





Mathematical Paper Folding Exhibit
 Introduction
 Interview with Dr Robert Lang
 Single Cut Origami
 Curved Crease Origami
 Business Card Menger Sponge
 Links to Other Sites and Resources 
Menger
Sponge
Text and Diagrams by Dr. Jeannine Mosely


Photo
by Ravi Apte 
Menger's Sponge
Menger's sponge  named for its inventor Karl Menger (19021985)
and sometimes wrongly called Sierpinski's Sponge  is a fractal
solid that can be described as follows. Take a cube, divide it into
27 (3 x 3 x 3) smaller cubes of the same size; now remove the cube
in the center of each face plus the cube at the center of the whole.
You are left with a structure consisting of the eight small corner
cubes plus twelve small edge cubes holding them together. Now, imagine
repeating this process on each of these remaining 20 cubes. Repeat
again. And again, ad infinitum ...


More
information about fractals may be seen on the Institute’s
site at this link. 
The Business Card Menger
Sponge Project
The primary goal of the Business Card Menger Sponge Project was
to build a depth 3 approximation to Menger’s Sponge as shown
above, out of 66,048 business cards. This can be done by building
8000 business card cubes of 6 cards each, linking them together
and using the additional cards to panel the 18,048 exterior faces
of the sponge, giving a more pleasing finish to the final structure.
In order to build the sponge, I devised a decomposition of the overall
structure into simple units that almost anyone can learn to make,
which can then be assembled into the whole. The finished sponge
measures slightly over 54 inches (140 cm) on each side and weighs
about 150 pounds (70 kg).
The intention of the project was to be a communal effort: Initially
I made a level 2 sponge using 3456 cards. That took me about 30
hours and I imagined I could finish a level 3 sponge in about 600
hours. I'm pretty fast and could probably finish it by myself, but
then I couldn't claim workman's compensation when I got repetitive
motion syndrome. I decided I would rather do it with a little help
from my friends.
My idea was that a model of a level 3 Menger Sponge would be built
out of business cards one cube at a time, with many folders helping
by precreasing the business cards. As the structure got larger
there would be room for perhaps as many as 48 people to work on
it simultaneously. They would still be the major bottle neck, since
assembly and paneling alone take more than half the construction
time. I calculated it would take such a group working together around
50100 hours from start to finish, but I doubted I could find enough
dedicated volunteers to do it this way.
Instead, I determined to build the sponge with the help of many
volunteers around the country building subassemblies that could
be brought or shipped to a central location for final assembly.

Subassemblies of the Business
Card Sponge
It was not immediately obvious what form the subassemblies should
take, since they would have to be joined together after they were
built. Building level 1 sponges is easy and takes about an hour
to ninety minutes. 20 level 1 sponges must be assembled to make
a level 2 sponge, and you cannot just pretend that each level 1
is a simple cube that can be easily linked. Each face of a level
1 sponge has 8 cubes which have to be linked simultaneously to those
of its neighboring level 1. Trying to link level 2 sponges is even
worse.
I realized that I would have to devise a way to join the subassemblies
together by building linking cubes between them and, after several
false starts, I devised such a decomposition based on a subunit
that I call a "tripod".


A tripod is a block of four linked cubes, where a corner cube
has three cubes attached to three adjacent faces. A sponge of
any level can be broken into pieces that are combinations of one,
two, three, four, six or eight tripods joined together by additional
cubes.



A level 1 sponge can be made by linking 4 tripods together with
4 additional cubes.



A
level 2 sponge can be made by linking 32 tripod modules: 4 single
tripods, 12 double tripods, 12 triple tripods and 4 quadruple tripods,
with 80 additional cubes. 


A level 3 sponge can be made by linking 448 tripod modules: 4
single tripods, 84 double tripods, 156 triple tripods, 84 quadruple
tripods, 48 double double tripods and 72 double triple tripods,
with 1600 additional cubes.


This tripod decomposition has many
desirable features. The modules are small and can be completed quickly,
making each volunteer's contribution relatively small. The modules
are regular and easy to teach, understand and build correctly. They
can be paneled after construction. Linking modules together is easy
and does not require reaching into small, confined spaces. After assembly,
the faces of the linking cubes are easily accessible for paneling.
The tripods comprise a large percentage (80 percent) of the final
sponge.


photo by Ravi Apte 
Where did I get all the business
cards?
A lot of people are curious as to where I got the 66,000 business
cards needed to build the Menger's Sponge model. In February, 1995
the company I worked for changed its name. My colleagues gave me their
business cards. In August, 1995, my employers moved. My colleagues
gave me their business cards. After we arrived at our new location,
the printers sent us business cards with the new address  and the
old zipcode. I got to keep them. This accounts for the 48,000 cards
needed for the base, and some to spare.
The 18,048 colored cards needed for the exterior paneling were provided
by Jay Meddaugh of American Wholesale Thermographers of Woburn, Massachusetts,
who saved bad print runs of colored business cards for me for several
months.

Structural Considerations for
the Business Card Sponge
The depth 0 sponge is a single business card cube. When made from
standard American business cards, it measures 2" x 2" x
2". The depth 1 sponge measures 6" x 6" x 6",
the depth 2 sponge measures 18" x 18" x 18", and the
depth 3 sponge measures 54" x 54" x 54", or 4.5 feet
or 1.37 meters on a side. Not outrageously large! (But it won't fit
through my doorway or in my van!)
Early on I resolved to "panel" the exterior faces of the
sponge. I wanted to do this for several reasons: It would look better
with all the flaps hidden. Since I would use six different colors
for each of the six different planes the faces lie in, this structure
would be more visible. The finished structure would be stronger, though
also heavier. And, finally, it would let the individual builders incorporate
their own business cards, adding their "signatures" to the
work. Paneling would require an extra 18,048 cards, for a total of
66,048 cards.
My largest concern was for structural integrity. 66,000 business cards
are heavy. The cards that I had were of slightly higher quality than
the average business card, and so a little heavier, but also a little
stronger and easier to fold. They weigh about 18 ounces for a box
of 500. This would make the projected weight of the finished sponge
about 150 pounds (68 kg). Once built, it would not be possible to
lift it off its table. I would have to build a pallet to hold it.
But if it was that heavy, would the model even support its own weight?
Some calculations suggested that it would. The bottom layer consists
of 512 cubes which must support the weight of the entire structure.
This would make the average load born by the bottom cubes of 8000/512
= 15.625 cubes. Further consideration revealed, however, that the
middle layer is the sparsest layer, containing only 64 cubes, and
that these must bear the weight of whole half the structure above.
That would be a load of slightly more than 1 pound (.5 kg) per individual
cube.
I noted, however, that many of the cubes on the bottom layer would
have empty space above them and wouldn't supporting anything at all.
Only 64 cubes on the bottom are connected through unbroken columns
to the top of the sponge. Of course, they could still distribute load
laterally to their neighboring cubes (which chiefly serve to keep
them from "exploding"), but as a worst case, we can imagine
that these 64 cubes must carry the weight of the entire structure,
with a load of around 2 pounds (1 kg) each.
So I built a cube and started stacking 1 pound weights on it. (Okay,
so they were 18 ounce weights.) After the stack was 10 boxes high,
the cube began to show signs of stress.
The other potential structural problem involved shear. Think of the
level 3 sponge as a ring of eight level 2 sponges on the bottom with
four level 2 sponges connecting them to another ring of eight level
2 sponges on top. Four of the top sponges are supported by a column
of two sponges beneath them, but the other four are supported only
through their *vertical* connections to the others. Their weight might
tend to cause them to crack off. Fortunately, each of these suspended
sponges, weighing around 7.5 pounds (3.5 kg), are joined to their
neighbors by 64 mutually connected faces on each side.
I believed the structure could support itself.
After some nine years of effort, involving several hundred folders
around the country, the Business Card Menger Sponge was completed.
In June 2005 it was exhibited at the Origami USA annual convention
in Manhattan. Since then it has continued to reside in my garage,
though I hope some day to donate it to a museum.
A level 4 sponge would require almost a million cards and weigh over
a ton. I do not believe it could support its own weight  so a level
3 is the biggest sponge we can hope to build. 

photo by Ravi Apte 
Make Your Own Business
Card Cubes
Download
full instructions for how to build your own business card sponge.
To make a cube out of six business cards, first take
two cards and place them across each other at right angles, centering
them as nearly as possible. Fold the flaps of the bottom card down
over the top card. Turn them over and repeat. Pull the two cards
apart. Six of them can be assembled as shown below to make a cube.
All flaps must be on the outside of the finished cube.
Two cubes can be linked by positioning them as shown. Take the two
flaps on one face and tuck them under the corners of the two flaps
on the adjacent face of the other cube. The joint is surprisingly
strong! You can keep adding cubes like this to build any structure
you can think of. 

Go
to Dr Mosely’s website for more information 
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