Geodesic Grids/Breakdown structures

From Wikibooks, open books for an open world
Jump to navigation Jump to search
(3,2) triangle lattice
(3,2) hex lattice

In a grid , the breakdown structure the part that determines the arrangement of triangles within each base triangle. This arrangement is first described on a planar surface, then stretched to fit the surface of the sphere. The topology remains the same, though the exact measurements change.

Normally, is taken to be an integer no less than 1, and is taken to be an integer no less than 0. Values outside this range define breakdowns already covered by values within the range, so are usually not used. Each breakdown is associated with a value that determines how many vertexes, edges, and faces will be in the final grid. Note that may not be unique: for example, and both have .

The mirror image of the breakdown is . The breakdowns and are exactly the same; usually that breakdown is written as . is its own mirror image, as is : other breakdowns are chiral, i.e. different from their mirror images.

The easiest way to draw the breakdown structure is to use a planar, triangular grid. Choose a point to start, and designate it . From that point, go right steps along a line, then turn 60 degrees towards the upper right and take steps along that line. Designate that point . The third corner of the breakdown structure is equidistant from the other two. It is useful to apply a skewed coordinate system to the triangular grid, where the first axis is horizontal, and the second axis is a 60-degree line to the upper right. Then the coordinates of the three corners are , and . The breakdown structure of a Goldberg polyhedron is drawn nearly the same way, except that the triangle is overlaid on a hex lattice instead of a triangle lattice.

(3,2), other method

Another method that is conceptually useful is to draw an equilateral triangle in the plane. Along one edge, mark evenly spaced vertices (including the base vertices). Going clockwise, mark points along the next edge, then along the last. Number the vertices along the first side, then along the remaining two sides sequentially (counting the point shared between the other two sides only once). Draw lines between points with the same number. Then take those lines and make copies rotated 120° and 240° about the center. The vertices of the breakdown structure correspond to the points where 3 lines intersect (not necessarily the marked points).

Breakdown structures are commonly separated into three classes.

  • Class I breakdowns are of the form (n,0). In Class I breakdowns, the lines of the small triangles are parallel to the lines of the large triangle. This is also called the "alternate" breakdown.
  • Class II breakdowns are of the form (n,n). In Class II breakdowns, the lines of the small triangles are instead perpendicular to the lines of the large triangle. This is sometimes referred to as the "triacon" breakdown.
  • Class III breakdowns are of the form (n,m), where m is not n or 0. Class III grids account for all other breakdowns. Class III grids are chiral; that is, the mirror image of a class III grid is not the same as the original grid.

Below are some example breakdown structures.

Class I
Subdivided triangle 01 00.svg
Subdivided triangle 02 00.svg
Subdivided triangle 03 00.svg
Subdivided triangle 04 00.svg
Class II
Subdivided triangle 01 01.svg
Subdivided triangle 02 02.svg
Subdivided triangle 03 03.svg
Subdivided triangle 04 04.svg
Class III
Subdivided triangle 01 02.svg
Subdivided triangle 02 01.svg
Subdivided triangle 03 01.svg
Subdivided triangle 03 02.svg