The Soma Cube
A TelExperiment by Sebastian Marius Kirsch
[soma-e.tex,v 1.3 1997/06/17 17:08:30 skirsch Exp skirsch]
c 1996,1997 Sebastian Marius Kirsch. All rights reserved. Repro
duction and distribution is allowed under the terms of the GNU
General Public License. For personal/non-commercial use only.
The Soma Cube was
invented in 1936 by Piet
Hein, a Danish poet and
puzzle inventor.
It repPSfrag replacements
resents all possibilities of ◦
combining three or four 90◦
cubes in a ‘not-straight’360◦
way.
Its seven pieces180
crease
can be assembled to form
edge
thousands of figures;◦ it
360
crease
is a kind of three-dimensional
A
Tangram in this aspect.
A first origami version of B
the Soma Cube was intro- 1
duced by Steve Biddle in 2
16 squares
his book ‘The New Origami’. Steve Biddle uses 6–8
Sonobé units to produce2 ×
the pieces and arrives at the impressive number of 120 units. Maarten
van Gelder has developed a Soma Cube whose pieces are each folded from
a single square; he uses box pleating techniques developed by Max Hulme.
I am presenting a version that is folded from ticker tape. My version gets
by with seven 1.1 m long strips of ticker tape; each piece requires between
40 and 60 squares.
All seven elements are folded from the same principles. Instead of describing each element step-by-step, I will therefore describe the basic elements
and the methods of extending them. These methods are then presented
with four of the pieces; the remaining pieces are only described verbally,
but are constructed from the elements that have already been described.
Before I start, I would like to thank two people: Philip Noble, whose
FlexiCube has contributed much inspiration to this model, and Heinz
Strobl, who has worked his way through the whole instruction and who
has helped by removing many blunders and mistakes.
This is a translation of a German article that originally appeared on
no. 22 of der falter, the magazine of Origami Deutschland. I dedicate
this translation to Jan Polish, whom I tried to teach the model at the
Würzburg ’97 Convention at one o’clock at night, but in vain. I hope
that my diagrams are clearer than my verbal instructions.
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360◦
◦
180
crease
edge
◦
90
360◦
180◦
◦
360 crease crease
1 Some
A edge
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replacements
90◦
PSfrag replacements
360◦
180◦
crease
edge
introductory
remarks
360◦ crease
◦
360B crease
A
1.1 ASymbols
1
B
A
2 The
B diagrams distinguish
1
between aB
16 squaressimple1 fold and the edge of2 the strip. For 1
◦ squares
6–8folds (180
2 ◦ and 9016
), a thin line is used 2
216
×(squares
16 squares
6–8strip and
). For the edge of the
360◦
90◦
360◦
180◦
crease
edge
360◦ crease
A
180◦
B
◦
1
90
2
16 squares
edge
crease
360◦
2 ×(
6–8 a thick line is used
6–8
creases,
). An open
◦
2
×
6–8 360 crease
2×
arrow (
) is used to indicate that the end
2 ×( ) is used
of the strip is to be laid across a frame, and a filled arrow
to indicate that the strip is to be pulled through under a frame.
PSfrag replacements
1.2
The ‘Witches’ staircase’
Before beginning,
360◦
the strip must 180
be ◦
A
divided into squares.
crease
edge
This is done easiA
90◦
est with a 360
fold◦ that
crease
B
A
is called a ‘HexenB
treppe’ in German.
1
Some people may
still know this from
B
16 but
squares
kindergarten,
I
2
will briefly repeat
6–8
how to produce 2it.× PSfrag replacements
You begin by folding a valley fold of 45◦◦ in the middle of the strip. Both
90 to each other. Now you fold B
end of the strip are now perpendicular
◦
360
along the edge of A to the left, A along◦ the edge of B down, and so on
180 the resulting staircase.
till the end of the strip. Then you undo
crease
edge
1.3
Locking the ends
360◦ crease
The ends are locked by shorten-A
ing the remaining strip to two squares,B
sharpening it a little, and tucking it 1
into the opposite slot. Long fingernails 2
16 squares
and/or tweezers can be helpful for the
6–8
last bit.
2×
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◦
90
2360
◦The basic unit
180◦
Length of the strip = 42 × width
crease
edge
I will now describe how to make a basic unit with three cubes. The
360◦ crease
variation
with two cubes is easy to produce: Just make each frame two
A
squares
shorter.
B
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1 12◦ Valley fold field no. 11 down. (Field no. 9 for a unit with two cubes.)
90
360◦
16 squares
◦
180
6–8
crease
2×
edge
360◦ crease
2 Form a loose horizontal frame, based on the shown valley folds.
A
B
1
2
16 squares
6–8
PSfrag replacements
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2×
90◦
90
3 ◦◦ Lay the tape one time around 360
4 ◦ Valley fold the tape to the back
360 the frame. This is the vertical
180◦ and to the right.
180◦ frame.
crease
crease
edge
edge
360◦ crease
◦
360 crease
1
A
A
2
B
B
1
2
16 squares
16 squares
6–8
6–8
2×
2×
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PSfrag replacements
3a
4a
90◦ Variation of step 3 for piece 3, 4,
90◦ Variation of step 4 for piece 4, 6
360◦ 6 and 7: Pull the tape through
360◦ and 7: Let a loop stick out on
180◦ under the frame on the right, lay
180◦ the underside.
it
over
the
frame
on
the
left.
crease
crease
edge
edge
360◦ crease
360◦ crease
1
A
A
2
B
B
1
2
16 squares
16 squares
6–8
6–8
2×
2×
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90◦
5 ◦ Pull the tape through under the
6 Valley fold and mountain fold
360
◦ horizontal frame on the back and
the tape: a right-angled triangle
180
is produced.
creasein front.
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edge
90◦
360◦ crease
360◦
A
180◦
B
crease
1
edge
2
360◦ crease
16 squares
A
6–8
B
2×
1
2
16 squares
6–8
PSfrag replacements
2×
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90◦
7 ◦ Tuck this triangle behind the ho8 Pull the tape through under the
90
360◦
rizontal
frame.
vertical frame underneath and
180◦
360◦
over.
◦
crease
180
1
edge
crease
◦
edge
360 crease
A
360◦ crease
A
B
B
1
2
16 squares
2
16 squares
6–8
PSfrag replacements
2 ×◦
6–8
90
2×
360◦
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9 Repeat steps 6–8 and pull the 10
180◦ Lock the ends.
90◦ tape through under the horicrease
360◦ zontal frame one more time.
edge
◦
180
◦
360 crease
crease
A
edge
6–8 B
360◦ crease
1
A
2
B
16 squares
1
6–8
2
2×
16 squares
2×
3
Extension of the basic unit
The described basic unit can be extended by allowing a loop to stick out
in step 3 or 8, and by weaving around it afterwards. If you let the loop
stick out on the second frame, the woven unit is perpendicular to the basic
unit. If the loop sticks out while weaving the basic unit (steps 8–9), the
woven unit is in the same direction as the basic unit.
Before continuing, you should first try to make a basic unit with two
cubes, since five of the seven pieces are based on this unit.
4
PSfrag
replacements
The pieces of the
cube
90◦
Using the basic unit, the seven pieces of the
360◦
cube can now be folded. They can be assembled
180◦
into thousands of figures—into arches, walls,
crease
monuments, snakes, ships and much more. An
edge
example is shown on the left.
360◦ crease
While pulling the tape through under a cube,A
the tape can occasionally be caught. You canB
help it by first putting a small strip of paper 1
in as guidance. Thin tweezers or a thin metals 2
strip can also be used as guidance for the
16 tape.
squares
After the first steps, the diagrams usually omit
6–8
the beginning of the strip, because it would get
2×
in the way during later steps. Please do not be alarmed by this omission.
lacements
90◦
360◦
180◦
crease
edge
60◦ crease
A
B
1
2
16 squares
6–8
2×
crease
edge
4.1
Piece 1
360◦ crease
A
B
1
2
16 squares
6–8
Length of the strip = 55PSfrag
× width
replacements
2×
90◦
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1 Fold a basic unit with three 360
2 ◦ Mountain-fold the tape to
right and lay it once around
90◦ cubes. Let a loop stick out
180◦
◦ during the repetition of step 8.
loop.
360
crease
180◦ Pull the tape through under the
edge
creaseframe one more time.
360◦ crease
edge
A
360◦ crease
B
A
1
B
2
rag replacements
1
16 squares
902◦
6–8
16 squares
360◦
2×
6–8◦
180
PSfrag replacements
2×
crease
90◦
edge
3
Mountain-fold the tape to the
4 Pull the tape through under
360◦
360◦ creasetop.
loop on the top and between
180◦
A
two cubes in the middle.
crease
B
edge
1
◦
360
crease
2
A
16 squares
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B
6–8
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2×
901◦
90◦
3602◦
16 squares◦
360◦
180
6–8
180◦
crease
2×
crease
edge
edge
5 Pull the tape through under the
6 Lock the ends.
360◦ crease
◦
360 creaseloop on the top.
A
A
B
B
1
1
2
2
16 squares
16 squares
6–8
6–8
2×
2×
the
the
the
the
crease
edge
4.2
Piece 2
360◦ crease
A
B
1
2
16 squares
6–8
rag replacements
Length of the strip = 58PSfrag
× width
replacements
2×
90◦
◦
90
1 ◦ Leave 16 squares at the begin2 ◦ Valley-fold the beginning twice:
360
360
A triangle forms.
Tuck this
180◦ ning, then fold a basic unit with
180◦ triangle into the loop.
three
cubes.
Let
a
loop
stick
crease
edgeout during step 8 of the ba- crease
edge
◦
360 creasesic unit. Then complete the unit. ◦
360 crease
A
A
B
B
16 squares
1
1
2
PSfrag replacements
2
16 squares
90◦
6–8
rag replacements
6–8◦
360
2 ×◦
2
×◦
180
90
◦
crease
360
3 ◦ (new position) Lay the tape one ◦ edge
4 (old position) Mountain-fold the
180
360 creasetape to the top.
creasetime around the loop. (cf. piece
edge1/2–5)
A
B
360◦ crease
1
A
2
B
PSfrag replacements
16 squares◦
1
90
6–8
2
360◦
2×
16 squares
180◦
6–8
crease
2×
edge
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◦
5 Pull the tape through under the
6 Lock the ends.
360 crease
90◦ loop on the top, between the two
A
360◦ units in the middle, and once
B
180◦ more under the loop on the top.
1
crease
2
edge
16 squares
360◦ crease
6–8
A
2×
B
1
2
16 squares
6–8
2×
crease
edge
360◦ crease
A
B
1
PSfrag replacements 2
4.3 Piece 3
16 squares
90◦
6–8
Length of the strip = 53 × width
360◦
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2
×
◦
180
90◦
crease
1 Fold a basic unit with two cubes.
2 Lay a two cube wide frame
360◦
edgearound the loop, as indicated by
◦ Let a loop stick out during step
180
360◦ creasethe dotted lines.
8. (cf. piece 1/1)
crease
A
edge
B
◦
360 crease
1
A
2
B
16 squares
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1
6–8
902◦
2
×
16 squares
360◦
PSfrag replacements
◦
180
6–8
90◦
2×
crease
360◦
edge
3
Mountain-fold the tape to the
4 Lay the tape once around the
180◦
360◦ creasetop.
new frame.
crease
A
edge
B
◦
360
crease
1
A
2
PSfrag replacements
B
16 squares
6–8
901◦
2×
3602◦
16 squares◦
rag replacements
180
6–8
crease
90◦
2×
◦
edge
360
5 ◦ Mountain-fold the tape to the
6 While pulling it through.
360◦ crease
180
left
and
pull
it
through
under
the
A
crease
B
edgeoriginal loop.
◦
1
360 crease
2
A
16 squares
B
6–8
1
2
×
2
16 squares
6–8
2×
PSfrag replacements
90◦
◦
360
90◦
180◦
360◦
crease
180◦
edge
crease
7 Pull the tape throughPSfrag
between ◦
8 Lock the ends.
360
crease
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edgethe unit, on the right under the
A ◦
360◦ creasenew frame, and under the loop.
90
B ◦
A
360
1 ◦
B
180
2
1
crease
16 squares
2
6–8edge
16 squares
×
360◦2 crease
6–8
A
2×
B
1
2
16 squares
4.4 Piece 4
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Length of the strip = 60 × width
6–8
2×
This piece is one of the
most
difficult.
Two
distinctive loops have to
PSfrag replacements
stick out. This makes the first steps particularly
unstable. Persevere!
◦
90
Fold a basic unit with two cubes,
2 ◦ Tuck this triangle under the
360
rag replacementsand let the vertical frame stick
180◦ frame.
90◦ out (cf. steps 3a, 4a of the ba- crease
edge
360◦ sic unit). Pull the tape through
◦ under the frame twice and fold a ◦
360 crease
180
A
creasetriangle.
edge
B
1
360◦ crease
2
A
16 squares
B
6–8
1
2×
2
16 squares
6–8
2×
rag replacements
3 Pull the tape through
on replacements
the
4 (new position) The loop on the
PSfrag
back is a result of the last step.
90◦ top and on the bottom under
90◦ Valley-fold the tape twice.
360◦ the vertical frame, let a loop
◦
360
180◦ stick out in front (as indicated
180◦
creaseby the dotted lines).
crease
edge
edge
◦
360 crease
360◦ crease
A
A
B
B
1
1
2
2
16 squares
16 squares
6–8
6–8
2×
2×
1
rag replacements
PSfrag replacements
90◦
90◦
360◦
360◦
180◦
180◦
5
Tuck
the
triangle
into
the
loop.
6 Lay the tape once around the
crease
crease
edge
edgeloop.
360◦ crease
360◦ crease
A
A
B
B
1
1
2
2
16 squares
16 squares
6–8
6–8
PSfrag replacements
2×
2 ×◦
90
360◦
rag replacements
7 (new position) Mountain-fold 180
8 ◦ Mountain-fold the tape to the
90◦ the tape to the top, pull it
left and lay it once around the
360◦ through under the loop on the creaseloop.
edge
180◦ top and in the middle twice, so ◦
creasethe vertical crease is hidden. 360 crease
A
edge
2×
B
◦
360 crease
1
A
2
B
16 squares
PSfrag replacements
1
6–8◦
2
290
×
16 squares
360◦
6–8
180◦
crease
edgeLock the ends.
9 (new position) Mountain-fold 10
rag replacements
◦
it
◦ the tape to the left and pull 360
crease
90
A
360◦ through on the right and in the
◦ middle three times.
B
180
1
crease
2
edge
16 squares
◦
360 crease
6–8
A
2×
B
1
2
16 squares
6–8
2×
crease
edge
360◦ crease
PSfrag replacements
A
90B◦
3601◦
1802◦
4.5 Piece 5
16 squares
crease
edge
6–8
Length of the strip = 44 × width
◦
2×
360 crease
A
This is a variation of piece 1. You begin with
a basic unit with two
B
cubes, let a loop stick out during step 8 and continue
as in piece 1.
1
2
4.6 Piece 6
16 squares
Length of the strip = 54 × width
6–8
2×
This piece consists of a basic unit with two cubes, where the vertical
frame has been formed into a loop, and a unit with two cubes that has
been woven onto it.
PSfragfor
replacements
You can follow the directions
piece 4 until step 5, but you may not
let a loop stick out during step 3! Just imagine
90◦ that the loop in step 3–4
is not there.
360◦
◦
Then you do not weave a single cube onto
the
loop, as in pieces 4/6–7,
180
but you weave two cubes onto it as in piece
3. You lay the tape to the
crease
edge long frame to stick out
back around the loop and allow a three-squares
◦ ◦
behind. Then you turn the cube for 90360clockwise
crease around its vertical axis.
Now you can follow the directions for piece 3.A The loop is in the same
position as in piece 3/2.
B
4.7
Piece 7
Length of the strip = 56 × width
1
2
16 squares
6–8
2×
This is a mirror-image version of piece 6.
You follow the directions for piece 4 until step 5, but do not let a
loop stick out during step 3! Now you turn the basic unit around its
vertical axis again, but in the opposite direction as in piece 6, that is,
counter-clockwise.
Now the tape will stick out to the back. You lay it once around the
loop to the left, so it lies on the front side. Then you can follow the
directions for piece 3 and weave two cubes onto it.
In the pictures of pieces 6 and 7, the two cubes at the bottom are the
basic unit, as you see it in piece 4/4. The two cubes on the top are the
cubes that are woven onto the loop.
Ticker tape? What’s that?
META: This page is to contain a short article on ticker tape,
where to get it, etc. Unfortunately, it is not written yet. Expect
it for the final revision, which is to come out in a few weeks.
(Just before the OUSA Convention, to be precise.)
META: I can only judge from my experience here in Germany, where you can occasionally find a few rolls of ticker tape
still on stock in some office supply stores. I’d very much welcome any tips from people in other countries regarding where
to find rolls of ticker tape or any other foldable paper tape in
their country.