What Is The Minimum Number Of Planar Reflections Required To Transform A Given 3D Object With Icosahedral Symmetry Into Its Mirror Image, Assuming The Object Is Centered At The Origin And The Reflections Are Performed About Planes That Pass Through The Origin And Are Perpendicular To One Of The Object's 12 Vertices, 20 Faces, Or 30 Edges?

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To determine the minimum number of planar reflections required to transform a 3D object with icosahedral symmetry into its mirror image, we need to consider the symmetries and reflections involved.

  1. Understanding the Problem: The object has icosahedral symmetry, meaning it has 12 vertices, 20 faces, and 30 edges. Reflections must be performed about planes passing through the origin and perpendicular to one of these features.

  2. Symmetry Group: The icosahedral group includes both rotations and reflections. The full symmetry group is isomorphic to A5×Z2{A_5 \times \mathbb{Z}_2}, with 120 elements. The mirror image is an improper rotation.

  3. Reflections and Inversion: Reflecting across a plane inverts the object. Inversion, which maps each point (x,y,z){(x, y, z)} to (−x,−y,−z){(-x, -y, -z)}, is a different transformation. In 3D, inversion can be achieved by three reflections across mutually perpendicular planes.

  4. Icosahedral Reflection Planes: The icosahedron has reflection planes, but they are not orthogonal. Reflecting across planes perpendicular to vertices, faces, or edges does not necessarily result in orthogonal planes.

  5. Composition of Reflections: In 3D, the composition of three reflections can result in inversion. Since the mirror image is an improper rotation, it can be achieved by composing three reflections.

  6. Conclusion: Given the need to achieve an improper rotation (mirror image) and the structure of the icosahedral group, the minimal number of reflections required is three.

Thus, the minimum number of planar reflections required is 3{\boxed{3}}.