我們知道阿基米德多面體和卡塔蘭多面體互為對偶，當他們被嵌在一起時，會有一個球切過所有的邊，此時球和多面體的每個面會相交出一個圓。我們討論這些圓會有的性質，並以這些圓為原型做出動態圓，使的這些動態圓能滿足以下條件：
1.任何時間下，動態圓都會在一個球面上，而且他們會在某個時間點和其原型重疊。
2.動態圓的其中兩個圓在任何時間下都和其原型重疊並且自轉。
3.動態圓會保持著其原型所擁有的性質，及相切性、正交性及組合性。
我們的作圖過程都用Cabri 3D來處理。

We want to discuss the incircles of the 11 dual polyhedrons, and construct an animation which satisfy the following conditions:
1. All dynamic circles are always on the sphere, and they overlap the incircles at a time.
2. Two of the circles remain invariant in appearance while each rotating about its axis.
3. All dynamic patterns preserve mutual tangency, orthogonality and contact graph.
We want to see and discuss the animation. We use Cabri 3D to do the patterns.

[1] https://en.wikipedia.org/wiki/Catalan_solid
[2] https://en.wikipedia.org/wiki/Circle_packing_theorem
[2] http://georgehart.com/virtual-polyhedra/vp.html
[3] http://mathworld.wolfram.com/CatalanSolid.html
[4] 阮賢彧 (2010),　“卡塔蘭多面體的動態幾何作圖法”.　中央大學數學系碩士論文
[5] 謝智潁 (2007),　“卡塔蘭多面體的綜合作圖”.　清華大學數學系碩士論文
[6] 顏琇婷 (2008),　“對偶多面體”.　清華大學數學系碩士論文