Drawn and Quartered

[Dean Ballard]

A geometry dissection puzzle:

I have a two-dimensional shape that I cut into
four identical pieces, each of which is similar
to the original shape. (all corresponding angles
equal, all corresponding sides in proportion)

For what set of shapes can this be done?

- Dean Ballard

[mike]

In article <u4OdnUvYTJLCn2bX...@giganews.com>,
deanb...@comcastxxx.net says...

To start with:

Any triangle
any paralellogram
a three-cube 'L' shape
a six-triangle 'sphinx' shape
numerous fractals and fractal sponges

That should be enough to go on for now, although I have probably
forgotten some others...

Mike

[Dean Ballard]

Mike wrote:
>> A geometry dissection puzzle:
>>
>> I have a two-dimensional shape that I cut into
>> four identical pieces, each of which is similar
>> to the original shape. (all corresponding angles
>> equal, all corresponding sides in proportion)
>>
>> For what set of shapes can this be done?
>>
> To start with:
>
> Any triangle
> any paralellogram
> a three-cube 'L' shape
> a six-triangle 'sphinx' shape
> numerous fractals and fractal sponges
>
> That should be enough to go on for now, although I have probably
> forgotten some others...
>
> Mike

Sounds like you've encountered this problem before.

I found all of these but the sphinx (and a couple of others as well),
but have not found a fractal shape that fits the bill. Even ignoring
the two-dimensional requirement, could you suggest a fractal
that works?

- Dean

[g.r...@iit.cnr.it]

On Nov 11, 7:28 pm, "Dean Ballard" <deanbal...@comcastxxx.net> wrote:
> A geometry dissection puzzle:

> For what set of shapes can this be done?

Apart those already mentioned, we can add (using a monospace font):
the tetromino L:
XXX
X
the pentomino P:
XXX
XX
the hexomino:
XXXX
XX
the enneomino:
XXXXXX
XXX

If instead of 4 copies we request, say, 16 copies, then other shapes
can be used, like
for example:
X
XX
X
and
XXXX
X
and
XXXX
XXX
and so on.

giovanni
--
http://anagram.it anagrams, alphametics, arithmogriphs,..

[Philippe 92]

Dean Ballard a ï¿œcrit :

Look for "rep-tiles"

for instance
<http://www.meden.demon.co.uk/Fractals/reptiles.html>

Regards.

--
Philippe C., mail : chephip, with domain free.fr
site : http://mathafou.free.fr/ (mathematical recreations)

[Dean Ballard]

"Philippe 92" <nos...@free.invalid> wrote in message news:mn.5d4d7d9b9...@free.fr...
> Dean Ballard a �crit :

Thanks for the pointers!
- Dean

[Dean Ballard]

<g.r...@iit.cnr.it> wrote in message news:1ed3bfc4-ec72-434d...@c3g2000yqd.googlegroups.com...

Thanks!
- Dean

[Jon°]

really.

[Philippe 92]

Dean Ballard wrote :
> <g.r...@iit.cnr.it> wrote

>> "Dean Ballard" <deanbal...@comcastxxx.net> wrote:
>>> A geometry dissection puzzle:
>>> For what set of shapes can this be done?
>>
>> Apart those already mentioned, we can add (using a monospace font):
>> the tetromino L:

>> the pentomino P:

>> the hexomino:
>> XXXX
>> XX
>> the enneomino:
>> XXXXXX
>> XXX

These two are just stretch of the L triomino
You can go on and get an infinite family of these stretched solutions.

>> If instead of 4 copies we request, say, 16 copies, then other shapes
>> can be used, like for example:

>> T tetromino
>> L pentomino
>> an heptomino
>> and so on.

As 16 is 4^2, any of the 4-reps is also a 16-rep, then "other shapes
can be used", that is in addition to all the ones we know as 4-reps...
The gap from 4 to 16 can be filled, of course : find the 5-reps..

>>
>> giovanni

> Thanks!
> - Dean

Hi Dean,

I wondered what was you answer to giovanni, as there was not a word
added, till I realized that you answered after giovanni's usenet
signature !
The usenet signature is all what is after '-- ', and is not part of
the message body :
It should never be quoted,
It is rendered differently with true newsreaders
You have a problem with your OE then (doesn't quote properly).
I requoted to correct.

Back to the rep-tiles question.

I mentionned previously a web site mainly about fractal rep-tiles.
I have just added a page about rep-tiles in general on my own web site.
<http://mathafou.free.fr/pbm_en/sol218.html>
Gives many non fractal ones.