divide square into 5 equal parts?

[Darren]

I know that you can divide a square into 5 equal parts simply by
slicing it into equal vertical or horizontal rows, but I am curious if
there are any other solutions, perhaps ones that have a part not
touching the outer square?

[Michael Stemper]

By "equal parts" do you simply mean "having equal areas", or do they
need to be congruent?

--
Michael F. Stemper
#include <Standard_Disclaimer>
If it's "tourist season", where do I get my license?

[William Elliot]

Yes.

[Darren]

On Feb 26, 9:44 pm, mstem...@walkabout.empros.com (Michael Stemper)
wrote:

> In article <fc8bb6d7-ade6-4bcd-b1e9-eb397bfe2...@9g2000vbq.googlegroups.com>, Darren <anon5...@yahoo.com> writes:
>
> >I know that you can divide a square into 5 equal parts simply by
> >slicing it into equal vertical or horizontal rows, but I am curious if
> >there are any other solutions, perhaps ones that have a part not
> >touching the outer square?
>
> By "equal parts" do you simply mean "having equal areas", or do they
> need to be congruent?
>
> --
> Michael F. Stemper
> #include <Standard_Disclaimer>
> If it's "tourist season", where do I get my license?

I meant congruent.

[Darren]

Thanks for the informative reply William :)

Can you give an example?

[Frederick Williams]

I wonder if he'll reply "Yes."

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

[William Elliot]

I suppose, if you can't even with he encouragement I gave.

Fill a square with a central square and four surrounding rectangles.
Adjust the length of the side of the central square so its area
equals the area of each of the rectangles.

What size think cap do you wear? ;-)

[Darren]

Ah, well I suppose I didn't define what equal means. I meant congruent
shapes. For example a square can be divided into 4 congruent squares,
or 4 congruent triangles, etc.

I doubt that it is possible with 5 pieces, however I am left wondering
what the impossibility proof would look like?

[Peter Webb]

"Darren" <anon...@yahoo.com> wrote in message
news:e24bcbf1-48f5-4c9e...@m2g2000vbc.googlegroups.com...

_______________________________________________
Well, I can make a start. With 5 triangles you have 15 sides. Unless one
triangle's sides exactly overlap 2 or more other triangles sides, the shape
that results when you put them together will have an odd number of sides. I
don't think (but cannot easily prove) that you can't have one side of one
triangle lining up with two separate sides of different triangles. The two
overlapping sides must be of equal length, as they must be the same side of
the triangle (or else they would be too short to overlap the longest side of
another, due to the triangle inequality). This very significantly reduces
the possible solutions, and given we have already used 3 of the 5 triangles
it should be possible to show the other two cannot be fitted in.

A way of proving that a square cannot be constructed is to prove that the
figures cannot be assembled in such a ways as to produce 4 right angles
corners. Unless at some corner there are three triangles touching, the only
possible solutions involve right angled triangles (as if there is only one
triangle at the 90 degree corner, it has to be right angled, and if two
different corners of a triangle add up to 90 degrees then the remaining
angle has to be 90 degrees).

Similar arguments apply shapes other than triangles; you have to have four
combinations of angles which each add up to 90 degrees. This is a
combinatoric rather than geometrical argument, and it may well be
sufficient.

So I don't think an impossibility proof would be that hard. Not that I have
one.

[Darren]

On Mar 1, 5:02 am, "Peter Webb" <r.peter.webb...@gmail.com> wrote:
> "Darren" <anon5...@yahoo.com> wrote in message

It sounds like you are restricting yourself to polygons, I was
thinking of more general shapes that you could cut out of a piece of
paper, but not sure how to define that mathematically?

[Michael Stemper]

In article <jimvts$rc8$1...@news.albasani.net>, "Peter Webb" <r.peter...@gmail.com> writes:
>"Darren" <anon...@yahoo.com> wrote in message news:e24bcbf1-48f5-4c9e...@m2g2000vbc.googlegroups.com...

>> On Mon, 27 Feb 2012, Darren wrote:

>> > > > I know that you can divide a square into 5 equal parts simply by
>> > > > slicing it into equal vertical or horizontal rows, but I am curious
>> > > > if there are any other solutions, perhaps ones that have a part not
>> > > > touching the outer square?

I was wondering if it might be possible to do with some Polyominoes:
<http://mathworld.wolfram.com/Polyomino.html>, but your argument makes
me think that wouldn't work. Even it it's not rigorous, it's an ingenious
approach to the problem.

--
Michael F. Stemper
#include <Standard_Disclaimer>

The name of the story is "A Sound of Thunder".
It was written by Ray Bradbury. You're welcome.

[Peter Webb]

"Darren" <anon...@yahoo.com> wrote in message

news:6ddc132a-6f36-4c2e...@hs8g2000vbb.googlegroups.com...

________________________________________
I suspect it is sufficient to prove just for polygons. If you have a shape
with curved sides with this property, you can construct a polygon with the
same property. If the shapes don't overlap, and they fill the square, then
the curved sides must exactly line up and must be on the inside of the
square. Just replace these curved segments with straight line segments and
you can construct a polygon with the desired property.