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I assume everyone here is familiar with the Hoberman Sphere. Fun toy, by the way. My question is this : The sphere is an icosododecahedron. Suffice it to say, it has a set of isometries which define it. Now, when the sphere is expanded, it has clearly undergone a transformation, yet none of the symmetries have been broken. Also yet, no translation, rotation or reflection has been performed. Has the H. sphere undergone a global transformation ? Poincare ?

Forgive me if this question seems elementary. I see much more sophisticated questions being discussed here. Just thought I'd ask. I pulled this one on a crystallographer friend of mine and I want to make sure I am Right. I want to get revenge for his " what's the symmetry of a tennis ball " jab.

Selene Routley
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MarkLeFont
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    I'm not sure the question " what's the symmetry of a tennis ball " is a jab. It's a really valid question, albeit perhaps not worded fully precisely. It's "symmetries" are (assuming our notion of invariance is isometry) the rotations of $SO(3)$. If our notion of invariance is relaxed to conformalness, then the Möbius transformations are its symmetries. Your notion of invariance is probably going to be preservation of the icosahedral shape and dilation is quotiented away. So your group is the group $A_5 \times Z_2 \times (\mathbb{R}^+,,*)$. The direct product of a discrete point group ... – Selene Routley Mar 15 '18 at 22:51
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    ...$A_5 \times Z_2$ of icosahedral symmetries and the one dimensional Lie group $(\mathbb{R}^+, ,*)$ of uniform dilations. – Selene Routley Mar 15 '18 at 22:51

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I don't know the proper technical term for it, but it seems to me that the expansion and contraction of the Hoberman Sphere are processes where different parts of the sphere expand (or contract) at different rates depending on their position. Thus, the transformation could perhaps be describe as an anisotropic enlargement or anisotropic reduction. If something is the same in all directions it is isotropic. If it is not it is anisotropic.

Mathematically we might say that if the origin is the centre of the sphere and we work in spherical polar coordinates that on transformation

$$\phi_{new} = \phi_{old}$$ $$\theta_{new} = \theta_{old}$$ $$r_{new} = f(r_{old}, \phi_{old}, \theta_{old})r_{old} $$

where $f(r_{old}, \phi_{old}, \theta_{old})$ gives an expansion/reduction factor as a function of $\phi$, $\theta$ and $r$.

tom
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