubtedly pose interesting puzzles as they are now, but I just haven't achieved the shape I'm after.

    Prof. Hart aptly described his dissection as "a cube sliced by the propeller of an airplane flying through it." I'm trying to achieve a simimlar effect, but with a "3-bladed propeller whose blades have been swept back." The resulting pieces go together only in one direction, unlike Prof. Hart's symmetric pieces.

    BTW: Another closely-related puzzle is based on a trisection by Robert Ried, modeled by Oskar van Deventer at

  3. What software do you use to create your models?

    It looks like the axis of your helix goes corner to opposite corner, is that correct?
    In that two piece model, George Hart aligns his axis from middle of edge to middle of opposite edge.

    He also showed us a large tetrahedron about 8 inches tall that he had sliced into two identical pieces in a similar manner, and a 3-piece cube. All this was at George Miller's house a year ago.

  4. Yes, the cut runs from corner-to-corner. Your tipoff led me to a multitude of other discussions, particularly Martin Gardner's description of a similar method given to him by Joe E. Morse that exploits the intrinsic 3-fold symmetry in the corner-to-corner orientation.

    In hindsight, George Hart's choice of bisecting from edge-to-edge makes intuitive sense: that orientation has intrinsic 2-fold symmetry, and hence it bisects elegantly. (Perhaps I should try my oddball cuts on such a bisection, too.)

    I gave away all the trisection models pictured, but I plan to build more. And I will certainly make another post about it, since I've found a lot of other discussions worth sharing!