## 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.