4-piece dissection of Juel's pyramid to a triangular prism



Izidor Hafner
Faculty of Electrical Engineering, University of Ljubljana
Trzaska 25, 1000 Ljubljana, Slovenia
e-mail: izidor.hafner@fe.uni-lj.si

Abstract. We provide 3D illustrations for a dissection of Juel's pyramid to parallelepiped. More simple dissection to rectangular triangular prism exists.
Juel's pyramid has as base the base of a cube and as the apex the centre of the cube. In [4, pgs. 211-214] a dissection of the pyramid to parallelepiped is described. The construction consists of cutting the pyramid into three layers. The top layer consists of a single Juel's pyramid with the edges that are 1/3 of the original edge length. The second layer consists of 5 such pyramids and 4 tetrahedra D. Two tetrahedra D. can be obtained from one small Juel's pyramid by vertical cuts. The base layer is a truncated pyramid. This is a 14-piece dissection of Juel's pyramid to parallelepiped.

But there is a simpler dissection using only 4 pieces. We could cut the Juel's pyramid to 2 pieces and reassemble them to Hill (or Sydler's Hill) tetrahedron [1, pg. 92, 2, pg. 234]. There is a 4-piece dissection of the solid to a triangular prism obtained by Sydler [2, pg. 234].
But there is an even nicer hinged dissection of the tetrahedron to a different triangular prism, using construction found independently by P. Schobi and A. Hanegraaf [2, pg. 235]. This is a 3-piece dissection. This dissection yields a 4-piece dissection of Juel's pyramid to the prism.

Make paper models using the following nets.

 

References


[1] V. G. Boltjanskii, Tretja problema Hilberta, Nauka, Moskva 1977.
[2] G. N. Frederickson, Dissections: Plane & Fancy, Cambridge U. Press, 1997.
[3] Martin Kraus' Live3D applet
[4] H. Meschowski, Grundlagen Der Euklidischen Geometrie (Croatian edition), Skolska Knjiga, Zagreb 1978