It's without permission
from me and from H. Strobl forbidden
to commercially use the models and/or diagrams on this page.
If you use these instructions please let me know.
I would like to know if the instructions are clear enough and
if there are things that could be better. I'm always open to suggestions.

Preface
Knotology is the name given by Heinz Strobl to the process of
making 3Dshapes from strips of paper, for example old fashioned
ticker tape. Why is it called knotology? Here is the answer from
the inventor himself:

I (ironically) introduced the word Knotologie (Knotology in
English) when I started folding with strips making real knots
with tape that form into flat regular pentagons.
In later techniques with equilateral triangles the tape loops
into threedimensional knots. I kept the name for the latest
models with isosceles triangles although there is no knot involved
at all.

If you like to fold real knots, you'll enjoy
Sphere94. (496 kB)
Sphere94 folded by Rosa
Click on the picture to see instructions.

I had the privilege to meet Heinz at the Dutch Origami Convention
2000 in Veldhoven, Holland.
The origami convention in Veldhoven, 1416 april 2000
Select the pictures for a bigger version
He made great things of this tape, an taught everyone who wanted
three of his models. I was so enthousiastic that I also asked to
learn a fourth model. With trial and error we succeeded this.
When I came home from the convention I started to make notes
of the things I learned and made a fifth model by myself. I had
also heard from someone that Heinz did not have the time to write
his techniques down. I asked Heinz if it was o.k. to write something
about it on this website, and he agreed.
Here you find the answer of Heinz when I asked him if he created
all the models by himself.
All the models I exhibited and taught in Veldhoven
were "created", designed and folded by me.
But ... especially the spherelike models are mostly geometry,
and everybody who has an understanding of platonic solids will come
to the same solution(s), when experimenting with strips folded into
isosceles triangles.
So I consider these kind of models rather as "discovered"
than as "created".
I know that some of the spherelike models were discovered independently
by other folders. For instance, the model made of 6 differently
colored strips of length 12 that I taught in Veldhoven (and some more)
was (were) independently discovered by Tomoko Fuse and/or
Wilhelm Möller (both of them were inspired by me to focus
attention on folding tapes).
As far as I know, only Tomoko Fuse published instuctions for
origami folded with tape in a book (cf. ISBN 448087272647, (C)1995,
it is printed in Japanese, so I can give you no title). I only published
"Knotologie" in some origami magazines and convention
books.
I collected models for the exhibition that I did not see anwyhere else.
Of course I cann not be sure that other folders have dicovered some of them
independently.
The yellow and lightblue "spheres" belong to
one family. They are made of 12 strips, assembled in the same
manner and differ only in the orientation of some creases and/or
which parts are sunk or not. Tomoko Fuse has published some models
in the above mentioned book that belong to this family (I did not show
these in my exhibition).
The black and darkblue spheres belong to another family made of 10 strips
and are harder to assemble (but not as hard as "the blue little
object with visually 4 cubes" where they are derived from. These and
all the other Knotologie that is not sperelike no one else has discovered
yet (I am rather sure. Or should I say created, because they can not be
derived from platonic solids).

Here you find general instructions for knotology and also
a few projects. Because I'm not so good in drawing diagrams for
this (lots of 3D) I made some pictures with my normal camera
and scanned them. Because I don't want to have the pictures too
many bytes, I compressed them. I hope they are still good enough
to work with.
If you use these instructions please let me know. I would
like to know if the instructions are clear enough and if there are
things that could be better. I'm always open to suggestions.

Back to contents
Method
How does Knotology work?
Knotology is the creation of threedimensional object by weaving
or plaiting of strips of paper. These strips are folded in squares
and triangles (half a square). In most cases these objects are
shaped as spheres or cylinders. In the pictures you can see a
few examples of these objects. Click on the pictures if you like
to see a bigger version.
The origami convention in Veldhoven, 1416 april 2000
To weave/plait these objects you need strips of paper of equal
width. De best material for this, is the oldfashioned ticker
tape. You can also make your own strips from heavier paper, for
instance 150 gr/m2. I recommend strips of 2cm width.

Back to contents
Method
How can I make a
strip with squares?

 The length of the strip is the desired amount of squares multiplied
by the width of the paper. (eg. For a strip with 12 squares you
need 2cm x 24cm)
 Cut 2 strips at the desired length.
 Put them together at a right angle and wrap them around each other.
You have to be sure that the width of the blue strip is just something
wider than the distance to the first fold of the red strip,
and also the other way around.
 Keep on folding the strips around each other. With every fold you
create a square, first a red, then blue, red, blue etc.*)
 If everything's fine, the strip at the end will be just a
little bit too short to make the last square. The first and last
square of the strip is hidden in the model. This is easier if
these are just a little bit shorter than the rest of the
squares.

*) In Holland this is called a "muizentrapje"
(mouses' staircase), in Germany a "hexentrappe"
(witches' staircase).
Sebastian Kirch calls it in English "witches' staircase"
and Heinz told me he heard English and American people call
it "witches ladder". I will call it "witches
ladder". Please let me know if you know this is the right
word (or the wrong one).

How do I cut a strip?
It's also possible to create very long witches ladder and cut
the strips later to the right length. The method used by Heinz
is this: Fold the strip at the place where you wish to cut. Cut
the fold  two layers of paper  from the strip.
The advantage of this method is that the first and last square
of the strips are automatically a little bit smaller than the
rest of the squares.

How do I make the plaiting
easer?
When you have a difficult slit and you can't get the end
of the strip through it, you can take a spare strip and put it
in the slit from the other side. You can use that strip as a
guide for the real strip.
It's also possible to narrow the beginning and/or the end of
a strip: the first and the last square are not visible in the
end, and it makes the plaiting easier.

Back to contents
Projects
The first project: A Wobbling Wall
of 9 cubes.
This wall is made of 3 cubes wide by 3 cubes long. These cubes
are connected with each other with strips, in such a way that
the wall is an action model. The 9 cubes can be close to each
other, but it's also possible that there are 4 cubeshaped holes
in the wall. The animation explain better what I want to say.
The animation
can be stopped by selecting the stopbutton of your browser.
I don't see any animation
To make this wall you need 3 strips of 6 squares for every cube. You
need 3 strips to connect the cubes with each other, one of length
10 and two of length 7.
You can choose the length of the strips, fold them into witches
ladders and cut them at the desired length.
If you use ticker tape, I recommend to use 8 strips of 18 squares and
2 strips of 23 squares. You can cut the first 8 of these strips in 3 parts
to make 8 cubes (8 times 666), the 2 strips of 23 squares you can divide
like this: 6710 and 667, material for the ninth cube and the
connecing strips.
If you start with A4paper (29.8cm), you have to make the
strips shorter (or thinner). At maximum you can get a strip of
14 length if you like to have the width at 2 cm.
 Fold the required witches ladders
and unfold them.
 Make valleyfolds of every fold.
 Cut 27 strips of 6 squares, as described above.
 Make 9 cubes of 3 strips each. Every first and last square
is used to finish the cube.
 The folds of the strips used to connect the cubes, have to
bend both ways easily. Take the strip of length 10, and connect
3 cubes with it. Connect more cubes in such a way that you get
the pattern shown in the picture. You will need the other 2 strips
to connect the last 2 cubes to the wall.
 You can open and close the wall.
You can make the wall as big as you please by adding more cubes to it.

Back to contents
Projects
NEW
Challenge: A Stamp machine
To make this "Stampfmaschine" you can follow these
directions and try to figure it out from these instructions. It's like a wobbling wall, but then in 3 directions.

Back to contents
Projects
The second project: A sphere of colored strips.

This sphere is a regular icosahedron, with on
top of each plane
a pyramid with three equal sides. These sides are triangles made
by dividing a square in half. If you connect the tops of the
pyramids, you get a dodecahedron.
A picture and explanation of this sphere, assembled from Sonobeunits,
you find on the page of Helena.

Take 6 strips with a length of 12 times the with.
In our example we will use 6 strips of 2cm x 24cm in the colors
purple, blue, green, yellow, orange and red.
Because of the fact that these strips also need diagonal folds,
you can make witches ladders at a different way. We will make
one witches ladder from one strip.
Fold the strip in half. With the fold away from you, take the
top part and fold the diagonal from the existing fold to the
right when you're righthanded and to the left when you're lefthanded.
Turn around and fold a witches ladder. Do this with all the strips.

Unfold the witches ladders and make
of all the folds mountainfolds, exept the diagonal valleyfold.
Make valleyfolds at 12 diagonals, just so that when one fold creases
from the topleft to the bottomright,
the next fold creases from the bottomleft to the topright.
(If you would draw a line over all the diagonal creases, you
wouldn't have to take your pen from the paper).

Take 5 strips, leave the 6th. Let the 6th be the one that can get
stained the easiest. In our case that's the yellow strip.
First we are going to make the "northpole".
Take 5 strips and let the middles of the strips meet in one point.
This point is the northpole. If we see the sphere from the outside,
de diagonal folds are valleyfolds and the straight folds are
mountainfolds. Use exactly five little pegs or clips to prevent
the strips from moving. Peg the strips together, there where
two straight creases meet, but not the diagonal creases.
You will get 5 pyramids, with triangular bases, who will meet in one point.
The 6th strip we call the equator. Plait the equator between the strips and
be sure that the beginning of the strip is hidden under the end
of the strip, and the end is hidden under one of the other 5
strips. Don't forget each time to pinch the pegs at a different
location. You will need no more and no less than 5 pegs.



Go on with plaiting until you reach the southpole. If you want
to insert a hangingcord, you have to add it now. Join the end
with the beginning of each strip and finish them neatly. Be sure
not to "lose" a strip inside the sphere. You can fold
the last strip before you tuck it in, or you can cut a triangle
to let it fit easier.



Back to contents
Projects
General instructions to make
a knotologysphere
First make the sphere from Sonobemodules (or in another way).
Look closely at it and look how many squares you need to go round
with one strip. In general a strip returns to it's beginning
point. Determine which squares have to be divided with a diagonal
fold. A strip needs as many squares as you counted going round
the sphere plus two for the finish. Make one strip and check
it by holding it against the testsphere.
Make a drawing of the strip and place the valley and mountainfolds.
Determine how many strips you need.
Cut in different colors strips of the right length. The colors
help you to keep the strips apart. If you use different colors,
you can easier see if the end of the strip is returning to its
beginning.
Make witches ladders, take them apart and crease the valley
and mountainfolds according to your own drawing.
In general you begin to make the sphere from the middles of the
strips. Sometimes it won't work. Then you can look for another,
easier beginning. Use little pegs to hold everything in his place.
Go on plaiting, be sure that every strip is exactly one square
at a time at the survace of the sphere. If there is an end of
a strip hanging loose, let it hanging until the sphere gets its
form. At the end we can tuck the loose ends in. If the last square
has a diagonal fold, you can fold it and tuck it in double, or
cut the surplus.

Back to contents
Projects
What you need to make a dodecahedron
This is one variation of the black and dark blue family Heinz told us about.
Sonobeelements:
You need 10 strips of 20 squares, folded like this:
Diagonals are valleyfolds, the straight lines are mountainfolds.
another picture


Back to contents
Projects
What you need to make
"4 cubes".
This model is a tricky one, because you have to weave with strips
that are not yet available for a specific cube. It's a good idea
to make this one first from sonobemodules, you'll need 12 sonobemodules
without diagonal fold and 6 modules with diagonal fold.
You need 3 strips of 14 squares, folded this way. Diagonals
are valleyfolds, straight lines are mountainfolds. With the grey
squares you'll make the first cube.

First you make the first cube, for the second cube you have the green
and blue strips, but the orange strip is on the other side.


You have to imagine which part of the orange strip will be joining in
the second cube, and then with weaving try to make the third and fourth
cube. Realise that the end of the strip meets always his beginning!

 

Back to contents
Going further with Knotology
If you made all these projects, you can try making spheres without making
the sphere in any other way. For example spheres from the family
Heinz told us about, the yellow and light blue ones in the first
picture.
Here you see a picture of a sphere, folded by Rosalinda Sanchez.
She folded this one with the aid of the book of Tomoko Fuse and
the picture of the light blue sphere in the first picture on
this page.
You need 12 strips of length 22 with the diagonals like in
the colored sphere. First make 5 piramids with 5 strips. Then
surround them with 5 times 6 piramids (some of them sunken, reverse
the folds while making the sphere) with the next 5 strips. Use
the last two strips one at a time as equators.
There are many variations, you can "pop out" the stars,
you can reverse the individual piramids, you can turn the whole
sphere insideout, etc. etc.
If you like to do more with tickertape, you could read
Soma Cube from strips, written by
Sebastian Kirsch.
He describes another project, the soma cube, with
almost the same technique. He made good drawings of how to make
a witches ladder and how to make the pieces for the puzzle.
There are also models made from tickertape on
Thoki Yenn's site.

Back to contents
It's without permission from me and from H. Strobl
forbidden to commercially use the models and/or diagrams on this page.
If you use these instructions please let me know.
I would like to know if the instructions are clear enough and
if there are things that could be better. I'm always open to suggestions.

