Klein bottle, Geom. a one-sided figure consisting of a tapered tube the narrow end of which is bent back, run through the side, and flared to join the wide end, thereby allowing any two points on the figure to be connected by an unbroken line.

The idea that 3-D projections of 4-D objects are analogous to 2-D projections of 3-D objects is an old one, but the ability to represent and explore these 3-D images in detail was impossible before the advent of computers and 3-D modeling programs. These 3-D objects are called “hyperobjects.” A three dimensional Klein bottle can be considered a hyperobject, because there is an apparent self-inter- section in 3-D that would not exist in 4-D.

These objects in and of themselves are of little interest. However, when created and rendered from within using reflective materials, they produce quite remarkable images, containing fractals, recursions, and chaotic structures. The rendered images are highly conceptual in that they are a result of the rendering process, which could be set in motion, but not directly controlled.

Construction of the Klein bottle is usually described as the process of deforming an ordinary bottle by bending the neck around, putting it through the side and connecting it to an opening in the bottom. The relationship of the Klein bottle to tori is well known, but lends itself more to geometric description of the surface than to actual construction of the object. The need for a simple construction led me to discover that the bottom of the bottle could be made from half of a torus. This quickly led to the realization that the other half could be rearranged to form the top. A second torus twisted to form a question mark creates the neck. Except for the hole where they intersect they are both complete. The simplified Klein Bottle is, far as I have been able to determine, an original construction.

Having discovered and constructed the simplified Klein bottle, the next step is exploring the inside. This voyage of discovery led to a strange manuscript revealing the true origin of the simplified Klein bottle.