## 5 prime limit in geometry

To understand how harmonics and polygons are connected, you need to see this video !

If you join the corners in the above star shaped polygons, you will have the regular polygons. Regular polygons evenly divide a circle in 2D, in this dimension they behave in a nice linear manner, with no polygon behaving much differently to the others. Each harmonic has a polygon with the same amount of sides as its number in the harmonic series.

When you use these regular polygons to evenly divide a sphere in 3D however some interesting things happen.

There are actually only 5 different ways to divide a sphere exactly evenly, the 5 unique shapes you get when you do this are called the Platonic solids.

A Platonic solid is a regular, convex polyhedron. They are made from identical-in-shape-and-size, regular polygonal faces with the same number of faces meeting at each vertex. All corners must also touch the outside of a sphere, dividing it into even parts. Basically, every side and angle in each shape is exactly the same, making them perfectly uniform.

These are the simplest and most uniform 3 dimensional shapes that exist, and for this reason they have been studied and loved for thousands of years.

When you use these regular polygons to evenly divide a sphere in 3D however some interesting things happen.

There are actually only 5 different ways to divide a sphere exactly evenly, the 5 unique shapes you get when you do this are called the Platonic solids.

A Platonic solid is a regular, convex polyhedron. They are made from identical-in-shape-and-size, regular polygonal faces with the same number of faces meeting at each vertex. All corners must also touch the outside of a sphere, dividing it into even parts. Basically, every side and angle in each shape is exactly the same, making them perfectly uniform.

These are the simplest and most uniform 3 dimensional shapes that exist, and for this reason they have been studied and loved for thousands of years.

The Platonic solids are made from only triangles, squares and pentagons. See how the amount of sides in each polygon, and the amount of polygons used in each solid are always regular numbers.

The first few regular numbers are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, ...

In number theory, these numbers are called

5 limit just intonation scales have ratios containing only regular numbers.

The first few regular numbers are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, ...

In number theory, these numbers are called

**5-smooth**, because they can be characterized as having only 2, 3, or 5 as prime factors.5 limit just intonation scales have ratios containing only regular numbers.

If we look back at the levitated drop of water video, 3, 4 and 5 make an inverted major chord.

If you use more than one type of regular polygon together in one shape, you can make the Archimedean solids. Obviously these are named after Archimedes and only 13 of them exist. The Archimedean solids are referred to as the semiregular polyhedra, because they are not 100% regular like the platonic solids are. Although they are not 100% regular, all of the edges in these are also exactly the same length.

As you can see, Archimedean solids contain regular number amounts of only triangles, squares, pentagons, hexagons, octagons and decagons (3, 4, 5, 6, 8 and 10 sided polygons), which themselves have regular number amounts of sides. So these are also basically "5 limit geometry".

Regular numbers again: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, ...

A hexagon is an octave of a triangle, which is why it contains 2 triangles. In the same way an octagon is an octave of a square, and a decagon is an octave of a pentagon. So the basic building blocks of these solids can also be said to be triangles, squares and pentagons.

Regular numbers again: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, ...

A hexagon is an octave of a triangle, which is why it contains 2 triangles. In the same way an octagon is an octave of a square, and a decagon is an octave of a pentagon. So the basic building blocks of these solids can also be said to be triangles, squares and pentagons.

In the following video of a micro-tuned piano / synth you can hear the 3 - 4 - 5 - 6 - 8 - 10 chord.

You cannot build much with prime numbers larger than 5. Polygons with these amounts of sides just won't fit together, and regular polygons in these amounts also don't fit together to make such good shapes. Ratios with prime numbers larger than 5 tend to sound quite odd. That is not to say they are bad, they just represent tension more.

Here are some intervals with 7, 11 and 13 in their ratios:

Here are some intervals with 7, 11 and 13 in their ratios:

Here are some 5 limit intervals, these sound a lot smoother:

Here is some super soothing meditation music made using 5 limit just intonation.

So it would seem that the 5 limit used to make smooth intervals also makes amazing 3 dimensional forms. Many other amazing objects are made with this "5 limit geometry":

If you want to learn more, you really should read "Music, Geometry and Mathematics". It is available in paperback and kindle formats on Amazon. All of the images and info on this page actually come from it, so if you read this far you will definitely enjoy the whole book.

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