bergerud
07-21-2012, 11:13 AM
I posted this in techniques because it is an interesting way, using miters, to glue up a solid. The first solid I made is called the Reuleaux tetrahedron. Look out, here comes a math lesson!! It is formed by taking a tetrahedron and replacing the four flat sides with curved patches of spheres whose centers lie on the opposite vertices. If the Reuleaux tetrahedron is placed between two parallel planes, it smoothly rolls like a sphere.
The solids I am making are called solids of constant diameter. The Reuleaux tetrahedron, however, has a slight flaw. The distance between two edges is slightly greater than the distance from the vertices to the sphere faces. In 1911, a mathematician named Meissner, figured a way to round three of the six edges and fix the problem. The problem of how to symmetrically round all of the edges has been an open problem for over 100 years. I recently found the formula for the symmetric solution (following a suggestion it be an envelope of small spheres along the edge) and so had to carve it.
A jig which would hold a tetrahedron like shape as it is carved boggled my mind so I decided to carve it in parts and then glue them together. The main reason for this post is to show how well this actually worked. You would imagine that these would be miters from hell! Not so. I left little nibs on the vertices which I used to hook on rubber bands during the glue up. The result was a nearly perfect set of joints. After the rubber bands came off, I just had to sand off the nibs to get nice sharp points.
This technique could be used to carve and glue up quite large solids. The accuracy of the carved miters is the key.
The solids I am making are called solids of constant diameter. The Reuleaux tetrahedron, however, has a slight flaw. The distance between two edges is slightly greater than the distance from the vertices to the sphere faces. In 1911, a mathematician named Meissner, figured a way to round three of the six edges and fix the problem. The problem of how to symmetrically round all of the edges has been an open problem for over 100 years. I recently found the formula for the symmetric solution (following a suggestion it be an envelope of small spheres along the edge) and so had to carve it.
A jig which would hold a tetrahedron like shape as it is carved boggled my mind so I decided to carve it in parts and then glue them together. The main reason for this post is to show how well this actually worked. You would imagine that these would be miters from hell! Not so. I left little nibs on the vertices which I used to hook on rubber bands during the glue up. The result was a nearly perfect set of joints. After the rubber bands came off, I just had to sand off the nibs to get nice sharp points.
This technique could be used to carve and glue up quite large solids. The accuracy of the carved miters is the key.