本文研究了五個中空正四面體、正立方體、正八面體、正十二面體和正二十面體之間的幾何關係，從而創建了兩個動畫：
1.正十二面體–六十面體–菱形三十面體–六十面體–正二十面體動態轉換
2.俄羅斯套娃是俄羅斯特產木製玩具，一般由多個空心木娃娃一個套一個組成。本文透過中空正多面體的性質，設計成一組相互套合的中空正多面體，其關係由外而內為正十二面體、正立方體、正四面體、正八面體、正二十面體。
本文在學術上得應用有數學實驗、三維動態幾何和尺規作圖；在應用方面上有數學玩具之設計、科展題材之發展、數學競賽題目研發和3D印表機建模實作等。

This work studies the geometric relationship among the five hollow tetrahedron, cube, octahedron, dodecahedron and icosahedron leading to the creation of two animations:
1. The Dodecahedron-Hexecontahedron-Rhombic Triacontahedron-Hexecontahedron-Icosahedron Transition
2. Formation of Nesting Hollow Platonic Solid
These animations have theoretical applications in mathematical experiments, 3D dynamic geometry, synthetic construction and practical applications in designing of mathematical toys, science project, mathematical competition and model construction with 3D printer.

Contents
Abstract.........................................................2
0. Software used in the construction and the animation..........5
1.Platonic solid.................................................5
1.1 Platonic solid...............................................5
1.2 Relationship among Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron..................................................5
2.Cube 5-Compound, Tetrahedron 5-Compound and Octahedron 5- Compound........................................................17
2.1 Cube 5-Compound, Tetrahedron 5-Compound and Octahedron 5- Compound constructed based on Dodecahedron......................17
2.2 Rhombic Triacontahedron as convex hull of Dodecahedron and Icosahedron ....................................................33
3.Nesting Hollow Platonic Solid.................................36
3.1 Construction of Cube 5-Compound, Tetrahedron 5-Compound and Octahedron 5-Compound based on the same Dodecahedron............36
3.2 Animation of Formation of Nesting Hollow Platonic Solid.....43
References......................................................48

References
1.Cundy, H.Martyn, Rollett, A.P. Mathematical models. Oxford University, USA; 1961
2.Wells, David G. The penguin dictionary of curious and interesting geometry. Penguin, USA; 1991
3.Cabri 3D, a dynamic mathematic software in space. Cabrilog, 2004 https://www.cabri.com/en/enterprise/cabri-3d/
4.Cube 5-Compound http://mathworld.wolfram.com/Cube5-Compound.html
5.Tetrahedron 5-Compound http://mathworld.wolfram.com/Tetrahedron5-Compound.html
6.Octahedron 5-Compound http://mathworld.wolfram.com/Octahedron5-Compound.html