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#1 2013-04-19 20:28:37

anna_gg
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Registered: 2012-01-10
Posts: 113

Regular concave polygons

What is the regular concave polygon with the smallest number of sides?
All its sides must be equal and all its internal or external angles must be equal as well (i.e. all its internal angles must be either equal or negative, that is, either θ or -θ).

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#2 2013-04-20 00:39:34

Agnishom
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From: The Complex Plane
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Re: Regular concave polygons

A concave quadrilateral.

You can show that a triangle is never concave using the angle sum property


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#3 2013-04-20 02:08:28

anna_gg
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Registered: 2012-01-10
Posts: 113

Re: Regular concave polygons

Can you design it?

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#4 2013-04-20 02:36:31

mathgogocart
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Registered: 2012-04-29
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Re: Regular concave polygons

here Quadrilaterals.gif


Hey.

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#5 2013-04-20 02:44:44

Agnishom
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From: The Complex Plane
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Re: Regular concave polygons

I am sorry, I didnt notice 'Regular'.

I don't think any regular polygon can be concave
Do you want me to prove it?

Last edited by Agnishom (2013-04-20 04:24:13)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#6 2013-04-20 03:51:57

anna_gg
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Registered: 2012-01-10
Posts: 113

Re: Regular concave polygons

I said concave, not convex. By "regular concave", I mean a concave polygon of which all the sides are equal and all its internal or external angles equal (that is, all the angles must be either θ or 360-θ).

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#7 2013-04-20 04:09:23

Nehushtan
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From: London
Registered: 2013-03-09
Posts: 615
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Re: Regular concave polygons

anna_gg wrote:

What is the regular concave polygon with the smallest number of sides?
All its sides must be equal and all its internal or external angles must be equal as well (i.e. all its internal angles must be either equal or negative, that is, either θ or -θ).

I think it’s a concave dodecagon

ConcaveDodecagon.png

Last edited by Nehushtan (2013-04-21 00:18:45)


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#8 2013-04-20 23:27:02

anna_gg
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Registered: 2012-01-10
Posts: 113

Re: Regular concave polygons

This was also my solution but it seems there is a polygon with fewer sides that meets the criteria.

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#9 2013-04-20 23:40:14

Mrwhy
Member
Registered: 2012-07-02
Posts: 52

Re: Regular concave polygons

There are stars
5 pointed
6 pointed
7 pointed

I reckon the 5 pointed has 10 sides

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#10 2013-04-20 23:52:23

anna_gg
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Registered: 2012-01-10
Posts: 113

Re: Regular concave polygons

Yet the stars cannot have all their internal angles equal to θ or to 360-θ...

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#11 2013-04-21 00:34:10

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
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Re: Regular concave polygons

What is the answer anyway?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#12 2013-04-21 02:57:10

Nehushtan
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From: London
Registered: 2013-03-09
Posts: 615
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Re: Regular concave polygons

anna_gg wrote:

This was also my solution but it seems there is a polygon with fewer sides that meets the criteria.

Why, of course! A concave decagon.

ConcaveDecagon.png

NB: I think I’ve proved that a regular concave polygon (i.e. a polygon with equal sides and internal angles either θ or 360°−θ, at least one of which is reflex) must have at least 9 sides. If my proof is correct, it remains to check whether there exists any regular concave nonagon.


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#13 2013-04-21 04:21:38

anna_gg
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Registered: 2012-01-10
Posts: 113

Re: Regular concave polygons

Correct smile smile

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