## definition

An **ordinal ruler** has these features:

- It has line (segments).
- It has graduations.
- It is graduated with ordinals.
- The ordinals are in order. Smaller ones in the left and larger ones in the right.

(note that an ordinary ruler is an ordinal ruler)

## how to make an ordinal ruler

There are many ways to make one, so I will write one of the simple ways.

- First, write two ordinals at the ends. By the definition, the left one must be smaller.
- For every line segment between and with no graduation in between,
- If, write nothing between.
- If is included in the fundamental sequence of, write the next term of the fundamental sequence at halfway between them.
- Else, write the smallest term of the fundamental sequence of which is bigger than.

(note that the graduations can be infinitely many)

### modified version

As many people make rulers with the procedure above, I found something problematic: 1. However big the ending ordinal is, if the first element of the fundamental sequence is

, The first quarter of the total ruler is left blank. 2. On the other hand, the latter half of the ruler is pretty dense.

To solve these problems. I suggest a new upgraded way:

Take an arbitrary ratio a:b.

- First, write two ordinals at the ends. By the definition, the left one must be smaller.
- For every line segment between and with no graduation in between,
- If, write nothing between.
- If is included in the fundamental sequence of, write the next term of the fundamental sequence
**at the point internally dividing in the ratio a:b**. - Else, write the smallest term of the fundamental sequence of which is bigger than.

If you take something like 2:1 or 3:1 as the ratio, it should solve the problems (but it will be harder to analyze the structure of it).

## examples

The image is two of the ordinals made by above recipe with the ratio of 1:1.

## use

You may be able to use it for googology graph.