# Geodesic Grids/Breakdown structures

(3,2) triangle lattice
(3,2) hex lattice

In a grid ${\displaystyle GG_{k}(n,m)}$, the breakdown structure the part ${\displaystyle (n,m)}$ 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, ${\displaystyle n}$ is taken to be an integer no less than 1, and ${\displaystyle m}$ 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 ${\displaystyle (n,m)}$ is associated with a value ${\displaystyle T=n^{2}+nm+m^{2}}$ that determines how many vertexes, edges, and faces will be in the final grid. Note that ${\displaystyle T}$ may not be unique: for example, ${\displaystyle (5,3)}$ and ${\displaystyle (7,0)}$ both have ${\displaystyle T=49}$.

The mirror image of the ${\displaystyle (n,m)}$ breakdown is ${\displaystyle (m,n)}$. The breakdowns ${\displaystyle (n,0)}$ and ${\displaystyle (0,n)}$ are exactly the same; usually that breakdown is written as ${\displaystyle (n,0)}$. ${\displaystyle (n,n)}$ is its own mirror image, as is ${\displaystyle (n,0)}$: 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 ${\displaystyle (0,0)}$. From that point, go right ${\displaystyle n}$ steps along a line, then turn 60 degrees towards the upper right and take ${\displaystyle m}$ steps along that line. Designate that point ${\displaystyle (n,m)}$. 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 ${\displaystyle (0,0),(n,m)}$, and ${\displaystyle (-m,n+m)}$. 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 ${\displaystyle n+m+1}$ evenly spaced vertices (including the base vertices). Going clockwise, mark ${\displaystyle n+1}$ points along the next edge, then ${\displaystyle m+1}$ 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 (1,0) (2,0) (3,0) (4,0) Class II (1,1) (2,2) (3,3) (4,4) Class III (1,2) (2,1) (3,1) (3,2)