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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 113

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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A concave quadrilateral.

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

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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 113

Can you design it?

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**mathgogocart****Member**- Registered: 2012-04-29
- Posts: 1,434

here

Hey.

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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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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 113

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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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 its a concave dodecagon

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

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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 113

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

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

There are stars

5 pointed

6 pointed

7 pointed

I reckon the 5 pointed has 10 sides

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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 113

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

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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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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.

NB: I think Ive 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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**anna_gg****Member**- Registered: 2012-01-10
- Posts: 113

Correct

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