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Topic: rational n-gon inscribed in a unit circle
Replies: 59   Last Post: Dec 19, 2013 12:56 AM

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quasi

Posts: 10,184
Registered: 7/15/05
Re: rational n-gon inscribed in a unit circle
Posted: Dec 11, 2013 5:59 PM
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quasi wrote:
>quasi wrote:
>>quasi wrote:
>>>
>>>I'll start with the previous question, recast as a conjecture,
>>>along with some related conjectures.
>>>
>>>Call an n-gon rational (or edge-rational to be more precise)
>>>if all edge lengths are rational.
>>>
>>>Conjecture (1):
>>>
>>>If n > 6, there does not exist a rational n-gon which can be
>>>inscribed in a unit circle.
>>>
>>>Conjecture (2):
>>>
>>>If a rational hexagon is inscribed in a unit circle, then
>>>it's a regular hexagon.
>>>
>>>Conjecture (3):
>>>
>>>If a rational pentagon is inscribed in a unit circle, then
>>>a = b = c = 1 and d^2 + e^2 = 4 for some permutation
>>>a,b,c,d,e of the edge lengths.
>>>
>>>Conjecture (4):
>>>
>>>If a rational quadrilateral is inscribed in a unit circle,
>>>then a^2 + b^2 = 4 and c^2 + d^2 = 4 for some permutation
>>>a,b,c,d of the edge lengths.

>>
>>Conjecture (4) [revised]:
>>
>>If a rational quadrilateral is inscribed in a unit circle,
>>then either
>>
>> a = b = c = 1 and d = 2
>>
>>or
>>
>> a^2 + b^2 = 4 and c^2 + d^2 = 4
>>
>>for some permutation a,b,c,d of the edge lengths.
>>

>>>Remarks:
>>>
>>>I'm not very confident of the truth of any of the above
>>>conjectures.
>>>
>>>In the quest for proofs or disproofs, Conjecture (4) might
>>>be a good place to start.

>
>Conjecture (4) is false.
>
>I've found lots of counterexamples.
>
>Perhaps the spirit of the conjecture can be saved by weakening
>the conclusion, but I'm on my way out, so I'll have to leave
>it for now -- I'll take another look at it later tonight or
>tomorrow.
>
>I haven't yet looked for counterexamples to conjectures
>(1),(2),(3), but the failure of conjecture (4) suggests that
>those conjectures are probably also false. We'll see.


Ok, some of my counterexample for conjecture (4) adapt easily
to get counterexamples for conjectures (2) and (3).

If I revise them along the lines of my latest revision of
conjecture (4), they might live again, but I'll look at
that later. As of now, conjectures (2),(3) are dead.

Conjecture (1) is still alive, but probably not for long.

quasi


Date Subject Author
12/10/13
Read rational n-gon inscribed in a unit circle
quasi
12/10/13
Read Re: rational n-gon inscribed in a unit circle
ross.finlayson@gmail.com
12/10/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Helmut Richter
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
scattered
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
fom
12/11/13
Read Re: rational n-gon inscribed in a unit circle
fom
12/11/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/11/13
Read Re: rational n-gon inscribed in a unit circle
fom
12/11/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/11/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Brian Q. Hutchings
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Brian Q. Hutchings
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Thomas Nordhaus
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/12/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/13/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/13/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/15/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/15/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/15/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/15/13
Read Re: rational n-gon inscribed in a unit circle
David Bernier
12/15/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/18/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/18/13
Read Re: rational n-gon inscribed in a unit circle
ross.finlayson@gmail.com
12/19/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/14/13
Read Re: rational n-gon inscribed in a unit circle
quasi
12/14/13
Read Re: rational n-gon inscribed in a unit circle
Richard Tobin
12/14/13
Read Re: rational n-gon inscribed in a unit circle
ross.finlayson@gmail.com
12/15/13
Read Re: rational n-gon inscribed in a unit circle
Brian Q. Hutchings

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