# 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