About this 3D model

This model is featured in figure 3.31 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com). This sculpture was inspired by a question of [Vi Hart](http://www.youtube.com/user/Vihart). As far as we know, this is the 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](http://archive.bridgesmathart.org/2014/bridges2014-143.html), or Evelyn Lamb's [blog post](http://blogs.scientificamerican.com/roots-of-unity/2014/05/19/a-hypercube-of-monkeys-quaternion-group/) at Scientific American. This is joint work with [Will Segerman](http://www.willsegerman.com). Also available from [Shapeways](http://shpws.me/t2sb). Also check out the interactive animated online version at [monkeys.hypernom.com](http://monkeys.hypernom.com). (Use the WASD keys, arrow keys, numbers 1-6.)