3. The uniform truncated dodecahedron is constructed by cutting the vertices off

Question

3. The uniform truncated dodecahedron is constructed by cutting the vertices off a dodecahedron in such a way that: . All of the faces created by this process are regular polygons. . The cuts do not touch each other. (This distinguishes it from a rectified polyhedron - see slide 11/14 of Lecture 11). (a) How many faces does the uniform truncated dodecahedron have? Explain your answer. 2 marks] (b) Do all the faces of the uniform truncated dodecahedron have the same number of edges? List the possible numbers of edges that a face of the uniform truncated dodecahedron could have. 3 marks]

Explanation / Answer

(a). Uniform Truncated Dodecahedron has 12 Regular Decagonal Faces, 20 Regular Traingular Faces.

(b). No,not all faces have same number of edges because we have decagonal as well as triangular surfaces. Decagon has 10 edges while Triangle has 3 edges.

Calculating the number of edges by number of faces -

12(10) + 20(3) = 180.

But, we should remember that it is a closed and symmetrical figure, so the edges will be shared which will make the actual number of edges half of the theoritical value we calculated, making it -

180/2 = 90.

So, a Uniform Truncated Dodecahedron has 90 Edges, of which 60 are countable at Decagonal Faces, and 30 at Triangular Faces.

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