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What is Geodesic Dome Frequency?

An Explanation

 

 

What is the difference between a 2v, 3v, or 4v Geodesic Dome, and what is "dome frequency"?

 

The "Short Answer"

 

Dome Frequency is denoted by the letter "v".

"2v" is shorthand for "2 frequency".

A 2v dome is a 2 frequency dome, and a 3v dome is a 3 frequency dome.

 

The short answer for dome frequency is, the higher the "v", or frequency, the more triangles there are in the geodesic dome.

A higher frequency dome with more triangles will be stronger and more spherical than a lower frequency dome.

And the higher frequency dome will be more complicated to build, as it will have more struts.

Here are some examples:

 

 

 

Geodesic Dome Frequency Examples

 

 
1v Dome
2v Dome
3v Dome
 

 
4v Dome
5v Dome
 

 
6v Dome
 

 

 

 

1 Frequency Dome

2 Frequency Dome

3 Frequency Dome
 

 

 

4 Frequency Dome

5 Frequency Dome

6 Frequency Dome
 

 

The "Long Answer"

 

 

Almost all Geodesic Domes are based on the Icosahedron.

 

Dodecahedron
Octahedron
Tetrahedron

 

But Geodesic Domes can be made from other Platonic Solids such as a dodecahedron, octahedron, or tetrahedron.

 

 

 

 

 

The Icosahedron is a geometric solid that has 20 sides, called "faces".

Each "face" of the Icosahedron is made from an equilateral triangle.

 

 

 

A 1 Frequency, or "1v" Dome is an Icosahedron.

 

 

1v Icosahedron Dome
When the Icosahedron has the bottom section removed and is placed on the ground, it becomes a "1v", or 1 Frequency Dome.
All of the struts (edges) in a 1v Dome are the same length.
 

A Good Way to Use a 1v Dome


A 1v Icosahedron Dome Shelter
by Geodesic Buildings, Malaga, Spain.

 

 

But the problem with a 1v Dome is...

 

It is kinda "boxy" and not very spherical like a dome should be.
You can not make a 1v Geodesic Dome very big and still be strong.
A 20' wide 1v Geodesic Dome will need struts over 10' long, which become flimsy at that length.
Longer struts are not as strong as shorter struts. The best length for a Geodesic Dome strut is around 5' long.
But a 5' long strut will make a 1v dome that is less than 10' wide.

 

The Answer to Building a Larger Dome with Shorter Struts is:

 

Break Each of the 1v Triangles into Smaller Triangles through "Tessalation".

 

"Tessalation" Lets You Build a Larger Dome with Shorter Struts, and Makes the Dome More Spherical.

 

And...

 

"Tessalation" Increases Geodesic Dome Frequency.

 

"Tessalation" means that you are breaking up a flat surface into smaller "tiles" with no gaps or overlaps.

The tessalation of the triangular face of an Icosahedron into smaller triangles increases the Geodesic Dome Frequency.

 

 

 

1 Frequency
1 Edge, 1 Triangle
(No Tessalation)

2 Frequency
2 Edges Each Side,
4 Triangles

3 Frequency
3 Edges Each Side,
9 Triangles
 

 

 

4 Frequency
4 Edges Each Side,
16 Triangles

5 Frequency
5 Edges Each Side,
25 Triangles

6 Frequency
6 Edges Each Side,
36 Triangles
 

 

 

 

 

Let's Tessalate a 1v Dome into a 2v Dome.

 

 

 

The 1v Dome is made up of 15 triangles, with each triangle as the face of an icosahedron.

 

Each triangle of the 1v dome is "equilateral", which means all sides are the same length.

1v / 1 Frequency Dome

A 1v Dome Uses Equilateral Triangles.
All sides are the same length.

 

 

 

 

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

In addition, the tessalation 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.

A 1v Triangle with no Tessalation.

All edges are the same length


A 1v triangle becomes a 2v triangle when tessalated into 4 triangles.

The red and blue edges (or struts) represent different lengths.

 

 

 

A Higher Frequency Geodesic Dome is More Than Just Dividing the Face of an Icosahedron Into Smaller Triangles Through Tessalation!

 

 

 

And Here is the Key...

 

 

 

 

The Lengths of the Edges (or Struts) of the New Triangles Are Changed In a Mathematical Way to Make the Dome More Spherical.

 
 

 

During tessalation, 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.

 

 

 

Here Are Some Examples Of How The Tessalation Of Each Face Of An Icosahedron Into Higher Frequencies Makes A Dome More Spherical:

 

 

 

1 Frequency Geodesic Dome

 

 

 

 


The 1 Frequency Geodesic Dome Has 1 Edge for Each Edge of the Icosahedron.

 

 

 


The 1 Frequency Dome has 1 Edge and 1 Triangle for each Face of the Icosahedron.


The 1 Frequency Dome is an Icosahedron.

 

 

 

 

2 Frequency Geodesic Dome

 

 

 

 


The 2 Frequency Geodesic Dome Has 2 Edges for Each Edge of the Icosahedron.
1 Vertex On Each Edge is Pushed Outward.

 

 

 


The 2 Frequency Dome has 2 Edges on Each Side and 4 Triangles for each Face of the Icosahedron.


This is how the 2 Frequency Dome is derived from the Icosahedron.

 

 

 

 

3 Frequency Geodesic Dome

 

 

 

 


The 3 Frequency Geodesic Dome Has 3 Edges for Each Edge of the Icosahedron.
2 Vertices On Each Edge Are Pushed Outward.

 

 

 


The 3 Frequency Dome has 3 Edges on Each Side and 9 Triangles for each Face of the Icosahedron.


This is how the 3 Frequency Dome is derived from the Icosahedron.

 

 

 

 

4 Frequency Geodesic Dome

 

 

 

 


The 4 Frequency Geodesic Dome Has 4 Edges for Each Edge of the Icosahedron.
3 Vertices On Each Edge Are Pushed Outward.

 

 

 


The 4 Frequency Dome has 4 Edges on Each Side and 16 Triangles for each Face of the Icosahedron.


This is how the 4 Frequency Dome is derived from the Icosahedron.

 

 

 

 

5 Frequency Geodesic Dome

 

 

 

 


The 5 Frequency Geodesic Dome Has 5 Edges for Each Edge of the Icosahedron.
4 Vertices On Each Edge Are Pushed Outward.

 

 

 


The 5 Frequency Dome has 5 Edges on Each Side and 25 Triangles for each Face of the Icosahedron.


This is how the 5 Frequency Dome is derived from the Icosahedron.

 

 

 

 

6 Frequency Geodesic Dome

 

 

 

 


The 6 Frequency Geodesic Dome Has 6 Edges for Each Edge of the Icosahedron.
5 Vertices On Each Edge Are Pushed Outward.

 

 

 


The 6 Frequency Dome has 6 Edges on Each Side and 36 Triangles for each Face of the Icosahedron.


This is how the 6 Frequency Dome is derived from the Icosahedron.

 

 

 

 

Here is the "Secret" of How the Lengths of the Tessalated Triangle Cause the Vertices to be "Pushed Outward" into a Sphere.

 

 

 

 

 

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!


2v Dome Panel

2v Dome Edge Lengths
for a 16' Geodesic Dome

 

 

 

The Other "Secret" is...

 

 
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.


2v Dome Panel with Angles

2v Dome Panel applied to a Spherical Surface
 

 

 

 

Here is a Way to Imagine This Process.

 

 

 

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.

 

 

 

Here Are 6 Examples of the Curvature of Dome Panels
When the Length of the Struts are Changed.

(Notice the pattern that the shortest struts are always on the outside of the triangle, and the longest struts in the center of the triangle.)

 

 

 

2v Dome Example

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

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


Curved 2v Dome Panel on a spherical surface.

Flat 2v Dome Panel Diagram

 

 

 

 

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"


Curved 3v Dome Panel on a spherical surface.

 


Flat 3v Dome Panel Diagram

 

 

 

 

 

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"


Curved 4v Dome Panel on a spherical surface.

 


Flat 4v Dome Panel Diagram

 

 

 

 

 

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"


Curved 5v Dome Panel on a spherical surface.

Flat 5v Dome Panel Diagram

 

 

 

 

 

6v Dome Example

For building a 46' Diameter Dome, the edge (strut) lengths will be:

Red (A): 3' 9"
Yellow (B): 4' 4-3/4"
Brown (C): 4' 2-3/8"
Green (D): 4' 8-1/8"
Orange (E): 4' 3-7/8"
Black (F): 4' 6-7/8"
Gray (G): 4' 9"
Purple (H): 4' 11-5/8"
Blue (I): 5'


Curved 6v Dome Panel on a spherical surface.

6v Dome Panel
Flat 6v Dome Panel Diagram
 

 

Did you notice any patterns for the strut lengths?
And can you see how a higher frequency dome is more spherical?

 

 

 

 

 

How to Determine the Frequency of A Geodesic Dome.

 

 

 

5-way Hub Connection.
Only 6 in every dome.

No Matter How Big the Dome Is...

Or What Frequency the Dome Has...

Every Geodesic Dome has only six 5-way hub connections.

All other connectors will be 6-way hub connections.


6-way Hub Connection.
 

 

 

One of the Six 5-way Hub Connection is always at the very top, or "North Pole" of the dome. The other Five 5-way Hub Connections are evenly spaced around the sides of the dome, about 1/3 up the side of the dome.

 

The Six 5-way Hub Connections represent the corners of the original Icosahedron, and the struts between each of these 5-way connections is the edge of the face of the original Icosahedron.

 

 

 

 

2v edge of the original Icosahedron face.
(2 struts)

3v edge of the original Icosahedron face.
(3 struts)

4v edge of the original Icosahedron face.
(4 struts)

5v edge of the original Icosahedron face.
(5 struts)

6v edge of the original Icosahedron face.
(6 struts)
 

 

 

 

This means you only have to find two of the the 5-way hub connections in a Geodesic Dome, and then count the number of struts between the 5-way connection hubs to determine the frequency.

 

 

 

 

 

Here is Some Examples of How to Determine Geodesic Dome Frequency Using This Method:


2v Dome
 

 

 

 
3v Dome
4v Dome
 

 

 

 
5v Dome
6v Dome
 

 

 

 

So Here Is The Test...

 

What are the Geodesic Frequencies of These Domes?

 

 

 

 

1v   2v   3v   4v   5v   6v  

1v   2v   3v   4v   5v   6v  
 

 

 

 

1v   2v   3v   4v   5v   6v  

1v   2v   3v   4v   5v   6v  
 

 

 

 

1v   2v   3v   4v   5v   6v  

1v   2v   3v   4v   5v   6v  
 

 

 

 

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