# Dissection of small stellated dodecahedron and great stellated dodecahedron to rhombic triacontahedron and hexecontahedron

Izidor Hafner

Faculty of Electrical Engineering, University of Ljubljana

Trzaska 25, 1000 Ljubljana, Slovenia

e-mail: izidor.hafner@fe.uni-lj.si

We observe that the combination of the small stellated dodecahedron and of the great stellated dodecahedron
can be dissected to combination of the rhombic triacontahedron and of the hexecontahedron.
In [2] we have shown that the combination of the icosahedron, the dodecahedron and
the icosidodecahedron can be dissected to combination of the triacontahedron and the hexecontahedron.
In this way we answered the question posed in [1, pg. 231].
If we consider the small stellated dodecahedron as built from the dodecahedron
and of a layer of 12 pentagonal pyramids, and consider the great stellated dodecahedron
as built from the icosahedron and of a layer of 20 triangular pyramids, then
these pyramids should form the icosidodecahedron (or the small icosihemidodecahedron and
the small dodecahemidodecahedron).
But we could observe the surplus of 60 (non regular) tetrahedra of small stellated
dodecahedron over the triacontahedron and deficit of these solids of
the great stellated dodecahedron relative to the hexecontahedron.
The mentioned tetrahedra are 1/6 (in volume) of prolate golden rhombohedron.

References

[1] J. H. Conway, C. Radin, and L. Sadun, On angles whose squared trigonometric functions are
rational, Discrete & Computational Geometry, 22 (1999), pages 321-332.

[2] I. Hafner, Live3D Animation to Solution of Conway-Radin-Sadun problem, Visual Mathematics, Volume 9, No. 1, 2007 ,1