David Hall
I enjoy woodworking and geometric constructions, so naturally, the
recent discovery of the "Chord Factor Construction" technique for
generating affine polygons on conic sections which form quadric
surfaces led me to build some interesting and beautiful models. The
two models I have submitted are based on the same shape - an affine
hexagon faceted paraboloid, yet the media for the two models are very
different. Developing the math behind the shape was as fun as the
physical modeling of it.
I am pleased to submit these models in diverse media from simple to
complex.
I am honored to co-author the paper with Scott Vorthmann and David
Richter regarding the math behind the new geometric construction
technique that is featured in these models.
This paraboloid is made of standard zometool components. Only the vertices of the paraboloid and hexagonal facets are present (no edges). Struts in each of the three directions are the same length and color. They form the units of X, Y and Z axes of a transformed 3D coordinate system bounded by the paraboloid. The model shows that the paraboloid surface can be generated using only an integer point set. This surprising discovery is what initially prompted me to develop this model. This model is based on the "Chord Factor Construction" method for generating conic sections and quadric surfaces as described in the paper that I co-authored with Scott Vorthmann and David Richter for this year's conference.
This affine hexagon faceted paraboloid was made from a forked hickory tree branch. It was made by hand with just a rotary table on an old school 3-axis milling machine (no CNC). Some of the bark between the forked branches was retained at the base and back of the model to contrast the natural features of the tree from which it was carved with the sharp lines of the hexagonal facets.
The shape was inspired by similar models that I created with vZome. It is based on the "Chord Factor Construction" method for generating conic sections and quadric surfaces. This model combines the geometry behind the new technique with some artistic tools and skills of a disappearing era.