Jul 4, 2017 | By David
A mathematician at Oklahoma State University has been expanding 3D printing technology itself into a whole new dimension. He published a paper on the different kinds of symmetry of physical objects, explaining the ‘quaternion’ group as a particular symmetry group. He goes on to explain how 3D printed objects can be used to demonstrate the symmetrical possibilities of so-called ‘four-dimensional’ objects.
The paper in question is entitled ‘The Quaternion Group as Symmetry Group’, and was co-authored by Henry Segerman from Oklahoma State’s Department of Mathematics and Vi Hart from the Communication Designs Group at San Francisco’s SAP Labs. It stresses the potential utility of being able to physically represent certain types of symmetry, in particular the quaternion group, as a way to enable a better conceptualization of what is referred to as the fourth dimension.
In an accompanying video, Segerman shows a 3D printed spherical structure, which when lit from the inside casts a square grid shadow on the ground. This shows a form of symmetry across the plane between the edges of the sphere and the ground that the sphere is standing on. This complex symmetry would otherwise be very difficult if not impossible to conceive of, let alone to visualize. The use of 3D printing to easily and cheaply create such an intricate geometry means that we are able to picture this dimension in a physical form.
A similar but more complex 3D printed object is based on an example described in the paper itself. It describes a selection of ‘monkey blocks’, cubes with pictures of monkeys on them, being assembled to form what is referred to as a ‘hypercube’.
The blocks themselves have no symmetry due to the asymmetrical patterns of the monkeys faces and body parts. However, through their being assembled in a specific way, with each monkeys hand on the foot and tail of another monkey, and each foot on the head of another, they can be conceived as having a different kind of rotational symmetry. The axes for rotation can be thought of as lines that go through the connections between monkey’s hands, feet, tails and heads, and in this way the whole structure is identical when rotated. It has symmetry, just one that functions in a dimension other than the 3 that we are accustomed to picturing and thinking of physical objects existing in.
This may be all very abstract and difficult to wrap your head around, but luckily there are some trippy 3D computer visualizations of the structure that make the transformations a little easier to understand. The resulting 3D printed monkey sculpture itself can be used to get a firmer grasp on this perplexing and potentially mind-blowing extra-dimensional geometry.
Posted in 3D Printing Application
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