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I'm sure you know the baby game about sorting shapes by putting them in holes:

enter image description here

I'm wondering if such a game could exist in 4 dimensions? I imagine the shapes would have 4 dimensions and the holes 3 dimensions...

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Essentially what you need is a cross-section of a 4D polytope, representing a hole through which it can pass. These cross-sections can be quite complex. For example, the great icosahedral 120-cell, a regular star 4-polytope,


         
          (Orthogonal projection. Wikipedia image created by Robert Webb's Stella software.)
has this cross-section, "a 3D cross-section through that 4D polytope, taken 18% of the way through the model, along one of the 60 icosahedral symmetry axes":
          Gofix2
          (Image from Robert Webb's software3d.com.)
I don't know if the full polytope could pass through this cross-section, but at least from this you can see the beautiful intricacy of the cross-sections.

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    $\begingroup$ This is not a cross-section of the 120 cell since it is not convex and a 120 cell is convex. This is a cross-section of the great icosahedral 120 cell which is different from the 120 cell and is not convex. $\endgroup$ Sep 16, 2015 at 0:18
  • $\begingroup$ @KristalCantwell: Thank you for the correction! $\endgroup$ Sep 16, 2015 at 0:24
  • $\begingroup$ I suspect that if there are any 4-dimensional babies out there then they are playing with rather simpler shapes than these. $\endgroup$ Sep 16, 2015 at 2:28
  • $\begingroup$ @GerryMyerson: not if that baby wants to get into the right school! $\endgroup$ Sep 16, 2015 at 4:41
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    $\begingroup$ To me, a simpler answer that captures all the features of the 3D toy is to take 4D blocks that are right prisms with any 3D solid as base. This would avoid the problem of having a cross-section of an object that won't allow the object to pass through. $\endgroup$
    – Ken Fan
    Sep 16, 2015 at 5:19

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