Geodesics at the MonkeyHouse

Dave Anderson

original papers written and published between January 1998 and August 1999

Knot a Braid of Links #149
Cool Math Site of the Week
March 2, 1999

Geodesics have become something of a fixation for me since early 1998. The idea of projecting a mesh of triangles onto a curved surface has inspired me to do mathematics, create models, and build structures. Many others also seem to be interested: some have enjoyed my work, some have contacted me for specific advice, and a special few have even pushed me into new directions and expanded my life.

Here are my papers. I have written them to being useful and technically correct, while not impenetrable for the average reader. If you have any comments on them, please contact me directly!

Calculating the Chords of a Geodesic Patch - The formulation of the triangular mesh and the projection of an icosahedral geodesic patch is described. An analytic solution of the icosahedron is developed.

Calculating the Fastening Angles between Geodesic Subfaces - The fastening angle, pairs of which make up the dihedral angles between subfaces, is developed. These are important to prefabicated geodesic construction, allowing panels to be created to precise specifications and bolted together on site.

A Technique for Building Inexpensive Geodesic Models - Numbers and drawings are nice, but models are better. Here I struggle with the method of creating prototypes - after a few false starts, I rediscover a wonderful technique.

A Simple Bolted PVC Geodesic Strut Design - Math is solved. Prototypes are made. Now the rubber meets the road. I design and build a 2-frequency half-spherical icosahedral geodesic frame which may someday become a greenhouse.

Planning a Bigger Dome, Part I - With a success under my belt, I have a pipe dream. A big dome - 10 frequency - 50 feet in diameter - made out of PVC struts each less than 4 feet long. Oh rapture. Oh bliss.

Planning a Bigger Dome, Part II - High frequency dome shells distribute distrubing forces throughout their structure - but locally applied radial forces tend to make them dimple. I develop the warped octet truss - a thing of beauty and elegance that allows much larger geodesic structures to be built. The truss is applied under the geodesic surface to strengthen and support it.

Planning a Bigger Dome, Part III - With the math done and the warped octet truss in place, I dive into VRML and do the next best thing to buying PVC and constructing the dome: I build a virtual patch out of cyberstuff in virtual reality.

Tetrahedral Geodesics - Icosahedral shells are not the only game in town. Tetrahedrons are much simpler, but the subfaces are not so uniformally sized as with the icosahedron's subfaces - the cental faces are bigger since they have to bulge so much more. The tetrahedron is analytically solved and internal warped octet truss bracing is applied.

Octahedral Geodesics - To round things out, I solve the octahedral geodesic. The octahedron has the exciting property of being right-angled, which potentially makes it available to be wedded to traditional rectangular-box architecture. The octahedron is analytically solved and internal warped octet truss bracing is applied.

Externally Octet-Trussed Geodesic Patches - Since the warped octet truss bracing is equally effective externally as internally, a sign is reversed and external bracing is produced. This preserves the geodesic inside, important in some applications.

Visualizing the Bulge in Geodesic Projections - Since a patch is deformed when projected on a sphere, it bulges in the center. The tetrahedral bulge is greater than the octahedral, which is greater than the icosahedral. VRML renderings illustrate this property.

Geodesic Constructions - People had been asking for pictures, so I delivered. Photos of some models and constructions are presented, including two connected patches of a 6-frequency icosahedral geodesic with internal octet truss bracing.

Constructing Geodesic Patch Surfaces, Part I - My direction now turned to producing skins for my geodesic skeletons. I explore some of the properties of coverings, including waste minimization, symmetry, and cut-template growth.

Constructing Geodesic Patch Surfaces, Part II - The topic of waste when creating geodesic coverings rears its formidable head, and a derivation of flattened geodesic boundaries is made. Coming up with the results was good fun and an nice exercise in three dimensional analytic geometry.

Constructing Geodesic Patch Surfaces, Part III - Finally some finished patterns for the first 5 frequency patches of the tetrahedron, octahedron, and icosahedron are produced. The results can be printed, cut, folded, and assembled into geodesics.

Calculating Geodesic Radial Vertex Angles - Figuring out the angles needed below the normal at a vertex is needed in many dome projects and construction techniques in order to create hubs.

Superprojected, Semiprojected, and Subprojected Geodesics - By parameterizing the projection of the geodesic surface, puckered and bulbous domes may be created. VRML and paper model renderings of the geodesics are provided.

Connecting Superprojected and Subprojected Geodesic Patches - A new parameterization keeps the edges of different bulging and dimpled patches the same - they can be joined in the same geodesic seamlessly. VRML and paper model renderings of the geodesics are provided.

Blending Functions and Projecting Geodesic Patches - An exploration into the blending functions which are used to make connecting superprojected and subprojected patches. Blending functions can take many forms and create interesting shapes.

Flat-edged Projected Geodesic Patches - A new kind of geodesic structure is introduced, the use of a geodesic patch with a blending function and projection parameter as total enclosing structure.

Full Sphere Geodesic Surfaces - Turning the patch into a full geodesic surface; the icosahedron, octahedron, and tetrahedral structures are developed and blending functions are applied.