New Geodesic Structures
Geodesic structures are self supporting geometric frameworks usually in
the form of a dome or sphere. The geometric components can be
equilateral triangles, making up a full or partial icosahedron, or
interlocking pentagons and hexagons; a pattern often made into a
football.
Strictly speaking, the term geodesic refers to a 'great circle' on a sphere, such as the equator or a line of longitude on earth. The edges of the faces on a structure align to approximate 'great circles'.
The geodesic structures in this article are arrived at by a process similar to that described in the Complementary Solids topic. Whereas the simpler structures described above have a limited number of geometric components, those based of this process can have an unlimited number of components making them suitable for much larger structures.
Strictly speaking, the term geodesic refers to a 'great circle' on a sphere, such as the equator or a line of longitude on earth. The edges of the faces on a structure align to approximate 'great circles'.
The geodesic structures in this article are arrived at by a process similar to that described in the Complementary Solids topic. Whereas the simpler structures described above have a limited number of geometric components, those based of this process can have an unlimited number of components making them suitable for much larger structures.
The process
Stage 1Take a tetrahedron and imagine filing away, equally, at all six edges.
Stop filing when the shape no longer changes. The result is a cube.
Four of the corners lie at the centres of the original faces of the
tetrahedron.
Stage 2
Take the cube and repeat the filing process on the edges.
Take the cube and repeat the filing process on the edges.
Instead of getting the original solid, as we did when filing the corners, we now get a new, more complex, figure.
This figure has twelve identical faces, each a deformed square, or rhombus.
Some properties of these structures
The process, for creating complementary solids, swapped faces for corners and corners for faces. This process, for creating new geodesic structures, results in the corners and the faces of the original figure becoming corners in the new figure. The edges in the original figure become the faces of the new figure.| Original figure | New figure | |
| Number of corners | Nc | Nc + Nf = Ne + 2 |
| Number of edges | Ne | Nc + Ne + Nf - 2 = 2Ne |
| Number of faces | Nf | Ne |
| Number of spaces | 2 | 2 |
For example, the cube has eight corners, six faces and twelve edges. The figure resulting from the process has fourteen corners, twelve faces and twenty-four edges.
The shape of the faces
The faces, of the resultant figures, are all four-cornered. Two corners correspond to the original corners, at each end of a filed edge, and two corners correspond to centres of the adjacent faces.
Starting with the tetrahedron, the result of first stage, the cube, has six square faces. The figure resulting from the second stage has rhombic faces. The next stage has kite-shaped faces, and from then on it's quadrilaterals, of various proportions, all the way.
Faces meeting at corners
If a face of the original figure has n edges then the corner of the new figure, corresponding to the centre of that face, is the meeting point of n new faces. For example, when the process is performed on a cube, a corner of the new figure, corresponding to a face of the cube, is the meeting point of four faces.
If a corner of the original figure is the meeting point of n edges then the corner of the new figure, corresponding to that corner, is the meeting point of n new faces. For example, when the process is performed on a cube, a corner on the new figure corresponding to a corner of the cube is the meeting point of three faces.
Memories
This means that if the process is repeated on a cube, and on the subsequent figures, any resulting figure will have eight corners where three faces meet and the rest of the corners where four faces meet. The eight corners, where three faces meet, are like memories of the corners of the original cube, or indeed of the four corners and four triangular faces of the tetrahedron from the stage before the cube.
Geodesic structures of unlimited complexity
The process can be repeated until a figure of the desired complexity is achieved.As the complexity increases, the structure tends towards the shape of a sphere. The faces become smaller and smaller compared with the size of the whole structure. The faces, furthest from the corners where other than four faces meet, tend towards the shape of a square. The angles between edges, at the corners where three faces meet, tend towards 2.pi/3, etc.
Any of the Platonic Solids can be used as a starting figure for the process, to achieve a spherical structure, as can many of the intermediate stages between Complementary Solids. Different starting figures produce different patterns on the surface of the structure.
A framework made up of triangular figures is very strong because triangles resist distortion. To create a stronger structure from a quadrilateral-based framework, simply add struts across the diagonals. For the strongest structure add struts across the shortest diagonals.
Mike Holden - Nov 2005