1. The Fourth Way Enneagram was constructed by an esotericist in the 1920's(??) and was derived from 142857 in 1/7.

Well, with 9 points he connected the numeric digits of the cyclic sequence to make the figure.

It seems it was for a spiritual/esoteric purpose, but we can apply mathematics to it.

"Ennea" meaning 9, for the 9 points he used. But we have 10 numeric digits we see in sequences.

That's how we get 1/7's:

How did we make that?

Remember that 1/7 is a repetitive cycle: 0.142857 142857 142857

But who says it has to be in decimal notation?

The sequence seems very different in duodecimal, but the figure doesn't.

Now what about decimal 1/19? And its 12-base equivalent 1/17?

In this cycle both the figures and the numbers look different.

It's interesting to me how the lowest digit (0) goes around around to the highest digit (9 or B) to create a SYMMETRIC pattern.

And the longer the cycle the more complex these patterns become.

This might have nothing to do with spirituality or esoteric enlightenment, but it shows how mathematics can show its beauty (look at fractal geometry)

2.

3. Wow that is totally WICKED!!How did you come across that

4. Thank you Bad Monkey. BTW, I'm Eonos. (I'm using my primary email for this and I'd prefer not to switch profiles and subscribe to every thread I post as Eonos)

When I was researching about 142857 I came across the Fourth Way Enneagram. It's used for spiritual purposes in occult I believe. I thought "How would this be used in mathematics?"

So, I added a zero and "mapped" a few repetends. And there they are, both in decimal and duodecimal notation.

I'm not sure if someone has already done this. Anyway, I tried another thread: Visualizing Repetends in Numbers

5. One thing I'm still wondering is...

This is an unnoticed method for visualization of numbers, something that I just made up.

So why are these repetends symmetrical? They would seem arbitrarily ordered, but not in this unexpected visualization.

Can anyone explain this?

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