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Topic review (newest first)

anna_gg
2013-04-22 02:21:38

Correct smile smile

Nehushtan
2013-04-22 00:57:10

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.

http://i602.photobucket.com/albums/tt105/GeorgeLaw5/ConcaveDecagon.png

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.

Agnishom
2013-04-21 22:34:10

What is the answer anyway?

anna_gg
2013-04-21 21:52:23

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

Mrwhy
2013-04-21 21:40:14

There are stars
5 pointed
6 pointed
7 pointed

I reckon the 5 pointed has 10 sides

anna_gg
2013-04-21 21:27:02

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

Nehushtan
2013-04-21 02:09:23

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

http://i602.photobucket.com/albums/tt105/GeorgeLaw5/ConcaveDodecagon.png

anna_gg
2013-04-21 01:51:57

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).

Agnishom
2013-04-21 00:44:44

I am sorry, I didnt notice 'Regular'.

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

mathgogocart
2013-04-21 00:36:31

here http://www.cut-the-knot.org/Curriculum/Geometry/Quadrilaterals.gif

anna_gg
2013-04-21 00:08:28

Can you design it?

Agnishom
2013-04-20 22:39:34

A concave quadrilateral.

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

anna_gg
2013-04-20 18:28:37

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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