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  • Cross sections of the curve in two direction create straight lines,
  • 2 other cuts show parabolic curves,
  • two other plane orientations cut out hyperbolae.

The close-packed crystal space matrix of 12 rays, 4 planes, delineating Tetrahedron and octahedron edges, is known in classic chemistry as the face-centered-cubic or cubic close-packed crystal structure.  In fact, this atomic packing matrix positions atoms at the vertices and center of the cubeoctahedron, Buckminster Fuller’s beloved Tensegrity structure, or vector equilibrium, defining the planes of cubic, tetrahedral and octahedral space.
The best way to stiffen a cube of sticks is with diagonals on the 6 faces which connect to make a simpler form, the tetrahedron.
Remove 2 of the 6 tetrahedral edges, leaving 2 pair of 90 degree oriented edges, which can be dipped into soap solution to form a beautifully balanced minimal surface, the hyperbolic paraboloid saddle curves of balanced curvature at every point.

Hyperbolic Paraboloids in Cubeoctahedron corner of diamond-refractive cube

The 90 degree hyperbolic paraboloid of a regular tetrahedron is most symmetric within the close-packed crystal matrix, which can be expanded to create a curved space tunnel system.

Tent_4 tetrahedral HP-1
Octahedral 4Hypar roof 10ft tall-poles
Hyperbolic Paraboloids in Cubeoctahedron

These drawings show how the regular tetrahedron fits into the cube, the hyperbolic paraboloid (hypar) generated from the tetrahedron, and the curved space created (in this case, 6 smoothly-connected tetrahedral hypars within the boundaries of the cubeoctahedron)

tetrahedral saddle tent-2

Tetrahedral hyperbolic paraboloids can be connected in other configurations to create pleasing forms such as this modular tent design.

Hyperbolic Paraboloid Ferrocement model-1

90 degree hyperbolic paraboloids of 4 regular octahedra
curvature integrals zero

Hyperbolic Paraboloids of octahedron-composite
icosa-hypar pod on greens

Small retreat pod formed with 4 hypar saddle curves of icosahedron.  Floor area = 120sq.ft., under building code concerns in most locals.


Hyperbolic Paraboloid cube-octahedron weavings

Hypar saddle surfaces created from intersection of cube and octahedron frameworks.  Would make excellent ferrocement roof!

cube-octahedron hypar house-3
cube-octahedron hypar house-2
icosa-hypar pod front
Hyperbolic Paraboloid Ferrocement Icosahedron-front

Pleasing saddle-curve roof structure over kitchen, built using space-age mud and wattle (fiberglass, burlap, acrylic, and cement on a wood frame) at a Rwandan orphanage

Successful technology transfer to Rwanda by Birambye includes bamboo wrap joints developed at

This hypar house is one of many renditions and orientations possible.

Floor area for house pictured is 1430 sq.ft.

cube-octahedron hypar house-4
cube-octahedron hypar house-5-R