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May 17th, 2019, 07:50 AM   #1
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Spirals

Hello,

So, I was challenging myself to find the length of a Spiral, when I noticed something interesting I wanted to share.

Here is some context to help you understand. Imagine a half circle, the base is equal to 1. The length of the half circle would then be Pi. Now put another half circle upside down underneath it to form a spiral. This half circles base is 2. Then add another and keep making the spiral, the base is 3 this time. Add another, its base is 4. finally, add one more for a base of 5.

I was trying to find the spiral length, when I noticed something interesting. The length is basically all the half circles added together, which would be 1pi+2pi+3pi+4pi+5pi, equaling 47.1338898......

Then, I had an idea to find the length, and this is where I noticed something interesting. Here is what I did:

5!pi.

It gave me 376.9911184. VERY far off. I then had a hunch. What if I divided this by the answer before, which was 47.1338898?

I tried it, and I got exactly 8. Here is what I noticed:

All the half circle bases added together equal 15. Take the answer before, which was 47.1338898, and divide it by all the bases combined, 15.

The answer is Pi.

Almost like a factorial, but with addition instead of multiplication. I mean 5+4+3+2+1 instead of 5x4x3x2x1.

So, Divide 376.9911184 by (5! / (5+4+3+2+1)), and you get the same answer as before, 47.1338898.

This got my attention, so I tried it with a different spiral. This time, it was 1+2+3+4, which should equal a length of 31.41592654.

10 times Pi.

I tried what I found, which was (4! Pi) / (4!/(4+3+2+1))

Same answer, 31.41592654.

So, to sum it up, 4! / (4+3+2+1) gives you the number you need to divide 4! Pi by to get the spirals length.

This is all of course if the spiral was entirely made up of half circles.

Well, that is what I found. It is quite interesting in my opinion.

Jared
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May 17th, 2019, 03:08 PM   #2
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As you say, these are not true spirals.

But it is really good to be pushing the boundaries of what you know, using your knowledge
so +1

As you gain wider experience and knowledge these ideas can become better and better so keep pushing.
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May 18th, 2019, 04:15 AM   #3
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Thanks so much for the input. You are right, it is not a true spiral. However, I still learned something from this, so this didn't go to waste!

You are also right, as I gain experience and knowledge, my ideas can get better and better. Believe it or not, I learned by experimenting and making mistakes along the way. It's one of the best ways I learn. This is especially true with mathematics and geometry.

I always experimented with math and geometry, from the first time I came to this Forum posting about what I called Pentagon pi, ( equaling approximately 2.37764,) to today, when I talk about spirals.

Jared
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May 18th, 2019, 11:23 AM   #4
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Spirals are particularly interesting and actually widely prevalent in Nature.
They range in size from galactic (spiral galaxies) to the microscopic dislocations in liquid crystals.

You can find out a lot, including experimental stuff, here.

The Self Made Tapestry
Pattern Formation in Nature

Philip Ball.
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