To make a 1v Dome into a 2v Dome, each triangle in the 1v Dome is tessalated into 4 triangles.

In addition, the tessellation of a Geodesic Dome will result in different lengths for the struts, or edges.

The Blue "A" and Red "B": struts (edges) of the 4 new triangles represent different lengths.

During tessellation, the 1v edges are subdivided into two edges, joined by a vertex.

And the lengths of the edges are changed so that each new vertex is pushed outward an equal distance from the center of the dome to make the 2v dome more spherical than the 1v dome.

The edges (or struts) on the outside of the tessalated triangles in the higher dome frequencies are always significantly shorter than the edges (struts) in the middle of the triangle.

In this 2v example, the Red "B" Struts on the outside of the triangle are only 88% of the length of the Blue "A" Struts on the inside of the triangle.

For a 16' Geodesic Dome, this means the Red Struts would be 4' 5" in length, and the Blue Struts would 5' in length. That is quite a difference!

Each corner of the original Icosahedron triangle is part of a 5-way connection, which if flattened, creates a 72 degree angle.
(360 degrees / 5 = 72 degrees.)

The interior angles of the triangle are part of a 6 way connection, which normally creates a 60 degree angle.
(360 / 6 = 60 degrees.)

Normally, the sum of the 3 angles of a Euclidian or "flat" triangle must equal 180 degrees.

But this Geodesic Dome triangle is being applied to a positive curvature of a sphere, and so must follow the rules of "spherical geometry".

One rule of spherical geometry is that every triangle applied to the positive curvature of a sphere must exceed 180 degrees.

This means that the 60 degree angles in this diagram are really greater than 60 degrees. These 60+ degree angles, along with the 72 degree angles in the corners will cause the triangles to bend away from a flat plane so that the vertices will follow the curved surface of a sphere.

So, it is a combination of these 5-way and 6-way connections and their 72 degree and 60+ degree angles, along with the shorter Red edges on the outside of the Icosahedron triangle, that will bend each Icosahedron face into a 3 dimensional curved surface to create a portion of the Geodesic Dome.

This is similar to when you take a flat sheet of paper in the shape of a circle, with a "pie slice" cut out of it.

When you move the edges of the "pie slice" together, the flat paper forms a cone.

In the same way, the triangular panel is deformed into a curved surface as the perimeter edges become shorter than the interior edges, and the 5-way and 6-way connections are enforced.

2v Dome Example

For building a 16' Diameter Dome, the edge (strut) lengths will be:

Blue (A): 5'
Red (B): 4' 5"

3v Dome Example

For building a 24' Diameter Dome, the edge (strut) lengths will be:

Blue (A): 5'
Yellow (B): 4' 10-3/4"
Red (C): 4' 2-3/4"

4v Dome Example

For building a 30' dome, the edge (strut) lengths will be:

Red (A): 3' 10-3/4"
Yellow (B): 4' 6-1/2"
Brown (C): 4' 6-3/8"
Black (D): 4' 9-3/4"
Blue (E): 5'
Orange (F): 4' 7-1/2"

5v Dome Example

For building a 38 Diameter Dome, the edge (strut) lengths will be:

Red (A): 3' 9-1/2"
Yellow (B): 4' 5-1/2"
Brown (C): 4' 3-3/4"
Green (D): 4' 8-3/4"
Orange (E): 4' 10-1/2"
Black (F): 4' 8-1/4"
Blue (G): 5'
Purple (H): 4' 5-1/8"
Gray (I): 4' 8-1/4"