Jump to content


Welcome to BrainDen.com - Brain Teasers Forum

Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account.
As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends.

Of course, you can also enjoy our collection of amazing optical illusions and cool math games.

If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top.
If you have a website, we would appreciate a little link to BrainDen.

Thanks and enjoy the Den :-)
Guest Message by DevFuse
 

Photo
- - - - -

Re-exploration of regular n-gon polygons.


  • Please log in to reply
10 replies to this topic

#1 BMAD

BMAD

    Senior Member

  • Members
  • PipPipPipPip
  • 1658 posts
  • Gender:Female

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.

 

130327_2.jpg

Then draw another 108 degree angle on the last segment.

130327_3.jpg

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

130327_4.gif

We can write a table:

130327_5.png

We can graph those values:

130327_6.png

We can also write an equation:

130327_7.gif

 

 

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.

  • 0

#2 BMAD

BMAD

    Senior Member

  • Members
  • PipPipPipPip
  • 1658 posts
  • Gender:Female

Posted 28 March 2013 - 05:09 PM

Spoiler for Hint:


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

  • 0

#3 ParaLogic

ParaLogic

    Junior Member

  • Members
  • PipPip
  • 22 posts
  • Gender:Male

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.

  • 0

#4 ParaLogic

ParaLogic

    Junior Member

  • Members
  • PipPip
  • 22 posts
  • Gender:Male

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.

  • 0

#5 ParaLogic

ParaLogic

    Junior Member

  • Members
  • PipPip
  • 22 posts
  • Gender:Male

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.

  • 0

#6 bonanova

bonanova

    bonanova

  • Moderator
  • PipPipPipPip
  • 5528 posts
  • Gender:Male
  • Location:New York

Posted 30 March 2013 - 03:51 AM

Spoiler for Looks like


  • 0
The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#7 ParaLogic

ParaLogic

    Junior Member

  • Members
  • PipPip
  • 22 posts
  • Gender:Male

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.

  • 0

#8 BMAD

BMAD

    Senior Member

  • Members
  • PipPipPipPip
  • 1658 posts
  • Gender:Female

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.
  • 0

#9 BMAD

BMAD

    Senior Member

  • Members
  • PipPipPipPip
  • 1658 posts
  • Gender:Female

Posted 30 March 2013 - 02:39 PM

Spoiler for something to chew over

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

  • 0

#10 ParaLogic

ParaLogic

    Junior Member

  • Members
  • PipPip
  • 22 posts
  • Gender:Male

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.

  • 0




0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users