It's hard to look at those inner lines and arrows on the Enneagram symbol and not think they must have some meaning. So how are they used with the personality types?
The arrows are not always shown on the Enneagram symbol and not all schools of the Enneagram types give meaning to the arrows. However, most schools do give meaning to the lines connecting the numbers.
Using the lines and arrows with the types
There are two schools of thought on applying the inner lines and arrows to the personality types that developed separately: stress / security points and directions of disintegration / integration. These two approaches are similar but have their differences as well. You can read more about them on the Enneagram stress and growth page here.
For many who don't use the arrows but only the lines, the types connected by a line are often described as connecting points. There are two connecting points for each type as indicated by the lines. The connecting point types are said to have an influence upon one another.
For example, type 1 has connecting points at 7 and 4. Types 7 and 4 are said to have an influence on type 1 in that a person who's primary type is 1 will take on some type 7 and type 4 characteristics at times.
When the arrows are used there is an explicit differentiation between how the type 7 and type 4 influence type 1. When the arrows are not used there is no differentiation.
Where did the lines and arrows come from?
The Enneagram symbol or diagram was not originally used with personality types. It was used by G.I. Gurdjieff and the Fourth Way School in a very different way.
The Enneagram hexad
The lines connecting points 1, 2, 4, 5, 7, and 8 form a six-pointed figure referred to as the hexad. The reason the lines are connected the way they are has to do with what Gurdjieff called the law of 7.
Divide 7 by any number between 1 and 6 and you get decimal results with the same repeating sequence of numbers.
1/7 = .142857...
2/7 = .285714...
3/7 = .428571...
4/7 = .571428...
5/7 = .714285...
6/7 = .857142...
This sequence of numbers not only connects the lines to the numbers but places the arrows on the lines in the direction of the sequence.
The Enneagram triad or triangle
The lines connecting points 9, 6, and 3 represent the law of 3 for Gurdjieff, mathematically represented by the following repeating decimals.
1/3 = .3333...
2/3 = .3333... + .3333... = .6666...
3/3 = .3333... + .3333... + .3333 = .9999...
Fallacy of the lines and arrows
The original use of the inner lines and arrows on the Enneagram by Gurdjieff was quite different from the way the personality types use them. Gurdjieff used the hexad for one purpose and the triangle for another. The personality types use all the lines and arrows in the same way. The problem is that all the lines are not the same.
For types 1, 3, 6, 8, and 9 the lines are the same (i.e., they connect to points that are two points to the left and right). This is not the case for types 2, 4, 5, and 7. To assign all the lines the same meaning ignores this difference.
Additionally, there is no underlying reasoning to support the idea that moving with or against the arrows should have different meaning.
When you take a symbol such as the Enneagram and use it for a different purpose, you can't have it both ways. You can't use it differently while at the same time justify how you use it through the logic of it's original use, but that's what happens sometimes with the Enneagram personality types.
The bottom line, like everything else for the Enneagram personality types, use the lines and arrows if they work for you, otherwise don't.