## week 3

Sometimes it is difficult to distinguish a piece of artwork from math and math from the a piece of artwork because at times art and math are embedded into one another. This picture or piece of work by Albert Durer about the methods of perspectives which made an important contribution to the polyhedra literature shows a mixture of art and math. Durer’s *Underweysung der Messung* contains a very interesting discussion of perspective and other techniques and typifies the renaissance idea that polyhedra are models worthy of an artist’s attention. More importantly, this book presents the earliest known examples of polyhedral *nets*, i.e., polyheda unfolded to lie flat for printing. The image at right is Durer’s drawing of the net of an icon while the net is correct, his techniques of perspective were still under development, and it is interesting to observe that the projection at the upper right has a number of inaccuracies.

Here is another of Durer’s nets. This is intended as a truncation of a truncated cube. While most of his nets are quite accurate, this contains a significant error, which you will notice if you study it for a few moments. Eight of the vertices (those at the top left and top right of the four central dodecagons) show 360 degrees worth of angles around them, and so can not fold as intended. This should serve as a reminder that the idea of a net is not as simple and obvious as one might suppose. The beauty of these artistic works is that they tend to establish an interconnection between all the sciences. For instance, the truncation of snub yields the chiral which is important chemistry terminology, especially in optics. The idea that math, art, and science are all interconnected hold in the extent that we can look at a piece and be able to see branches of all three disciplines being displayed. A theorem can be traced from this fact that literary works of abstract concepts all belong together and the ones of concrete concepts also are reached from the same beginning. I believe this is important for us as students and researchers to know because when we do interpret mathematics in art, it is sometimes evasive. But when we take into consideration that most of the concepts are in some way connected, we will be more likely reach a more concrete and accurate conclusion about what it is we are researching and interpreting.

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