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Origami Like Output

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  Origami Like Output Bogus@Embarq.com (Steve) - 2015-09-30 12:19 +0000
    Re: Origami Like Output jdaw1 <jdawiseman@gmail.com> - 2015-09-30 12:40 -0700
      Re: Origami Like Output Bogus@Embarq.com (Steve) - 2015-10-01 13:17 +0000
        Re: Origami Like Output jdaw1 <jdawiseman@gmail.com> - 2015-10-03 16:03 -0700
          Re: Origami Like Output Bogus@Embarq.com (Steve) - 2015-10-04 14:26 +0000
            Re: Origami Like Output mawisegirl@gmail.com - 2015-10-04 08:29 -0700
            Re: Origami Like Output jdaw1 <jdawiseman@gmail.com> - 2015-10-04 09:38 -0700
              Re: Origami Like Output Bogus@Embarq.com (Steve) - 2015-10-04 17:13 +0000
            Re: Origami Like Output Bogus@Embarq.com (Steve) - 2015-10-08 11:50 +0000

#2356 — Origami Like Output

Hi,

   Found an old PostScript program, and spent some time
analyzing it and commenting some of the findings.  If
you delete the new code indicated by lines near %+ marks,
you get black and white output of a quarter of the full
pattern.  Which can be folded to get a result.  The color
version can be modified to print on both sides.  304 as
opposed to 306 in the translate, my printer is a bit off.
It may be too small to fold up easily.  8.5 by 8.5 inch
output.

   I have forgotten why I was interested in this.  Some
sort of folding problem.  Probably from "Scientific
American", "American Scientist", a computer magazine, or
their ilk.  Map folding?  Does not seem to fit.  Origami?
Of what?  Plane folding?  Maybe, it does reduce the area
by two.  Ring any bells?

   However it is a cute folding problem that, in its simple
version, is a bit of fun.

Regards,

Steve N

%! Test page
%+ 28 September 2015
%+ Original file 7 March 1995.
%+    Try and find out what this was for.  And lose all of the
%+ edits...  Original written for a four way symmetry that was
%+ not used.  That explains the negative coordinates and the
%+ 2 2 scale.  Translate to center and .5 .5 scale to show the
%+ entire drawing.  Negate X scale and swap up and down labels
%+ for two sided printing.  Add color for good measure.
 
% 8.5 x 11 = 612 x 792

% Lock x y position to machine pixel

/LPixel { transform round itransform exch
          transform round itransform exch } def

%+ 16 x 16 grid on 612 x 612 point area.

/XYUnit { 38.25 mul exch 38.25 mul exch } bind def % Define grid spacing
/Unit { 38.25 mul } bind def % Define grid spacing

%+ Straight forward diagonal line routines.

/Diag-u { dup 1 add % x y y'
        3 1 roll   % y' x y
   exch dup 1 add  % y' y x x'
        3 1 roll   % y' x' y x
exch XYUnit moveto
exch XYUnit lineto stroke } bind def

/Diag-d { dup 1 sub % x y y'
        3 1 roll   % y' x y
   exch dup 1 add  % y' y x x'
        3 1 roll   % y' x' y x
exch XYUnit moveto
exch XYUnit lineto stroke } bind def

/Diag-U { dup 2 add % x y y'
        3 1 roll   % y' x y
   exch dup 2 add  % y' y x x'
        3 1 roll   % y' x' y x
exch XYUnit moveto
exch XYUnit lineto stroke } bind def

/Diag-D { dup 2 sub % x y y'
        3 1 roll   % y' x y
   exch dup 2 add  % y' y x x'
        3 1 roll   % y' x' y x
exch XYUnit moveto
exch XYUnit lineto stroke } bind def

/Box { XYUnit moveto  0 2 XYUnit rlineto  2 0 XYUnit rlineto
     0 -2 XYUnit rlineto  closepath stroke } bind def

% Origami Test 612 ÷ 16 = 38.25

 1 1 idtransform pop abs setlinewidth
%+ 16 x 16 degug grid.
% 38.25 38.25 LPixel 577.55 { dup 0 moveto 612 lineto stroke } for
% 38.25 38.25 LPixel 612.5 { dup 0 exch moveto 612 exch lineto stroke } for

   2 2 idtransform pop abs setlinewidth

gsave
%   8 8 XYUnit translate
%+
304 304 translate .5 .5 scale % Show four way symmetry.

2 2 scale
   newpath -8 -8 XYUnit moveto  -8 8 XYUnit lineto  8 8 XYUnit lineto
   8 -8 XYUnit lineto  closepath clip  newpath

   [10 3 3 3 3 3 ] 0 setdash
%+
0 1 0 setrgbcolor

   -6 Unit 4 Unit 6 Unit { dup -8 Unit moveto 8 Unit lineto stroke } for
   -6 Unit 4 Unit 6 Unit { dup -8 Unit exch moveto 8 Unit exch lineto stroke } for

   -3 7 Diag-u  5 7 Diag-u
   -5 7 Diag-d  3 7 Diag-d
   -1 5 Diag-u  7 5 Diag-u
   -7 5 Diag-d  1 5 Diag-d
   -2 4 Diag-d  6 4 Diag-d
   -8 2 Diag-u  0 2 Diag-u
   -4 2 Diag-d  4 2 Diag-d
   -6 0 Diag-u  2 0 Diag-u

   180 rotate
   -3 7 Diag-u  5 7 Diag-u
   -5 7 Diag-d  3 7 Diag-d
   -1 5 Diag-u  7 5 Diag-u
   -7 5 Diag-d  1 5 Diag-d
   -2 4 Diag-d  6 4 Diag-d
   -8 2 Diag-u  0 2 Diag-u
   -4 2 Diag-d  4 2 Diag-d
   -6 0 Diag-u  2 0 Diag-u

   180 rotate
   [10 4 ] 0 setdash
%+
1 0 0 setrgbcolor

   -2 8 Diag-D 6 8 Diag-D
   -3 7 Diag-D 5 7 Diag-D
   -7 5 Diag-U 1 5 Diag-U
   -6 4 Diag-U 2 4 Diag-U
   -7 3 Diag-D 1 3 Diag-D
   -8 2 Diag-D 0 2 Diag-D
   -4 2 Diag-U 4 2 Diag-U
   -3 1 Diag-U 5 1 Diag-U

   -5 7 Box 3 7 Box
   -9 3 Box -1 3 Box 7 3 Box
   -5 -1 Box
   180 rotate
   -2 8 Diag-D 6 8 Diag-D
   -3 7 Diag-D 5 7 Diag-D
   -7 5 Diag-U 1 5 Diag-U
   -6 4 Diag-U 2 4 Diag-U
   -7 3 Diag-D 1 3 Diag-D
   -8 2 Diag-D 0 2 Diag-D
   -4 2 Diag-U 4 2 Diag-U
   -3 1 Diag-U 5 1 Diag-U

   -5 7 Box 3 7 Box
   -9 3 Box -1 3 Box 7 3 Box
   -5 -1 Box
grestore

/Helvetica findfont 12 scalefont setfont

   [10 3 3 3 3 3 ] 0 setdash
%+
0 1 0 setrgbcolor

   4 19 XYUnit moveto 8 19 XYUnit lineto stroke
   9 19 XYUnit moveto (Fold down) show

   [10 4 ] 0 setdash
%+
1 0 0 setrgbcolor

   4 17 XYUnit moveto 8 17 XYUnit lineto stroke
   9 17 XYUnit moveto (Fold up) show

showpage

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#2357

There are two tessellations superimposed, and at the same frequency. That is a tool for a dissection: two identical squares ↔ square and octagon. It shows how to cut two squares into pieces that can be rearranged into the square and octagon.

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#2358

jdaw1 <jdawiseman@gmail.com> writes:
>There are two tessellations superimposed, and at the same frequency. That i=
>s a tool for a dissection: two identical squares =E2=86=94 square and octag=
>on. It shows how to cut two squares into pieces that can be rearranged into=
> the square and octagon.

Hi,

   Thank you for the reply.  Not sure I see that.  Do you have an
illustration of that process?  Okay, maybe I do see that.  Odd.
I would never have seen that.  Of couse that was not what I was
looking for.  Thanks again, a completely different insight.

   Also, did you fold the example?  No cutting was originally
mentioned.  A different operation.

Cheers,

Steve N.

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#2361

> Also, did you fold the example?  No cutting was originally
> mentioned.  A different operation.

No, I did not fold. The folding instructions make no sense: how can one fold up all eight sides of all the octagons? Perhaps the folding instructions are wrong?

I own a copy of ‘Dissections: Plane and Fancy’, by Greg N. Frederickson
http://www.amazon.co.uk/exec/obidos/ASIN/0521525829/jdaw 
http://www.amazon.com/exec/obidos/ASIN/0521525829/jdaw01 
which makes much use of the overlapping tessellation technique. It does seem to be Octagon and square ↔ two squares.

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#2362

Hi,

jdaw1 <jdawiseman@gmail.com> writes:
>> Also, did you fold the example?  No cutting was originally
>> mentioned.  A different operation.
>
>No, I did not fold. The folding instructions make no sense: how can
>one fold up all eight sides of all the octagons? Perhaps the folding
>instructions are wrong?

   If you print out the quarter size, black and white, version (not
using the code implied by %+ marks) it does fold up.  I have done it
a number of times.  It is a bit fiddy and relies on some give and take
in the paper.  It goes from a square sheet of paper to an odd square
object half the size of the starting square.  One side is four squares.
The other side has a square in the middle, two triangles, four rectangles
partially covered by the triangles, and two little squares left over.
The coloered, four times more complicated, version I have not tried,
though I guess I now will have to try it.  The folds look too small
to execute on an eight and a half inch square piece of paper.

   The fact that all edges of the octagon fold up and all edges of
the large squares fold down is what got me into trying this out.
I think, 1995 has faded quite a bit.  It looks impossible, but
isn't.  So you almost have to try it out.  Well, me back then
anyway.

>
>I own a copy of = E2= 80= 98Dissections: Plane and Fancy= E2= 80= 99,
>by Greg N. Frederickson
>http://www.amazon.co.uk/exec/obidos/ASIN/0521525829/jdaw=20
>http://www.amazon.com/exec/obidos/ASIN/0521525829/jdaw01=20
>which makes much use of the overlapping tessellation technique. It does see=
>m to be Octagon and square = E2= 86= 94 two squares.

   Interesting.  Will keep an eye out for that sort of thing now.

Regards,

Steve N.

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#2363

Later, I will try the original version for you. 

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#2364

Later I'll try the original code and folding. If you have it handy please post it to save my regenerating it from the amended code. 

(Previous post, deleted, from me while sitting at somebody else's computer. Hence deleted.)

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#2365

Hi,

jdaw1 <jdawiseman@gmail.com> writes:
>Later I'll try the original code and folding. If you have it
>handy please post it to save my regenerating it from the amended code. 

   Sure thing.  Hope you enjoy folding it up.

Regards,

Steve N.

%! Test page
% 8.5 x 11 = 612 x 792

% Lock x y position to machine pixel

/LPixel { transform round itransform exch
          transform round itransform exch } def

/XYUnit { 38.25 mul exch 38.25 mul exch } bind def % Define grid spacing
/Unit { 38.25 mul } bind def % Define grid spacing

/Diag-u { dup 1 add % x y y'
        3 1 roll   % y' x y
   exch dup 1 add  % y' y x x'
        3 1 roll   % y' x' y x
exch XYUnit moveto
exch XYUnit lineto stroke } bind def

/Diag-d { dup 1 sub % x y y'
        3 1 roll   % y' x y
   exch dup 1 add  % y' y x x'
        3 1 roll   % y' x' y x
exch XYUnit moveto
exch XYUnit lineto stroke } bind def

/Diag-U { dup 2 add % x y y'
        3 1 roll   % y' x y
   exch dup 2 add  % y' y x x'
        3 1 roll   % y' x' y x
exch XYUnit moveto
exch XYUnit lineto stroke } bind def

/Diag-D { dup 2 sub % x y y'
        3 1 roll   % y' x y
   exch dup 2 add  % y' y x x'
        3 1 roll   % y' x' y x
exch XYUnit moveto
exch XYUnit lineto stroke } bind def

/Box { XYUnit moveto  0 2 XYUnit rlineto  2 0 XYUnit rlineto
     0 -2 XYUnit rlineto  closepath stroke } bind def

% Origami Test 612 ÷ 16 = 38.25

 1 1 idtransform pop abs setlinewidth
% 38.25 38.25 LPixel 577.55 { dup 0 moveto 612 lineto stroke } for
% 38.25 38.25 LPixel 612.5 { dup 0 exch moveto 612 exch lineto stroke } for

   2 2 idtransform pop abs setlinewidth

gsave
%   8 8 XYUnit translate
2 2 scale
   newpath -8 -8 XYUnit moveto  -8 8 XYUnit lineto  8 8 XYUnit lineto
   8 -8 XYUnit lineto  closepath clip  newpath

   [10 3 3 3 3 3 ] 0 setdash
   -6 Unit 4 Unit 6 Unit { dup -8 Unit moveto 8 Unit lineto stroke } for
   -6 Unit 4 Unit 6 Unit { dup -8 Unit exch moveto 8 Unit exch lineto stroke } for

   -3 7 Diag-u  5 7 Diag-u
   -5 7 Diag-d  3 7 Diag-d
   -1 5 Diag-u  7 5 Diag-u
   -7 5 Diag-d  1 5 Diag-d
   -2 4 Diag-d  6 4 Diag-d
   -8 2 Diag-u  0 2 Diag-u
   -4 2 Diag-d  4 2 Diag-d
   -6 0 Diag-u  2 0 Diag-u

   180 rotate
   -3 7 Diag-u  5 7 Diag-u
   -5 7 Diag-d  3 7 Diag-d
   -1 5 Diag-u  7 5 Diag-u
   -7 5 Diag-d  1 5 Diag-d
   -2 4 Diag-d  6 4 Diag-d
   -8 2 Diag-u  0 2 Diag-u
   -4 2 Diag-d  4 2 Diag-d
   -6 0 Diag-u  2 0 Diag-u

   180 rotate
   [10 4 ] 0 setdash
   -2 8 Diag-D 6 8 Diag-D
   -3 7 Diag-D 5 7 Diag-D
   -7 5 Diag-U 1 5 Diag-U
   -6 4 Diag-U 2 4 Diag-U
   -7 3 Diag-D 1 3 Diag-D
   -8 2 Diag-D 0 2 Diag-D
   -4 2 Diag-U 4 2 Diag-U
   -3 1 Diag-U 5 1 Diag-U

   -5 7 Box 3 7 Box
   -9 3 Box -1 3 Box 7 3 Box
   -5 -1 Box
   180 rotate
   -2 8 Diag-D 6 8 Diag-D
   -3 7 Diag-D 5 7 Diag-D
   -7 5 Diag-U 1 5 Diag-U
   -6 4 Diag-U 2 4 Diag-U
   -7 3 Diag-D 1 3 Diag-D
   -8 2 Diag-D 0 2 Diag-D
   -4 2 Diag-U 4 2 Diag-U
   -3 1 Diag-U 5 1 Diag-U

   -5 7 Box 3 7 Box
   -9 3 Box -1 3 Box 7 3 Box
   -5 -1 Box
grestore

/Helvetica findfont 12 scalefont setfont

   [10 3 3 3 3 3 ] 0 setdash
   4 19 XYUnit moveto 8 19 XYUnit lineto stroke
   9 19 XYUnit moveto (Fold down) show

   [10 4 ] 0 setdash
   4 17 XYUnit moveto 8 17 XYUnit lineto stroke
   9 17 XYUnit moveto (Fold up) show

showpage

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#2366

Steve writes:

>The coloered, four times more complicated, version I have not tried,
>though I guess I now will have to try it.  The folds look too small
>to execute on an eight and a half inch square piece of paper.

   Well I got that version to fold up.  Parts of it look forced, so
there may be errors.  The small folds do make it harder than the
larger version.  It would probably be easier to fold four sections
that size, unfold then, tape them together and refold.

   General comments on printing out something to be folded, like
a paper airplane, origami, or a greeting card.  Printing the fold
up and fold down lines in contrasting colors does help.  Printing
on both sides of the paper helps when properly aligned.  When
misaligned, it can make things look rather crude.  If I do something
like this again, I think I will reverse the line colors rather
than the folding instructions.  Keeping the colors constant is
prettier, but is probably more error prone.

Regards,

Steve N.

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