Henry Segerman takes a brisk stroll through the world of four-dimensional objects with some really cool 3D-printed sculptures, like this sphere that projects a square grid when lit:
Segerman coauthored an interesting paper on the topic with Vi Hart on physical objects in which the elements are combine in a manner to maintain rotational symmetry.
We briefly review the distinction between abstract groups and symmetry groups of objects, and discuss the question of which groups have appeared as the symmetry groups of physical objects. To our knowledge, the quaternion group (a beautiful group with eight elements) has not appeared in this fashion. We describe the quaternion group, both formally and intuitively, and give our strategy for representing the quaternion group as the symmetry group of a physical sculpture.
Rotational symmetry in groups of objects is a fascinating topic. A few years ago I whipped up a few Venn diagrams with rotational symmetry to demonstrate some of the fascinating math behing this branch of set theory, including this one showing all possible combinations of the seven dwarfs:
Segerman's extradimensional example of this is a monkey sculpture with rotational symmetry that may or may not be even cooler to ponder while high:
• 4D MONKEY DUST (YouTube / Numberphile)