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Guest Message by DevFuse

Re-exploration of regular n-gon polygons.

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Posted 28 March 2013 - 05:08 PM

A discrete function that takes the number of sides of a regular polygon and tells you the measure of one of its inner angles. A regular triangle has three sides and its inner angle is 60 degrees. A regular quadrilateral has four sides and its inner angle is 90 degrees. A regular pentagon has five sides and its inner angle is 108 degrees.

That's a recipe for a regular pentagon right there. Draw a 108 degree angle between two segments with the same length.

Then draw another 108 degree angle on the last segment.

And another, and another, until the segments reconnect and you have a regular polygon with five sides.

We can write a table:

We can graph those values:

We can also write an equation:

That equation perfectly describes the discrete values in that graph. But the equation is stupid. It doesn't know it's only supposed to describe those discrete values. We can put in other values and, like a sucker, it'll give us a number, even though it isn't supposed to and even though that number won't make any sense.

Like n = 3.5. A regular polygon with 3.5 sides? No such thing. But if we throw n = 3.5 into that function, it gives us the number 77.1 degrees.

Maybe that's just gibberish, the result of pushing this function machine beyond its warranty. But maybe it isn't.

What if we tried to draw a regular 3.5-gon in the same way we did the regular 5-gon up there?

When you make the shape, ask yourself the following question: But where is the 3.5 in that shape? Maybe you see how the number 3.5 turned into the number 77.1 and how the number 77.1 turned into that _____ shape. But where is the 3.5 in that shape?

Spoiler for (hint:

Edited by BMAD, 28 March 2013 - 05:17 PM.

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Posted 28 March 2013 - 05:09 PM

Spoiler for Hint:

Edited by BMAD, 28 March 2013 - 05:10 PM.

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#3 ParaLogic

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Posted 29 March 2013 - 09:33 PM

Just picturing it in my head, I'm guessing that the resulting shape will be a

Spoiler for ...

Edit:

Spoiler for The math checks out

Edit2:

Spoiler for Just pointing out the obvious

Edited by ParaLogic, 29 March 2013 - 09:43 PM.

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#4 ParaLogic

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Posted 29 March 2013 - 09:52 PM

Eugh editing time ran out just when I noticed my mistake in Edit2. So ignore that one.

Spoiler for What I really meant to say

Need to test more!

Edit: Yep, it seems to work with 5/2 and 9/2 as well!

Edit2: I think I have reached a conclusion!!. (post coming up)

Edited by ParaLogic, 29 March 2013 - 10:02 PM.

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#5 ParaLogic

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Posted 29 March 2013 - 10:37 PM

All right. I've cluttered up this thread very much, but this particular topic intrigued me very much. I believe my conclusion is correct, although I didn't actually prove it. (Someone else can handle that. )

Spoiler for Conclusion...?

So we've got that behind us.

Spoiler for But...?

Edit: Did I take the question in the OP too far? It was pretty vague...

Edited by ParaLogic, 29 March 2013 - 10:43 PM.

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#6 bonanova

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Posted 30 March 2013 - 03:51 AM

Spoiler for Looks like

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Vidi vici veni.

#7 ParaLogic

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Posted 30 March 2013 - 01:15 PM

Looking at bonanova's post, maybe I did take the question too far...

Either way, I just realized that for numbers like 7/4 and 11/6:

Spoiler for

Edited by ParaLogic, 30 March 2013 - 01:15 PM.

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Posted 30 March 2013 - 02:19 PM

Looking at bonanova's post, maybe I did take the question too far...

Either way, I just realized that for numbers like 7/4 and 11/6:

Spoiler for

I disagree paralogic. i believe having a negative angle would just mean that the angle is found in the "opposite direction" as the convention I used to make my pentagon.
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Posted 30 March 2013 - 02:39 PM

Spoiler for something to chew over

Edited by BMAD, 30 March 2013 - 02:40 PM.

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#10 ParaLogic

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Posted 30 March 2013 - 11:15 PM

I don't like the implications of negative angles, but if you use them, then

Spoiler for

Spoiler for Regarding the shapes in the above post

As for irrational numbers...well I don't (want to) know.

Edited by ParaLogic, 30 March 2013 - 11:24 PM.

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