Regarding that business up above about dividing a circle into 3 equal
parts using a square, blah blah blah ... maybe I don't remember my
geometry so well. Checked back in that table I mentioned in "Proven Shop
Tips" for dividing a circle into n equal parts, and sure enough, the
number to multiply the diameter of a circle by to divide into 6 equal
parts is ... exactly 0.5.
So anyone got the proof handy that a hexagon with sides of length s can
be inscribed in a circle whose radius equals s? I have my old algebra
and calculus books, but no geometry.
--
Found--the gene that causes belief in genetic determinism
"David Nebenzahl" wrote:
> Expand, please. Don't know exactly how this proof works.
=================================
This is another purely graphical soultion.
http://mathworld.wolfram.com/Hexagon.html
Lew
>
> Plug " Side-Side-Side" into Google, should keep you out of trouble
> for a couple of hours, especially the congruent triangle proofs.
>
> Lew
>
>
"David Nebenzahl" wrote:
> So anyone got the proof handy that a hexagon with sides of length s
> can be inscribed in a circle whose radius equals s? I have my old
> algebra and calculus books, but no geometry.
Remember the first three (3) plane geometry proofs?
1) Side-Angle-Side
2) Side-Side-Side
3) Angle-Side-Angle
Side-Side-Side works for me.
Lew
"Bill" <[email protected]> wrote:
>
>"David Nebenzahl" <[email protected]> wrote in message
>news:[email protected]...
>> On 5/21/2009 9:25 AM Len spake thus:
>>
>>> "David Nebenzahl" <[email protected]> wrote in message
>>> news:[email protected]...
>>>>
>>>> Like more than half the respondents to this thread, you completely
>>>> missed the point.
>>>>
>>>> I know how to do that. I wasn't asking for a demonstration; I was
>>>> asking for a *proof*.
>>>>
>>>> (Even though your description contains some of the elements of a
>>>> proof.)
>>>
>>> See:
>>> http://www.nvcc.edu/home/tstreilein/constructions/Inscribed/inscribe4.htm
>>>
>>> for the proof.
>>
>
>It seems to be based on calculation (360/6), but obscures that fact; I would
>expect better work from a math major.
LOL!
Obviously, as your previous post hinted, the OP didn't really want a
formal proof. (He may not realize that is not what he wanted, but the
fact that this question was raised because of not having an old
geometry text is a pretty good clue.) What he wanted was a logical
demonstration based on facts he accepted, with steps he didn't have to
figure out. Note the range of logically identical responses here that
have been dismissed as "mere demonstrations" or accepted as "proofs"
depending on the number and detail of the steps explicitly stated, and
whether the steps were numbered and labeled "proof" <g>.
--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.
On 5/19/2009 6:47 PM Tom Veatch spake thus:
> On Tue, 19 May 2009 18:07:58 -0700, David Nebenzahl
> <[email protected]> wrote:
>
> ...
>>So anyone got the proof handy that a hexagon with sides of length s can
>>be inscribed in a circle whose radius equals s? I have my old algebra
>>and calculus books, but no geometry.
>
> Set your compass to a convenient radius and draw a circle. Without
> changing the compass setting strike an arc from any point on the
> circle that intersects the circle. From that intersection, strike
> another intersecting arc. Continue around the circle ...
That's a demonstration, not a proof. But you knew that.
I'm interested in the proof. It can't be all that complicated.
--
Found--the gene that causes belief in genetic determinism
On 5/19/2009 7:02 PM Lew Hodgett spake thus:
> "David Nebenzahl" wrote:
>
>> So anyone got the proof handy that a hexagon with sides of length s
>> can be inscribed in a circle whose radius equals s? I have my old
>> algebra and calculus books, but no geometry.
>
> Remember the first three (3) plane geometry proofs?
>
> 1) Side-Angle-Side
> 2) Side-Side-Side
> 3) Angle-Side-Angle
>
> Side-Side-Side works for me.
Expand, please. Don't know exactly how this proof works.
--
Found--the gene that causes belief in genetic determinism
On 5/19/2009 7:18 PM alexy spake thus:
> David Nebenzahl <[email protected]> wrote:
>
>>Regarding that business up above about dividing a circle into 3 equal
>>parts using a square, blah blah blah ... maybe I don't remember my
>>geometry so well. Checked back in that table I mentioned in "Proven Shop
>>Tips" for dividing a circle into n equal parts, and sure enough, the
>>number to multiply the diameter of a circle by to divide into 6 equal
>>parts is ... exactly 0.5.
>>
>>So anyone got the proof handy that a hexagon with sides of length s can
>>be inscribed in a circle whose radius equals s? I have my old algebra
>>and calculus books, but no geometry.
>
> An equilateral triangle has equal sides and three 60-degree angles.
> Arrange six of them with sides s in an array with one common vertex,
> and you will have the inscribed hexagon.
I don't see any proof in there, only an assertion.
--
Found--the gene that causes belief in genetic determinism
On 5/19/2009 7:42 PM alexy spake thus:
> alexy <[email protected]> wrote:
>
>>David Nebenzahl <[email protected]> wrote:
>>
>>>Regarding that business up above about dividing a circle into 3 equal
>>>parts using a square, blah blah blah ... maybe I don't remember my
>>>geometry so well. Checked back in that table I mentioned in "Proven Shop
>>>Tips" for dividing a circle into n equal parts, and sure enough, the
>>>number to multiply the diameter of a circle by to divide into 6 equal
>>>parts is ... exactly 0.5.
>>>
>>>So anyone got the proof handy that a hexagon with sides of length s can
>>>be inscribed in a circle whose radius equals s? I have my old algebra
>>>and calculus books, but no geometry.
>>
>>An equilateral triangle has equal sides and three 60-degree angles.
>>Arrange six of them with sides s in an array with one common vertex,
>>and you will have the inscribed hexagon.
>
> Or, put another way:
>
> Consider a regular hexagon. Draw line segments from the center of the
> hexagon to each of the six vertices. These six equal angles at the
> center must add up to 360, so each is 60. Since the triangles are
> isosceles, and their angles add to 180, they are also equilateral. So
> the side of the hexagon is equal to the length of the line form the
> center to a vertex on the hexagon, which is the radius of the circle.
That sounds better. (Don't know if it constitutes a rigorous proof or
not, but it satisfies my "itching".)
--
Found--the gene that causes belief in genetic determinism
On 5/20/2009 7:20 PM [email protected] spake thus:
> 1. Take the circle, and draw a radius.
> 2. Use a compass to measure from the intersection of the radius and
> circle, to the center.
> 3. Scribe a circle from that point.
> 4. connect the center to the 2 new intersections
> 5. You now have 2 equilateral triangles inside the circle (all 3 sides
> are equal. - they're radii)
> you now have a third of a circle (or 2 sixths)
> 6. continue all the way around for the hexagon
Like more than half the respondents to this thread, you completely
missed the point.
I know how to do that. I wasn't asking for a demonstration; I was asking
for a *proof*.
(Even though your description contains some of the elements of a proof.)
--
Found--the gene that causes belief in genetic determinism
On 5/21/2009 9:25 AM Len spake thus:
> "David Nebenzahl" <[email protected]> wrote in message
> news:[email protected]...
>>
>> Like more than half the respondents to this thread, you completely
>> missed the point.
>>
>> I know how to do that. I wasn't asking for a demonstration; I was
>> asking for a *proof*.
>>
>> (Even though your description contains some of the elements of a
>> proof.)
>
> See:
> http://www.nvcc.edu/home/tstreilein/constructions/Inscribed/inscribe4.htm
>
> for the proof.
Thank you. That was exactly what I was looking for.
There; was that so hard?
--
Found--the gene that causes belief in genetic determinism
Bill wrote:
> Your LOL is well taken. I'm not sure whether one needs the numbers in
> between the fractions (like sqrt(2)) for woodworking, nor any negative
> numbers, imaginary numbers, non-real complex numbers, nor probably any
> numbers bigger than 500. Maybe that's why those aren't marked on the ruler.
> :)
It might depend on what you're doing. The ribs at
http://www.iedu.com/DeSoto/Projects/Stirling/Heat.html
needed to be cut so that the length along the parabola was exactly four
feet (the mirror width) and with accuracy to provide a good optical
focus along the entire eight-foot length - and...
...the tenoned parts shown at the bottom of
http://www.iedu.com/DeSoto/Projects/Bevel/
were for silverware trays with diagonal dividers; these were the divider
blanks, and they needed to /exactly/ fit (on /both/ ends :) ).
And no, none of the numbers needed were marked on any of my rulers. :)
--
Morris Dovey
DeSoto Solar
DeSoto, Iowa USA
http://www.iedu.com/DeSoto/
-MIKE- wrote:
> Morris Dovey wrote:
>> And no, none of the numbers needed were marked on any of my rulers. :)
>
> Am I the only one would rather mark than measure?
>
> Like if I have a piece of trim that needs to fit between A and B,
> I don't measure A to B then measure that out on the trim.
> I hold up the trim between A and B and mark the trim.
If it'll help you feel better, I neither marked /nor/ measured for those
projects - everything was cut from unmarked stock and then assembled as cut.
I did have drawings for the silverware trays because the customer needed
something to sign off on, but the drawings for the parabolic trough came
along after the fact, to document what had been done.
For stuff I don't need to be fussy about, I've had to switch to a light
touch with a knife - my eyes just aren't good enough any longer to split
a pencil mark...
--
Morris Dovey
DeSoto Solar
DeSoto, Iowa USA
http://www.iedu.com/DeSoto/
On 5/21/2009 1:25 PM alexy spake thus:
> "Bill" <[email protected]> wrote:
>
>>"David Nebenzahl" <[email protected]> wrote in message
>>news:[email protected]...
>>
>>> On 5/21/2009 9:25 AM Len spake thus:
>>>
>>>> "David Nebenzahl" <[email protected]> wrote in message
>>>> news:[email protected]...
>>>>>
>>>>> Like more than half the respondents to this thread, you completely
>>>>> missed the point.
>>>>>
>>>>> I know how to do that. I wasn't asking for a demonstration; I was
>>>>> asking for a *proof*.
>>>>>
>>>>> (Even though your description contains some of the elements of a
>>>>> proof.)
>>>>
>>>> See:
>>>> http://www.nvcc.edu/home/tstreilein/constructions/Inscribed/inscribe4.htm
>>>>
>>>> for the proof.
>>
>>It seems to be based on calculation (360/6), but obscures that fact; I would
>>expect better work from a math major.
>
> LOL!
>
> Obviously, as your previous post hinted, the OP didn't really want a
> formal proof. (He may not realize that is not what he wanted, but the
> fact that this question was raised because of not having an old
> geometry text is a pretty good clue.)
Wrong. I asked for a proof, and that's exactly what I wanted. Didn't
have to be a "2-column" proof[1], nor a particularly rigorous one, but I
wanted a step-by-step proof, not just another redundant demonstration.
> What he wanted was a logical demonstration based on facts he
> accepted, with steps he didn't have to figure out. Note the range of
> logically identical responses here that have been dismissed as "mere
> demonstrations" or accepted as "proofs" depending on the number and
> detail of the steps explicitly stated, and whether the steps were
> numbered and labeled "proof" <g>.
Please don't make assumptions like this. You're not a very good mind-reader.
[1] Having run across this term repeatedly while searching for geometry
stuff on the web, it seems this has nothing whatever to do with formal
mathematics, but is just the latest fashion for educator's preferred
format for making schoolkids do geometry proofs.
--
Found--the gene that causes belief in genetic determinism
1. Take the circle, and draw a radius.
2. Use a compass to measure from the intersection of the radius and
circle, to the center.
3. Scribe a circle from that point.
4. connect the center to the 2 new intersections
5. You now have 2 equilateral triangles inside the circle (all 3 sides
are equal. - they're radii)
you now have a third of a circle (or 2 sixths)
6. continue all the way around for the hexagon
shelly
David Nebenzahl <[email protected]> wrote:
>On 5/21/2009 1:25 PM alexy spake thus:
>
>> "Bill" <[email protected]> wrote:
>>
>>>"David Nebenzahl" <[email protected]> wrote in message
>>>news:[email protected]...
> >>
>>>> On 5/21/2009 9:25 AM Len spake thus:
>>>>
>>>>> "David Nebenzahl" <[email protected]> wrote in message
>>>>> news:[email protected]...
>>>>>>
>>>>>> Like more than half the respondents to this thread, you completely
>>>>>> missed the point.
>>>>>>
>>>>>> I know how to do that. I wasn't asking for a demonstration; I was
>>>>>> asking for a *proof*.
>>>>>>
>>>>>> (Even though your description contains some of the elements of a
>>>>>> proof.)
>>>>>
>>>>> See:
>>>>> http://www.nvcc.edu/home/tstreilein/constructions/Inscribed/inscribe4.htm
>>>>>
>>>>> for the proof.
>>>
>>>It seems to be based on calculation (360/6), but obscures that fact; I would
>>>expect better work from a math major.
>>
>> LOL!
>>
>> Obviously, as your previous post hinted, the OP didn't really want a
>> formal proof. (He may not realize that is not what he wanted, but the
>> fact that this question was raised because of not having an old
>> geometry text is a pretty good clue.)
>
>Wrong. I asked for a proof, and that's exactly what I wanted.
I never disputed that. I stand by my conjecture that you did not want
a formal proof.
> Didn't
>have to be a "2-column" proof[1],
Never heard of it.
> nor a particularly rigorous one, but I
>wanted a step-by-step proof, not just another redundant demonstration.
An informal proof that is not rigorous is What I describe below.
>
>> What he wanted was a logical demonstration based on facts he
>> accepted, with steps he didn't have to figure out. Note the range of
>> logically identical responses here that have been dismissed as "mere
>> demonstrations" or accepted as "proofs" depending on the number and
>> detail of the steps explicitly stated, and whether the steps were
>> numbered and labeled "proof" <g>.
>
>Please don't make assumptions like this. You're not a very good mind-reader.
Sounds like I was spot-on.
A formal proof would be relatively worthless in this situation, and is
clearly not what you wanted, given your acceptance of non-rigorous
informal proofs.
--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.
On Tue, 19 May 2009 18:07:58 -0700, David Nebenzahl
<[email protected]> wrote:
...
>So anyone got the proof handy that a hexagon with sides of length s can
>be inscribed in a circle whose radius equals s? I have my old algebra
>and calculus books, but no geometry.
It's easy enough to show using Trigonometry, but that doesn't
constitute a geometric proof, which as I recall has to be done with
only compass and straight edge. But it's simple enough to demonstrate
with a compass and straight edge. I don't know whether demonstration
by construction constitutes a formal geometric proof, or not.
Set your compass to a convenient radius and draw a circle. Without
changing the compass setting strike an arc from any point on the
circle that intersects the circle. From that intersection, strike
another intersecting arc. Continue around the circle and, if done
carefully enough, the 6th arc will pass through the original point.
Since all arcs have the same radius, all the chords connecting the
intersections are the same length and equal to the radius of the arc
which is also the radius of the circle. Connect each point of
intersection with its neighbors using a straight line. By definition,
6 sides, all of the same length, constitute a regular hexagon.
Tom Veatch
Wichita, KS
USA
An armed society is a polite society.
Manners are good when one may have to back up his acts with his life.
Robert A. Heinlein
"David Nebenzahl" <[email protected]> wrote in message
news:[email protected]...
> On 5/20/2009 7:20 PM [email protected] spake thus:
>
> > 1. Take the circle, and draw a radius.
> > 2. Use a compass to measure from the intersection of the radius and
> > circle, to the center.
> > 3. Scribe a circle from that point.
> > 4. connect the center to the 2 new intersections
> > 5. You now have 2 equilateral triangles inside the circle (all 3
sides
> > are equal. - they're radii)
> > you now have a third of a circle (or 2 sixths)
> > 6. continue all the way around for the hexagon
>
> Like more than half the respondents to this thread, you completely
> missed the point.
>
> I know how to do that. I wasn't asking for a demonstration; I was
asking
> for a *proof*.
>
> (Even though your description contains some of the elements of a
proof.)
>
See:
http://www.nvcc.edu/home/tstreilein/constructions/Inscribed/inscribe4.htm
for the proof.
Len
"Bill" <[email protected]> wrote in message
news:[email protected]...
>
> "alexy" <[email protected]> wrote in message
> news:[email protected]...
> > "Bill" <[email protected]> wrote:
> >
> >>
> >>"David Nebenzahl" <[email protected]> wrote in message
> >>news:[email protected]...
> >>> On 5/21/2009 9:25 AM Len spake thus:
> >>>
> >>>> "David Nebenzahl" <[email protected]> wrote in message
> >>>> news:[email protected]...
> >>>>>
> >>>>> Like more than half the respondents to this thread, you
completely
> >>>>> missed the point.
> >>>>>
> >>>>> I know how to do that. I wasn't asking for a demonstration; I
was
> >>>>> asking for a *proof*.
> >>>>>
> >>>>> (Even though your description contains some of the elements
of a
> >>>>> proof.)
> >>>>
> >>>> See:
> >>>>
http://www.nvcc.edu/home/tstreilein/constructions/Inscribed/inscribe4.htm
> >>>>
> >>>> for the proof.
> >>>
> >>
> >>It seems to be based on calculation (360/6), but obscures that
fact; I
> >>would
> >>expect better work from a math major.
> >
> > LOL!
> >
> > Obviously, as your previous post hinted, the OP didn't really
want a
> > formal proof. (He may not realize that is not what he wanted,
but the
> > fact that this question was raised because of not having an old
> > geometry text is a pretty good clue.) What he wanted was a
logical
> > demonstration based on facts he accepted, with steps he didn't
have to
> > figure out. Note the range of logically identical responses here
that
> > have been dismissed as "mere demonstrations" or accepted as
"proofs"
> > depending on the number and detail of the steps explicitly
stated, and
> > whether the steps were numbered and labeled "proof" <g>.
> >
> > --
> > Alex -- Replace "nospam" with "mail" to reply by email. Checked
> > infrequently.
>
> Your LOL is well taken. I'm not sure whether one needs the numbers
in
> between the fractions (like sqrt(2)) for woodworking, nor any
negative
> numbers, imaginary numbers, non-real complex numbers, nor probably
any
> numbers bigger than 500. Maybe that's why those aren't marked on
the ruler.
> :)
>
> Bill
>
That's why I suggested getting a sashigane marked with for
shaku/sun/bu to begin with. Once you get used to it, it has the
markings on the back side for laying out this kind of stuff without a
lot of fuss.
Len
-MIKE- wrote:
> -MIKE- wrote:
>> Morris Dovey wrote:
>>> And no, none of the numbers needed were marked on any of my rulers. :)
>>>
>> Am I the only one would rather mark than measure?
>>
>> Like if I have a piece of trim that needs to fit between A and B,
>> I don't measure A to B then measure that out on the trim.
>> I hold up the trim between A and B and mark the trim.
>>
>
> Before some smarta$$ says, what do you do with a 12' piece of crown
> molding....
> I obviously call a couple friends to come over and hold it in place for
> me. duh.
What do you do with a 12' piece of crown molding on an outside corner
that isn't square?
Chris
Bill wrote:
> Your LOL is well taken. I'm not sure whether one needs the numbers in
> between the fractions (like sqrt(2)) for woodworking, nor any negative
> numbers, imaginary numbers, non-real complex numbers, nor probably any
> numbers bigger than 500. Maybe that's why those aren't marked on the ruler.
sqrt(2) is useful to find the length of the long side of a 45/45/90
triangle. Similarly, 1/2/sqrt(3) are the sides of a 30/60/90 triangle.
Numbers bigger than 500 are useful when working in millimetres.
I'm up in Canada and I know a guy who does everything in mm. Although
the initial conversion of regular North American lumber dimensions to mm
is a bit of a pain, it makes subsequent math a lot simpler. And of
course all the Euro stuff just works...
Chris
In article <[email protected]>, "Bill" <[email protected]> wrote:
>
>"David Nebenzahl" <[email protected]> wrote in message
>news:[email protected]...
>> On 5/19/2009 6:47 PM Tom Veatch spake thus:
>>
>>> On Tue, 19 May 2009 18:07:58 -0700, David Nebenzahl
>>> <[email protected]> wrote:
>>>
>>> ...
>>>>So anyone got the proof handy that a hexagon with sides of length s can
>>>>be inscribed in a circle whose radius equals s? I have my old algebra and
>>>>calculus books, but no geometry.
>>>
>>> Set your compass to a convenient radius and draw a circle. Without
>>> changing the compass setting strike an arc from any point on the
>>> circle that intersects the circle. From that intersection, strike
>>> another intersecting arc. Continue around the circle ...
>>
>> That's a demonstration, not a proof. But you knew that.
>>
>> I'm interested in the proof. It can't be all that complicated.
>
>It is well-known that one cannot trisect an angle with a straight-edge and
>compass, so I don't think you'll get a proof with that approach. On the
>other hand, using division you can divide 360 degrees, or 2*Pi radians by 6
>to get the angle for each slice of the pie The rest has already been
>discussed (side-side-side).
Nobody's attempting to trisect an angle in that approach; in fact, it's
essentially the same method as Euclid's proof, a link to which was already
posted up-thread.
On Tue, 19 May 2009 20:47:59 -0500, Tom Veatch <[email protected]> wrote:
>. I don't know whether demonstration
>by construction constitutes a formal geometric proof, or not.
>
Given your demonstration, I believe you can safely append:
QED
Regards,
Tom Watson
http://home.comcast.net/~tjwatson1/
In article <[email protected]>, alexy <[email protected]> wrote:
>Consider a regular hexagon. Draw line segments from the center of the
>hexagon to each of the six vertices. These six equal angles at the
>center must add up to 360, so each is 60.
OK so far...
> Since the triangles are
>isosceles, and their angles add to 180, they are also equilateral.
.. but you just went astray there.
That's not sufficient to prove that the triangles are equilateral, since the
angles add to 180 in *all* triangles.
This may be what you meant to say:
Since the angle at the vertex of each triangle is 60 degrees, the sum of the
angles at the base is 180 - 60 = 120 degrees. Since each triangle is
isosceles, the angles at the base are equal, and (since they add to 120)
therefore also 60 degrees. The triangles are therefore equiangular, and
therefore equilateral.
> So
>the side of the hexagon is equal to the length of the line form the
>center to a vertex on the hexagon, which is the radius of the circle.
David Nebenzahl <[email protected]> wrote:
>Regarding that business up above about dividing a circle into 3 equal
>parts using a square, blah blah blah ... maybe I don't remember my
>geometry so well. Checked back in that table I mentioned in "Proven Shop
>Tips" for dividing a circle into n equal parts, and sure enough, the
>number to multiply the diameter of a circle by to divide into 6 equal
>parts is ... exactly 0.5.
>
>So anyone got the proof handy that a hexagon with sides of length s can
>be inscribed in a circle whose radius equals s? I have my old algebra
>and calculus books, but no geometry.
An equilateral triangle has equal sides and three 60-degree angles.
Arrange six of them with sides s in an array with one common vertex,
and you will have the inscribed hexagon.
--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.
David Nebenzahl <[email protected]> wrote:
>On 5/19/2009 7:18 PM alexy spake thus:
>
>> David Nebenzahl <[email protected]> wrote:
>>
>>>Regarding that business up above about dividing a circle into 3 equal
>>>parts using a square, blah blah blah ... maybe I don't remember my
>>>geometry so well. Checked back in that table I mentioned in "Proven Shop
>>>Tips" for dividing a circle into n equal parts, and sure enough, the
>>>number to multiply the diameter of a circle by to divide into 6 equal
>>>parts is ... exactly 0.5.
>>>
>>>So anyone got the proof handy that a hexagon with sides of length s can
>>>be inscribed in a circle whose radius equals s? I have my old algebra
>>>and calculus books, but no geometry.
>>
>> An equilateral triangle has equal sides and three 60-degree angles.
>> Arrange six of them with sides s in an array with one common vertex,
>> and you will have the inscribed hexagon.
>
>I don't see any proof in there, only an assertion.
Look deeper. This is a proof of the "from which it can be clearly
seen..." type that occasionally drove me batty!
--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.
"David Nebenzahl" <[email protected]> wrote in message
news:[email protected]...
> On 5/19/2009 6:47 PM Tom Veatch spake thus:
>
>> On Tue, 19 May 2009 18:07:58 -0700, David Nebenzahl
>> <[email protected]> wrote:
>>
>> ...
>>>So anyone got the proof handy that a hexagon with sides of length s can
>>>be inscribed in a circle whose radius equals s? I have my old algebra and
>>>calculus books, but no geometry.
>>
>> Set your compass to a convenient radius and draw a circle. Without
>> changing the compass setting strike an arc from any point on the
>> circle that intersects the circle. From that intersection, strike
>> another intersecting arc. Continue around the circle ...
>
> That's a demonstration, not a proof. But you knew that.
>
> I'm interested in the proof. It can't be all that complicated.
It is well-known that one cannot trisect an angle with a straight-edge and
compass, so I don't think you'll get a proof with that approach. On the
other hand, using division you can divide 360 degrees, or 2*Pi radians by 6
to get the angle for each slice of the pie The rest has already been
discussed (side-side-side).
Bill
> This is another purely graphical soultion.
>
> http://mathworld.wolfram.com/Hexagon.html
>
> Lew
>
The illustration with the A, B, C, D and E circles gives the OP his 3
pie pieces, if he just puts the triangle inside the hexagon.
--
-MIKE-
"Playing is not something I do at night, it's my function in life"
--Elvin Jones (1927-2004)
--
http://mikedrums.com
[email protected]
---remove "DOT" ^^^^ to reply
"David Nebenzahl" <[email protected]> wrote in message
news:[email protected]...
> On 5/20/2009 7:20 PM [email protected] spake thus:
>
>> 1. Take the circle, and draw a radius.
>> 2. Use a compass to measure from the intersection of the radius and
>> circle, to the center.
>> 3. Scribe a circle from that point.
>> 4. connect the center to the 2 new intersections
>> 5. You now have 2 equilateral triangles inside the circle (all 3 sides
>> are equal. - they're radii)
>> you now have a third of a circle (or 2 sixths)
>> 6. continue all the way around for the hexagon
>
> Like more than half the respondents to this thread, you completely missed
> the point.
>
> I know how to do that. I wasn't asking for a demonstration; I was asking
> for a *proof*.
State the axioms you wish to start with, and we'll take it from there. I'm
sure someone here knows how to write a proof. A calculation based
explanation is a good proof if one axiomizes high school geometry. This
seems like the right approach here in rec.woodworking!
"David Nebenzahl" <[email protected]> wrote in message
news:[email protected]...
> On 5/21/2009 9:25 AM Len spake thus:
>
>> "David Nebenzahl" <[email protected]> wrote in message
>> news:[email protected]...
>>>
>>> Like more than half the respondents to this thread, you completely
>>> missed the point.
>>>
>>> I know how to do that. I wasn't asking for a demonstration; I was
>>> asking for a *proof*.
>>>
>>> (Even though your description contains some of the elements of a
>>> proof.)
>>
>> See:
>> http://www.nvcc.edu/home/tstreilein/constructions/Inscribed/inscribe4.htm
>>
>> for the proof.
>
It seems to be based on calculation (360/6), but obscures that fact; I would
expect better work from a math major.
> Obviously, as your previous post hinted, the OP didn't really want a
> formal proof. (He may not realize that is not what he wanted, but the
> fact that this question was raised because of not having an old
> geometry text is a pretty good clue.) What he wanted was a logical
> demonstration based on facts he accepted, with steps he didn't have to
> figure out. Note the range of logically identical responses here that
> have been dismissed as "mere demonstrations" or accepted as "proofs"
> depending on the number and detail of the steps explicitly stated, and
> whether the steps were numbered and labeled "proof" <g>.
>
Reminds me of a quote from a physics professor.
"You want an easy proof for the law of gravity? Step out of the window."
--
-MIKE-
"Playing is not something I do at night, it's my function in life"
--Elvin Jones (1927-2004)
--
http://mikedrums.com
[email protected]
---remove "DOT" ^^^^ to reply
Chris Friesen wrote:
> -MIKE- wrote:
>
>> Reminds me of a quote from a physics professor.
>>
>> "You want an easy proof for the law of gravity? Step out of the window."
>
> Obviously, that proof makes the assumption that the classroom in
> question is in an inertial reference frame. :)
>
> Chris
The window was in a big frame, on the 5th floor. :-)
--
-MIKE-
"Playing is not something I do at night, it's my function in life"
--Elvin Jones (1927-2004)
--
http://mikedrums.com
[email protected]
---remove "DOT" ^^^^ to reply
"alexy" <[email protected]> wrote in message
news:[email protected]...
> "Bill" <[email protected]> wrote:
>
>>
>>"David Nebenzahl" <[email protected]> wrote in message
>>news:[email protected]...
>>> On 5/21/2009 9:25 AM Len spake thus:
>>>
>>>> "David Nebenzahl" <[email protected]> wrote in message
>>>> news:[email protected]...
>>>>>
>>>>> Like more than half the respondents to this thread, you completely
>>>>> missed the point.
>>>>>
>>>>> I know how to do that. I wasn't asking for a demonstration; I was
>>>>> asking for a *proof*.
>>>>>
>>>>> (Even though your description contains some of the elements of a
>>>>> proof.)
>>>>
>>>> See:
>>>> http://www.nvcc.edu/home/tstreilein/constructions/Inscribed/inscribe4.htm
>>>>
>>>> for the proof.
>>>
>>
>>It seems to be based on calculation (360/6), but obscures that fact; I
>>would
>>expect better work from a math major.
>
> LOL!
>
> Obviously, as your previous post hinted, the OP didn't really want a
> formal proof. (He may not realize that is not what he wanted, but the
> fact that this question was raised because of not having an old
> geometry text is a pretty good clue.) What he wanted was a logical
> demonstration based on facts he accepted, with steps he didn't have to
> figure out. Note the range of logically identical responses here that
> have been dismissed as "mere demonstrations" or accepted as "proofs"
> depending on the number and detail of the steps explicitly stated, and
> whether the steps were numbered and labeled "proof" <g>.
>
> --
> Alex -- Replace "nospam" with "mail" to reply by email. Checked
> infrequently.
Your LOL is well taken. I'm not sure whether one needs the numbers in
between the fractions (like sqrt(2)) for woodworking, nor any negative
numbers, imaginary numbers, non-real complex numbers, nor probably any
numbers bigger than 500. Maybe that's why those aren't marked on the ruler.
:)
Bill
"Chris Friesen" <[email protected]> wrote in message
news:[email protected]...
> Bill wrote:
>
>> Your LOL is well taken. I'm not sure whether one needs the numbers in
>> between the fractions (like sqrt(2)) for woodworking, nor any negative
>> numbers, imaginary numbers, non-real complex numbers, nor probably any
>> numbers bigger than 500. Maybe that's why those aren't marked on the
>> ruler.
>
> sqrt(2) is useful to find the length of the long side of a 45/45/90
> triangle.
Do you think 1 53/128 would suffice (somebody with good eyesight might be
able to mark it off a ruler with 64th's). I'd do better with a micrometer.
I hope the wood is very stable. :)
Similarly, 1/2/sqrt(3) are the sides of a 30/60/90 triangle.
>
> Numbers bigger than 500 are useful when working in millimetres.
>
> I'm up in Canada and I know a guy who does everything in mm. Although
> the initial conversion of regular North American lumber dimensions to mm
> is a bit of a pain, it makes subsequent math a lot simpler. And of
> course all the Euro stuff just works...
>
> Chris
"Morris Dovey" <[email protected]> wrote in message
news:[email protected]...
> It might depend on what you're doing. The ribs at
>
> http://www.iedu.com/DeSoto/Projects/Stirling/Heat.html
>
> needed to be cut so that the length along the parabola was exactly four
> feet (the mirror width) and with accuracy to provide a good optical focus
> along the entire eight-foot length - and...
>
> ...the tenoned parts shown at the bottom of
>
> http://www.iedu.com/DeSoto/Projects/Bevel/
>
> were for silverware trays with diagonal dividers; these were the divider
> blanks, and they needed to /exactly/ fit (on /both/ ends :) ).
>
> And no, none of the numbers needed were marked on any of my rulers. :)
>
> --
> Morris Dovey
Interesting projects! My wife is supportive of amost any outlay for tools
as long as I build her some "bird-related" stuff (feeders, houses, etc).
Birds
don't tend to be particular beyond a 16th of an inch. :)
Bill
Morris Dovey wrote:
> And no, none of the numbers needed were marked on any of my rulers. :)
>
Am I the only one would rather mark than measure?
Like if I have a piece of trim that needs to fit between A and B,
I don't measure A to B then measure that out on the trim.
I hold up the trim between A and B and mark the trim.
--
-MIKE-
"Playing is not something I do at night, it's my function in life"
--Elvin Jones (1927-2004)
--
http://mikedrums.com
[email protected]
---remove "DOT" ^^^^ to reply
-MIKE- wrote:
> Morris Dovey wrote:
>> And no, none of the numbers needed were marked on any of my rulers. :)
>>
>
> Am I the only one would rather mark than measure?
>
> Like if I have a piece of trim that needs to fit between A and B,
> I don't measure A to B then measure that out on the trim.
> I hold up the trim between A and B and mark the trim.
>
Before some smarta$$ says, what do you do with a 12' piece of crown
molding....
I obviously call a couple friends to come over and hold it in place for
me. duh.
--
-MIKE-
"Playing is not something I do at night, it's my function in life"
--Elvin Jones (1927-2004)
--
http://mikedrums.com
[email protected]
---remove "DOT" ^^^^ to reply
Morris Dovey wrote:
> -MIKE- wrote:
>> Morris Dovey wrote:
>>> And no, none of the numbers needed were marked on any of my rulers. :)
>>
>> Am I the only one would rather mark than measure?
>>
>> Like if I have a piece of trim that needs to fit between A and B,
>> I don't measure A to B then measure that out on the trim.
>> I hold up the trim between A and B and mark the trim.
>
> If it'll help you feel better, I neither marked /nor/ measured for those
> projects - everything was cut from unmarked stock and then assembled as
> cut.
>
> I did have drawings for the silverware trays because the customer needed
> something to sign off on, but the drawings for the parabolic trough came
> along after the fact, to document what had been done.
>
> For stuff I don't need to be fussy about, I've had to switch to a light
> touch with a knife - my eyes just aren't good enough any longer to split
> a pencil mark...
>
Although I wasn't referring to your projects, but to the topic of
rulers, I have done several projects for which I don't remember using
any measuing or marking, but just sort of free-handing it as you go. I
think there ar many times when you need a thing, but have no parameters
or requirement the size of any part of it.
There's a bit of freedom and catharsis in it.
--
-MIKE-
"Playing is not something I do at night, it's my function in life"
--Elvin Jones (1927-2004)
--
http://mikedrums.com
[email protected]
---remove "DOT" ^^^^ to reply
Chris Friesen wrote:
> -MIKE- wrote:
>> -MIKE- wrote:
>>> Morris Dovey wrote:
>>>> And no, none of the numbers needed were marked on any of my rulers. :)
>>>>
>>> Am I the only one would rather mark than measure?
>>>
>>> Like if I have a piece of trim that needs to fit between A and B,
>>> I don't measure A to B then measure that out on the trim.
>>> I hold up the trim between A and B and mark the trim.
>>>
>> Before some smarta$$ says, what do you do with a 12' piece of crown
>> molding....
>> I obviously call a couple friends to come over and hold it in place for
>> me. duh.
>
> What do you do with a 12' piece of crown molding on an outside corner
> that isn't square?
>
> Chris
I can't tell is I "wooshed" you or you, me. :-)
--
-MIKE-
"Playing is not something I do at night, it's my function in life"
--Elvin Jones (1927-2004)
--
http://mikedrums.com
[email protected]
---remove "DOT" ^^^^ to reply
[email protected] (Doug Miller) wrote:
>In article <[email protected]>, alexy <[email protected]> wrote:
>
>>Consider a regular hexagon. Draw line segments from the center of the
>>hexagon to each of the six vertices. These six equal angles at the
>>center must add up to 360, so each is 60.
>
>OK so far...
>
>> Since the triangles are
>>isosceles, and their angles add to 180, they are also equilateral.
>
>.. but you just went astray there.
>
>That's not sufficient to prove that the triangles are equilateral, since the
>angles add to 180 in *all* triangles.
But I also pointed out that the triangle was isosceles. Isosceles and
one 60 degree angle are necessary and sufficient conditions for an
equilateral triangle (in Euclidean geometry).
>
>This may be what you meant to say:
>
>Since the angle at the vertex of each triangle is 60 degrees, the sum of the
>angles at the base is 180 - 60 = 120 degrees. Since each triangle is
>isosceles, the angles at the base are equal, and (since they add to 120)
>therefore also 60 degrees. The triangles are therefore equiangular, and
>therefore equilateral.
That is what I said. You just have to read between the lines <g>.
>
>> So
>>the side of the hexagon is equal to the length of the line form the
>>center to a vertex on the hexagon, which is the radius of the circle.
--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.
alexy <[email protected]> wrote:
>David Nebenzahl <[email protected]> wrote:
>
>>Regarding that business up above about dividing a circle into 3 equal
>>parts using a square, blah blah blah ... maybe I don't remember my
>>geometry so well. Checked back in that table I mentioned in "Proven Shop
>>Tips" for dividing a circle into n equal parts, and sure enough, the
>>number to multiply the diameter of a circle by to divide into 6 equal
>>parts is ... exactly 0.5.
>>
>>So anyone got the proof handy that a hexagon with sides of length s can
>>be inscribed in a circle whose radius equals s? I have my old algebra
>>and calculus books, but no geometry.
>
>An equilateral triangle has equal sides and three 60-degree angles.
>Arrange six of them with sides s in an array with one common vertex,
>and you will have the inscribed hexagon.
Or, put another way:
Consider a regular hexagon. Draw line segments from the center of the
hexagon to each of the six vertices. These six equal angles at the
center must add up to 360, so each is 60. Since the triangles are
isosceles, and their angles add to 180, they are also equilateral. So
the side of the hexagon is equal to the length of the line form the
center to a vertex on the hexagon, which is the radius of the circle.
--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.
On Tue, 19 May 2009 18:07:58 -0700, David Nebenzahl
<[email protected]> wrote:
>Regarding that business up above about dividing a circle into 3 equal
>parts using a square, blah blah blah ... maybe I don't remember my
>geometry so well. Checked back in that table I mentioned in "Proven Shop
>Tips" for dividing a circle into n equal parts, and sure enough, the
>number to multiply the diameter of a circle by to divide into 6 equal
>parts is ... exactly 0.5.
>
>So anyone got the proof handy that a hexagon with sides of length s can
>be inscribed in a circle whose radius equals s? I have my old algebra
>and calculus books, but no geometry.
Well, I havn't seen the proof but dividing a circle into thirds is
very easy with a compass. Set your radius, and keep it there. Draw
the circle, set the compass point on the circle and draw an arc inside
the circle. Repeat 5 times using an intersect as another pivot point.
You will get a perfect 6-petal flower.