## Tuesday, March 11, 2014

### Design Flexibility of triangular block

In this blog I have written much about the “design flexibility” inherent to a triangular block masonry design system.  What do I really mean by this “design flexibility”?

Let’s start with the basics: a sphere can be made.  This is done by approximating the pattern generated by a regular polyhedron, and filling the polygons (which comprise the polyhedron) with triangles.  For example, five triangles can form a pentagon, 6 triangles can form a hexagon; hexagons and pentagons can combine to form a truncated icosahedron (like a soccer ball).

Half of a sphere can be used to construct a dome, or hemisphere.

The spacing between blocks can be manipulated to “stretch” the contour or topography of a hemisphere into a catenary shape, like a catenary dome (thicker and/or thinner mortar joints can achieve this).

Furthermore, triangular blocks can be used to build a cylinder, as discussed here.  A cylinder –known to mathematicians as a right circular cylinder- can be bisected parallel to its axis of rotation, to form a rounded or Roman arch.

Sections of round cylinders can also be used to build arches with more than one center.  This adds extensive additional design flexibility, and gives the architect/designer many more options to choose from when building with triangular masonry.

Further still, the proportions of the triangular block can be chosen such that the helicity of the spiral edge allows round arches to intersect at right angles.  This is important because many people “don’t want to live in a dome.”  Domes are often associated with hippies chasing utopian dreams, faulty buildings left leaky, stinky and smelling of armpits and patchouli.  Many “simply could not live in a round dome, it’s so 1960’s and I’m stuck on this commune and it’s unsanitary and my God I have to take a shower now.”  The perceived failure of a counter-cultural revolution takes its toll on architectural design (sorry modern-day hippies, I don’t mean to offend you; I am talking about public perception).

My point is that most people in western society prefer to be in a square-cornered building with right angles and rectilinear orthogonal design.  It’s not called the wrong angle; it’s called the right angle.  We can accommodate the need for right angles in a building, and do so gracefully and beautifully, around a central hemisphere or dome.

What other things can we provide in terms of “design flexibility”?  How can domes be used to make buildings that are not necessarily round?  The key to this is to use many smaller domes to fill a larger floor plan.  If we consider domes (or spheres, for that matter) and how they can be placed close together, then there are essentially two types of closest packing which are useful for the architect and designer.  There is cubic packing and there is hexagonal packing.  Cubic packing is less efficient than hexagonal packing, as it uses up more free space, or creates a larger interstitial gap between adjacent domes.

The interstitial gap between domes is a curved 3 or 4-sided shape (hexagonal packing creates 3-sided shapes; cubic packing creates 4-sided shapes).  These shapes can be made into a “pedentive” which provides continuity between adjacent domes in a structure.  A pedentive is traditionally how a round domed roof is placed (for example) on a square or hexagonal building; it fills the gap between a round dome and the corners of the building.

If several smaller domes are used to make a roofing system for a large building, they may be arranged as cubic-packed or hexagonally-packed roofing units.  The pedentives between adjacent domes can be placed atop columns or posts, creating a beautiful open space with elegant symmetry and arches describing a high strength, symmetrical and robust roofing arrangement made entirely of triangular block.

This is a photograph of cubic-packed domes with arches and pedentives as designed and built by Guastavino, using his catalan arch method which he employed in the 19th century.  This particular example is at the State Education Building in Albany, New York.  One can see how elegantly smaller domes can be used to assemble into beautiful large buildings.

Summing up, we can make spheres, domes, catenary domes, cylinders, arches, many-centered arches, arches at right angles, and finally use a multiplicity of smaller domes, arches and columns to create a much larger floor plan.  All of these features taken together represent a very broad spectrum of “design flexibility”  provided by triangular masonry block.