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Topic: A ten-sided polygon has the following property ...
Replies: 40   Last Post: Jun 12, 2014 12:44 AM

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quasi

Posts: 10,307
Registered: 7/15/05
Re: A ten-sided polygon has the following property ...
Posted: Jun 11, 2014 4:29 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

quasi wrote:
>quasi wrote:
>>quasi wrote:
>>>
>>>For each integer n >= 3, let f(n) be the greatest integer m
>>>such that there exists an m-gon in the plane whose boundary
>>>can be covered by n lines.

>
>Clarification:
>
>The m-gons are required to be piecewise linear simple closed
>curves with no pair of adjacent edges lying on the same line.
>

>>>Some instant observations:
>>>
>>>(1) f(n) >= n, for all n.
>>>
>>>(2) f is non-decreasing.
>>>
>>>An easy observation:
>>>
>>>(3) f(n) >= 2n, for all n >= 5.
>>>
>>>Some tentative data:
>>>
>>> f(3)=3
>>> f(4)=4
>>> f(5)=10
>>> f(6)=12
>>> f(7)=18
>>> f(8)=21
>>> f(9)=32
>>> f(10)=36
>>> f(11)=48
>>>
>>>Corrections and/or extensions to the above are welcome.

>>
>>A few conjectures:
>>
>>(1) f is strictly increasing.
>>
>>(2) f(n) <= C(n,2), for all n.


Ok, I see now that conjecture (2) is, in fact, obvious. After
all, since each vertex is on two edges, each vertex must lie on
two distinct covering lines, thus the vertice are at the
intersection points of pairs of covering lines. But n distinct
lines have at most C(n,2) intersection points, hence
f(n) <= C(n,2).

>>(3) As n -> oo, f(n)/C(n,2) -> 1.

quasi


Date Subject Author
6/8/14
Read A ten-sided polygon has the following property ...
Port563
6/8/14
Read Re: A ten-sided polygon has the following property ...
James Waldby
6/9/14
Read Re: A ten-sided polygon has the following property ...
Port563
6/9/14
Read Re: A ten-sided polygon has the following property ...
William Elliot
6/9/14
Read Re: A ten-sided polygon has the following property ...
Robin Chapman
6/9/14
Read Re: A ten-sided polygon has the following property ...
Port563
6/9/14
Read Re: A ten-sided polygon has the following property ...
Peter Percival
6/10/14
Read Re: A ten-sided polygon has the following property ...
William Elliot
6/9/14
Read Re: A ten-sided polygon has the following property ...
Peter Percival
6/9/14
Read Re: A ten-sided polygon has the following property ...
Robin Chapman
6/9/14
Read Re: A ten-sided polygon has the following property ...
Phil Carmody
6/9/14
Read Re: A ten-sided polygon has the following property ...
Brian Q. Hutchings
6/10/14
Read Re: A ten-sided polygon has the following property ...
Port563
6/10/14
Read Re: A ten-sided polygon has the following property ...
Leon Aigret
6/10/14
Read Re: A ten-sided polygon has the following property ...
Port563
6/10/14
Read Re: A ten-sided polygon has the following property ...
Leon Aigret
6/10/14
Read Re: A ten-sided polygon has the following property ...
Port563
6/10/14
Read Re: A ten-sided polygon has the following property ...
Leon Aigret
6/10/14
Read Re: A ten-sided polygon has the following property ...
Port563
6/10/14
Read Re: A ten-sided polygon has the following property ...
Phil Carmody
6/10/14
Read Re: A ten-sided polygon has the following property ...
Phil Carmody
6/10/14
Read Re: A ten-sided polygon has the following property ...
Port563
6/10/14
Read Re: A ten-sided polygon has the following property ...
Phil Carmody
6/10/14
Read Re: A ten-sided polygon has the following property ...
Port563
6/10/14
Read Re: A ten-sided polygon has the following property ...
thenewcalculus@gmail.com
6/10/14
Read Re: A ten-sided polygon has the following property ...
Brian Q. Hutchings
6/10/14
Read Re: A ten-sided polygon has the following property ...
Brian Q. Hutchings
6/11/14
Read Re: A ten-sided polygon has the following property ...
Port563
6/10/14
Read Re: A ten-sided polygon has the following property ...
Waldek Hebisch
6/11/14
Read Re: A ten-sided polygon has the following property ...
quasi
6/11/14
Read Re: A ten-sided polygon has the following property ...
quasi
6/11/14
Read Re: A ten-sided polygon has the following property ...
quasi
6/11/14
Read Re: A ten-sided polygon has the following property ...
quasi
6/12/14
Read Re: A ten-sided polygon has the following property ...
quasi
6/12/14
Read Re: A ten-sided polygon has the following property ...
quasi
6/11/14
Read Re: A ten-sided polygon has the following property ...
Phil Carmody
6/11/14
Read Re: A ten-sided polygon has the following property ...
quasi
6/11/14
Read Re: A ten-sided polygon has the following property ...
Phil Carmody
6/11/14
Read Re: A ten-sided polygon has the following property ...
Port563
6/11/14
Read Re: A ten-sided polygon has the following property ...
Leon Aigret
6/11/14
Read Re: A ten-sided polygon has the following property ...
quasi

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