4-piece dissection of Juel's pyramid to a triangular prism
Abstract. We provide 3D illustrations for a dissection of Juel's pyramid to parallelepiped.
More simple dissection to rectangular triangular prism exists.
Faculty of Electrical Engineering, University of Ljubljana
Trzaska 25, 1000 Ljubljana, Slovenia
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.
 V. G. Boltjanskii, Tretja problema Hilberta, Nauka, Moskva 1977.
 G. N. Frederickson, Dissections: Plane & Fancy, Cambridge U. Press, 1997.
 Martin Kraus' Live3D applet
 H. Meschowski, Grundlagen Der Euklidischen Geometrie (Croatian edition), Skolska Knjiga, Zagreb 1978