KNOTOLOGY :: | sphere94 | stampfmaschine | :: KNOTOLOGIE

The of Heinz Strobl

Written by Paula Versnick
© 2000 P. Versnick & H. Strobl

Nederlandse versie

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.


knotology objects

  • Preface
  • Sphere 94
  • Method
  • How does knotology work?
  • How can I make a strip with squares?
  • How do I cut a strip?
  • How do I make the plaiting easer?
  • Projects
  • The first project: A Wobbling Wall of 9 cubes.
  • NEW!! A Stamp Machine.
  • The second project: A sphere of colored strips.
  • General instructions to make a knotology-sphere.
  • What you need to make a dodecaedron.
  • What you need to make "4 cubes".
  • Going further with knotology

  • Preface

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

    Sphere94 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 three-dimensional 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, 14-16 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 sphere-like 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 sphere-like 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 4-480-8727264-7, (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 light-blue "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 dark-blue 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 spere-like 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


    How does Knotology work?

    Knotology is the creation of three-dimensional 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.

     Hyper Icosaeder (25kB)

    The origami convention in Veldhoven, 14-16 april 2000

    To weave/plait these objects you need strips of paper of equal width. De best material for this, is the old-fashioned 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


    How can I make a strip with squares?

    1. 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)
    2. Cut 2 strips at the desired length.
    3. 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.
    4. Keep on folding the strips around each other. With every fold you create a square, first a red, then blue, red, blue etc.*)
    5. 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


    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 cube-shaped holes in the wall. The animation explain better what I want to say.

    The animation can be stopped by selecting
    the stop-button 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 6-6-6), the 2 strips of 23 squares you can divide like this: 6-7-10 and 6-6-7, material for the ninth cube and the connecing strips.

    If you start with A4-paper (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.

    1. Fold the required witches ladders and unfold them.
    2. Make valleyfolds of every fold.
    3. Cut 27 strips of 6 squares, as described above.
    4. Make 9 cubes of 3 strips each. Every first and last square is used to finish the cube.
    5. 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.
    6. 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


    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


    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 Sonobe-units, 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 top-left to the bottom-right, the next fold creases from the bottom-left to the top-right. (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 valley-folds and the straight folds are mountain-folds. 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 hanging-cord, 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


    General instructions to make a knotology-sphere

    First make the sphere from Sonobe-modules (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 test-sphere.
    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


    What you need to make a dodecahedron

    This is one variation of the black and dark blue family Heinz told us about.



    You need 10 strips of 20 squares, folded like this:

    Diagonals are valleyfolds, the straight lines are mountainfolds.

    another picture

    Back to contents


    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 sonobe-modules, you'll need 12 sonobe-modules 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 inside-out, etc. etc.

    If you like to do more with ticker-tape, 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 ticker-tape 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.

    KNOTOLOGY :: | sphere94 | stampfmaschine | :: KNOTOLOGIE

    Paula's Orihouse
    Gerard & Paula