Polyhedral nets are flat compositions of polygons capable of being cut and folded along prescribed lines and joined at their edges to form polyhedral solids. A polyhedral net shows both an outline of the polyhedral and fold lines for the polyhedral.
It is generally agreed that Underweysung der Messung by Albrecht Dürer (1471-1528) presents the earliest known examples of polyhedral nets. This was clearly an extension of the renaissance idea that polyhedra are models worthy of an artist's attention, and the two-dimensional representation of polyhedra was seen as one of the main problems of perspective geometry in Dürer's time. Further, Dürer produced an engraving, Melencolia I, in 1514 that includes an uncommon polyhedron which has since been the topic of serious discussion, mathematical interest, and various reconstruction attempts.
Erwin Panofsky, a German-American art historian and essayist, described Dürer's polyhedron as a “truncated rhomboid”. According to Panofsky, the polyhedron is simply a cube or rhombohedron which has been truncated at the upper vertex and possibly the lower vertex. If the polyhedron is not truncated at the lower vertex, the polyhedron penetrates the earth. Most researchers, however, have concluded that Dürer's polyhedron is a six-sided solid with each side shaped as a rhombus, and that the solid has been truncated so that it can be circumscribed in a sphere. Nevertheless, disagreements remain over the polyhedron and any possible meaning that it may convey, and a method of creating a polyhedral net corresponding to Dürer's polyhedron has heretofore been absent.
Later, polyhedral nets were commonly used in cartography to resolve the earth's surface into a polyhedron. Examples of this can be found in patents to J. M. Boorman (U.S. Pat. No. 185,889), G. W. Bacon (U.S. Pat. No. 1,050,596), B. J. S. Cahill (U.S. Pat. No. 1,081,207), and Buckminster Fuller (U.S. Pat. No. 2,393,676).
Constructing polyhedral nets requires precise computing, measuring, drawing, cutting, and constructing. As such, polyhedral nets may be used in teaching lessons related to math (especially algebra and geometry) and art. Polyhedra constructed from polyhedral nets are also known to have interesting strength characteristics (they can often carry loads many times larger than their own weight,) and may also be used in teaching lessons related to physics.
Through history, polyhedra have been closely associated with the world of art. The peak of this relationship was certainly in the Renaissance. For some Renaissance artists, polyhedra simply provided challenging models to demonstrate their mastery of perspective. For others, polyhedra were symbolic of deep religious or philosophical truths. For example, Plato's association in the Timaeus between the Platonic solids and the elements of fire, earth, air, and water (and the universe) was of great import in the Renaissance. This was tied to the mastery of geometry necessary for perspective, and suggested a mathematical foundation for rationalizing artistry and understanding sight, just as renaissance science explored mathematical and visual foundations for understanding the physical world, astronomy, and anatomy. For other artists, polyhedra simply provide inspiration and a storehouse of forms with various symmetries from which to draw on.
Also, geometry as art seems the most natural approach for teaching this subject, just as at the times of Dürer when he believed that:                . . . . Since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art . . . . Course in the Art of Measurement        
Norman Shapiro, M. A., who is a teacher and an artist with more than thirty years of teaching experience, with art works in many collections: The Museum of Modern Art, the archives of several universities (Idaho State at Boise, Indiana State at Bloomington), and in many private collections, says: The best way for children to learn about geometry is to take up pencil and straight edge (a ruler will do) and design geometrically on paper. Children need to feel about geometry the way the Egyptians did. Children learn best when they too see geometry as a means to an end. Children are artists at heart. The rationale of making something geometry into art doesn't need explanation. To children, beauty comes before logic and theories; therefore, Geometry Through Art can provide the means.
The relationship between geometry and art appears in different ways: proportion, perspective, symmetry, abstraction and symbolic subject matter. Mathematics is not just about formulas and logic, but also about patterns, symmetry, structure, shape and beauty. Courses are given to study connections between mathematics and art and architecture, study topics like tilings, polyhedra and perspective.
Also, it might help in popularizing Mathematics, in raising public awareness of mathematics effectively. It will help scientists identify shapes in nature, for example: the complex arrangements of macromolecules in the virus shell are minute marvels of molecular architecture. Specific requirements of each type of virus have resulted in a fascinating apparent diversity of organization and geometrical design.
It will help the art scholars be able to decipher the message hidden in Melencolia I, and understand the meaning that Dürer meant to convey. Art Scholars have wondered whether “the possibility arises that the odd-shaped block and its magic square are not just two iconographical elements, amongst many others in the engraving, but perhaps even the Raison D'être of the whole work”.
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