If a pentagonal pyramid (Johnson solid 2) is cut off the icosahedron , we get the gyroelongated pentagonal pyramid (Johnson solid 2: J2). If we cut off another pentagonal pyramid, we get either a pentagonal antiprism or a metabidinimished icosahedron (J62). If we cut three pentagonal pyramids, we get a tridiminished icosahedron (J63).Click on the picture to activate Live3D graphics .
It is obvious, that the pentagonal antiprism and the metabidinimished icosahedron have the same volume. But they have the same complement relative to the rhombic triacontahedron, namely the complement consists of 2 pentagonal cups and 10 triangular pyramids. This means they are equidecomposable .
To make a paper model use the following nets:
 V.G. Boltjanskii, Tretja problema Hilberta, Nauka, Moskva 1977.
 I. Hafner, Live3D Animation to Solution of Conway-Radin-Sadun problem, Visual Mathematics, Volume 9, No. 1, 2007 ,1
 Martin Kraus' Live3D applet http://www.vis.uni-stuttgart.de/~kraus/index.html