Alexa F. Volpe and Juan B. Gutierrez
Alexa F. Volpe - Student,
Juan B. Gutierrez - Graduate Student,
Florida State University, Tallahasse, Florida
"The polytope shadow boxes we created for the exhibition were done to explore an exciting mathematical concept while demonstrating the important relationship between mathematics and the arts. The project visually captures and describes the shadows of a four-dimensional polytope (a pentachoron) and an infinite-dimensional polytope. Though our sensory experience limits us to a three-dimensional world, the polytope shadow boxes seek to offer a creative and mathematically profound look into the possibility of reflected designs of objects in an infinite number of dimensions. While the project remains faithful to its mathematic intentions, it also follows the artistic principles of form, content, aesthetic, and craftsmanship. For example, the lightboxes were an ideal way to describe shadow from multiple sides. The glass conveyed the idea of transparency, yet its hand-frosting suggests a subtle opacity referencing the mysteriousness of the infinite-dimensional object within. Craftsmanship was a top priority and the project was designed with an eye for detail. The shadow boxes were made of a solid bamboo base and assembled without the use of nails. The two lightboxes are a set showing how radically the change of shadow can occur as the number of degrees of freedom increase."
“Polytope Shadow Box 1 ”
March 2008, Hand-frosted glass panels, Solid Bamboo base, 7.5 x 7.5 x 10.5
Despite living in a three-dimensional world, we know an object can exist in multiple dimensions. Thus it is possible to conceive of objects in higher dimensions whose shadows are observed in three dimensions; for instance, the shadow of a regular object in four dimensions, a polychoron, can be represented by a three-dimensional object, which in turn can be represented as a shadow in a surface. Escalating this concept ad infinitum, we can imagine an object of infinite dimensions with reduced representations in lower dimensions. It has been proved by Juan B.Gutierrez that the projection of an object of infinite dimensions into lower dimensions does not have a unique infinite series of moments, and therefore can result in arbitrary shapes. In general, any object with a large number of degrees of freedom can project a large number of shapes in lower dimensions.
This box is an example of a described 4-d polytope (a pentachoron) see: http://en.wikipedia.org/wiki/List_of_regular_polytopes
“ Polytope Shadow Box 2 ”
March 2008, Hand-frosted glass panels, Solid Bamboo base, 7.5 x 7.5 x 10.5
This box is an example of an object in an arbitrary/infinite number of dimensions and its hypothetical shadow array.