More fun than a hypercube of monkeys

prusaprinters

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