More fun than a hypercube of monkeys by T.R.I.X.I.E.

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3D model of henryseg. The models were repaired and checked for printability.

This sculpture was inspired by a question of Vi Hart. As far as we know, this is the...Show more first sculpture (in fact, physical object) with the quaternion group as its symmetry group. The quaternion group {1,i,j,k,-1,-i,-j,-k} is not a subgroup of the symmetries of 3D space, but it is very naturally a subgroup of the symmetries of 4D space. The monkey was designed in a 3D cube, viewed as one of the eight cells of a hypercube. The quaternion group moves the monkey to the other seven cells. Radial projection moves the monkeys onto the 3-sphere, the unit sphere in 4D space, then stereographic projection moves the monkeys to 3D space. The distortion in the sizes of the monkeys comes only from this last step - otherwise all eight monkeys are identical. For more details, see Vi's and my paper, or Evelyn Lamb's blog post at Scientific American.
This is joint work with Will Segerman.
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