Friday, March 9, 2012

Geodesic frequencies and masonry domes


Geodesic domes are made in various “frequencies”, or “orders.”  As discussed earlier, geodesics are typically made of regular polyhedra.  The polygons which comprise these polyhedra can be broken down into their constituent basic or unit triangular shapes.  These unit or base triangles can be further subdivided into smaller triangles.  These smaller triangles can be further subdivided into still smaller triangles, etc., ad infinitum.  Each progressive division is a “frequency.”

This sort of triangular subdivision is illustrated in the Sierpinski fractal.

Higher frequency geodesics allow a large dome to be made from a relatively small unit shape.

Each time the size of a dome (radius) is doubled, the surface area is increased by a factor of 4.  This means that a dome twice as big will need 4 times as many blocks.  This reflects Galileo’s Square Cube Law.  Thus doubling the size of a dome uses 4 times as many bricks (surface area is squared) and increases the volume by a factor of 8 (the volume is cubed).  Using 4 times as many bricks creates 8 times as much volume; this is very efficient.

There are some difficulties with higher frequency geodesics.  This is due to the difference between chord length and arc length.  Arc length (S, below) refers to the measurement of the curved surface of a sphere.  Chord length (l, below) refers to the straight line distance between two points on the surface of a sphere.  Arc length is longer than chord length as measured between two points on a sphere.  If a triangle projected onto a spherical surface is broken down into smaller and smaller triangles, then a number of different sized triangles will result.  They are not all the same; there is a maddeningly large number of different dimensions for various triangles.  Historically this has been a challenge for higher order geodesics, often resulting in weakened, leaky or poorly assembled domes. 

Mortar (or gaskets) fixes this problem easily.  The differences in size are accounted for by using more or less mortar (or gasket) between block.  It is a simple matter of aligning block within their geodesic pattern.

Higher frequency domes result in a change in proportions of block.  Because the block are assembling into a larger structure, the wall thickness also gets larger.  This occurs proportionally: if a structure is twice as big, its walls will be twice as thick.  The result is that unit triangular shapes look less like a thin plate or shell, and more like a thick block.  The thickness is around equal to edge length.  Higher frequency blocks are more “blocky.”  This is advantageous for ceramics or concrete to bear their compressive load.  A block bears compressive load better than a thin plate.

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