Tuesday, April 6, 2010

Arches at right angles

A cylinder made of triangular bricks can be assembled with a helical edge. “Helicity” refers to translation vs. rotation: thinking of a slinky, if you stretch out the coil, you decrease the helicity. The helicity of a masonry edge can be manipulated by the shape of the triangle.

If the translation is proportional to the radius of the cylinder, and the rotation is equal to 90 degrees; then this proportioned triangle can be assembled with a helical edge that allows arches to intersect at 90 degrees.

This proportioned triangle (in this case, with independent key) looks like this:

Arches made with these blocks can be turned in, like this:

Arches can also be turned out, like this:

Note the gap between the arches where they intersect. This gap can be filled with a larger spherical section or dome. If the cylinders are considered radius 1, then the larger sphere has a radius of 1.5.

The sphere filling this section does not fill it precisely. This is the ancient problem of “squaring a circle,” a fascinating and amusing story of dedicated mathematicians pursuing a Sisyphean task. It is impossible to square a circle, and this was not proven until 1882.

This way of putting together two barrel vaults made from triangular blocks, so that they intersect at a right angle into a larger dome (also made of triangular block) creates extensive additional design flexibility. There are many ways to configure this system.