Chapter II – Dome Frequencies – Travels and Adventures of Joyful Stephanie

What is a Frequency?

First, I had to figure out what a frequency is. According to Wikipedia, there are a variety of frequencies: temporal, spatial, and angular. ¹

Too technical for me.

Basically, a frequency is the number of times something occurs.

In domes – a frequency is how many times an icosahedron edge has been segmented – this makes the dome rounder. To identify the frequency is to know how many line segments/edges are between icosahedron vertices.

What is an Icosahedron Edge?

First – there are three important structural terms that define the platonic solid, Icosahedron:

Triangle Face – the actual two dimensional triangle
Edge/Line Segment – the lines that make up the triangle
Vertices – Where the edges/line segments meet

Photo Credit – Math.net

So, what is an Icosahedron edge? Essentially, it is the edge/line segments of a triangle face within an Icosahedron or, the edge/line segments between vertices.

Second – are these facts about this platonic solid:

An Icosahedron has 20 faces, 30 edges/line segments and 12 vertices.
Every one of the 12 vertices are the centre of a pentagon.
There are 12 vertices, yes, but not twelve individual pentagons because the pentagons of an Icosahedron overlap each other.
The imagine below illustrates how the pentagons overlap each other:

These facts are important because as you increase frequency, the Icosahedron is no longer a platonic solid. All platonic solids are polyhedrons, (meaning many sided shape). But, remember the definition of a platonic solid is that all its angles, edges and surfaces are equal. Once you start expanding the platonic solid by increasing frequency, the angle, edges and surfaces are no longer equal, thus it is no longer a platonic solid.

Most geodesic dome homes are based on the Icosahedron because it is the roundest of the platonic solids. Increasing the frequency makes the shape even rounder. This is why frequency matters.

Why Frequency Matters

It matters because the more frequencies, (meaning the more times an Icosahedron edge has been segmented), the rounder the structure will be.

Our dome will be a three frequency dome, as this is Michael’s favorite.

A one frequency Icosahedron has not been segmented – it is one of the five platonic solids.

A two frequency Icosahedron is dividing all of the triangle edges within an Icosahedron face into two parts – it is no longer platonic solid.

A three frequency Icosahedron is dividing all of the triangle edges within an Icosahedron face into three parts.

And so on….

The first row of images illustrate how a one dimensional triangle is segmented from 1 frequency to 2 frequencies and so on.

Next, the images highlight one triangle face of an Icosahedron, illustrating what happens when the three dimensional Icosahedron is segmented to create higher frequencies that build rounder geodesic domes.

Photo Credit – Pacific Domes

Counting Frequencies

The vertice of a pentagon is another fundamental concept of understanding frequencies. The number of line segments between pentagon vertices defines the frequency.

Photo Credit – Pacific Domes – Dome Book 2 – Page #24

Steps to locating the frequency in the above image:

Locate the two pentagons.
Starting with the left pentagon, find the vertice.
Then count the line segments to the vertice of the next pentagon. (I’ve taken artistic license with a picture from Pacific Domes – Dome Book 2 – Page #24 – I’ve highlighted the line segments between pentagon vertices – red, yellow then green.)
The number of line segments between the two vertices determines the frequency.
3 line segments = 3 frequency dome.

Conclusion

So, there it is!

Higher frequencies make a geodesic dome home rounder.

When you look at a geodesic dome home – to identify what “kind of” or “how many frequencies” it is – you count the edges/line segments between pentagon vertices!

The next chapter in my blog series on building a geodesic dome home is turning a 3 frequency icosahedron into a geodesic dome home!

S, 📐

References

1 – Wikipedia

Building A Geodesic Dome Home Series:

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