Five Intersecting Tetrahedra - Origami Model

Five Intersecting Tetrahedra - Origami Model
Violeta Vasilevska
Department of Mathematical Sciences,
University of South Dakota,
414 E. Clark Street,
Vermillion, SD 57069-2307, USA
Phone: (605) 677-5472; Fax: (605) 677-5263
e-mail: [email protected]
USD Math Days for Women
January 28, 2010
Section 1: History of the Five Intersecting Tetrahedra
• Francis Ow’s 60◦ unit, 1986;
• Tom Hull’s model of Five Intersecting Tetrahedra, 1996;
• This origami model was voted onto the British Origami Society’s list of ”Top 10 Favorite
Models”.
Section 2: Symmetries of the Five Intersecting Tetrahedra
1. The figure below shows an origami model of Five Intersecting Tetrahedra. Draw lines connecting nearby corners of the tetrahedra.
Drawing the lines will form a regular polyhedra that you are familiar with. This interesting
polyhedra - compound of five tetrahedra has been known for hundreds of years. Can you
recognize which regular polyhedra is formed?
Answer:
2. Why is this true?
In the figure of a dodecahedron below, can you find four mutually equidistant corners?
If so, draw lines connecting these corners. Which regular polyhedron do they form?
Answer:
Violeta Vasilevska, Ph.D.
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The figure below shows a tetrahedron inscribed in the dodecahedron.
How many regular tetrahedrons can you inscribe in a dodecahedron without using any corner
more than once? Explain!
Answer:
Compound of Five Tetrahedra
Five Intersecting Tetrahedra
Violeta Vasilevska, Ph.D.
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Rotational symmetries of a dodecahedron:
The Five Intersecting Tetrahedra model shares many symmetries with the dodecahedron. They
are called rotational symmetries. How many of them there are?
a) Draw an axis which passes through two opposite vertices.
b) How many such axes can you draw? Explain!
c) Draw an axis which passes through the centers
of two opposite faces.
d) How many such axes can you draw? Explain!
e) Draw an axis which passes through the midpoints of two opposite edges.
f) How many such axes can you draw? Explain!
What is the total number of rotational symmetries for a dodecahedron?
These rotational symmetries form a special structure in mathematics galled a ”rotational group”
of the dodecahedron.
Violeta Vasilevska, Ph.D.
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Section 3: Origami Five Intersecting Tetrahedra ([3])
Francis Ow’s 60o unit ([4])
To make one tetrahedra frame (of the same color) you will need two square origami paper, that
must be cut into 1 × 3 pieces.
Step 1. Start with a square origami paper. Fold
the square sheet into thirds and cut along the
creases.
Step 2. Take one 1×3 piece of paper made in step
1. Fold it in half, making a crease lengthwise, and
unfold.
Step 3. Fold the sides to the center crease made
in step 2.
Step 4. At the left end, make a short crease along
the half-way line of the upper side using the upper
flap.
Step 5. Fold the bottom-left corner to the crease
made in step 4 and at the same time make sure
the crease hits the midpoint of the side on the left
(as shown on the step 6 figure).
Step 6. Fold the upper-left corner over the flap
made in step 5.
Step 7. Undo the last two steps (i.e. unfold the
last two flaps).
Step 8. Make a reverse fold of the bottom-left
corner, using the crease made in step 5. The reversed flap should go inside the model (such that
part of the reversed flap is tucked under the upper
side paper).
Violeta Vasilevska, Ph.D.
Step 9. Fold the left edge of the upper side to the
crease line created in step 6. Then unfold.
This finishes one end of the unit.
Step 10. Rotate the unit 180◦ and repeat the steps
4 – 9 on the other end of the unit.
Step 11. Fold the whole unit in half lengthwise.
This finishes the unit.
Step 12. Repeat steps 2 – 11 with the other 1 × 3 units (total of 6 for one tetrahedra).
Locking Francis Ow’s units together:
Step 13. The end of each unit has a flap on one
side and a pocket on the other. Insert the flap of
one unit into the pocket of another unit, in such
a way that it hooks around the crease of that unit
(see step 14).
Step 14. After step 13 the lock should look like
on the figure to the right.
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Violeta Vasilevska, Ph.D.
Step 15. Note that to make one corner of a tetrahedral frame you need three units. Use the lock
from step 14 and insert the third unit using step
13.
Step 16. The figure to the right shows one corner
of the tetrahedron frame completed.
Step 17. To build on the tripod from step 16, add
two units to one of the tripod’s legs to make another corner. Then add the last unit to complete
the tetrahedron.
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Violeta Vasilevska, Ph.D.
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Linking The Five Origami Tetrahedra Together (Homework :-))
Step 18: Make twenty-four more Francis Ow’s units (using four different colors – six units of each
color) by repeating the steps 1 – 11.
Step 19. Make a corner of a second tetrahedral frame using three units of a same color (different
from the one used to make tetrahedron in step 17), using the steps 13 – 16.
Step 20. Use the corner of the second tetrahedron (constructed in step 19) and weave it into the
first one as follows: the three legs of the tripod should enter one ”face” of the first tetrahedron and
each of them should exit the interior of the first tetrahedron through a different ”face”. Then lock
the other three units into the second tetrahedron. (See Figure 1.)
In the Figure 1, notice that the left-most corner of the red tetrahedron is poking through a
”face” of the green one, and vice-versa, the right-most corner of the green tetrahedron is poking
through a ”face” of the red one. Notice the symmetry of this model.
It is worth noting that the Five Intersecting Tetrahedra model has this more general symmetry:
any two tetrahedra are intersecting in such a way that one corner of one tetrahedra is poking
through a ”face” of the other and vice versa.
Step 21. Use step 19 to make a corner of the third tetrahedron and use the Figure 2 to help you
insert the units for the third tetrahedra in the model formed in step 20.
Notice how the three units in the center of the Figure 2 are inserted forming a triangular pattern.
The same pattern is formed on the opposite side of the model. This is another symmetry that the
Five Intersecting Tetrahedra model has.
Use these two symmetrises of the model when inserting the units for the fourth and fifth tetrahedron in steps 22 and 23.
Step 22: Use step 19 to make a corner of the fourth tetrahedron and use the Figure 3 to help you
insert the units for the fourth tetrahedra in the model formed in step 21.
Step 23: Use step 19 to make a corner of the fifth tetrahedron and use the Figure 4 to help you
insert the units for the fifth tetrahedra in the model formed in step 22.
Violeta Vasilevska, Ph.D.
Figure 1: Step 20
Figure 3: Step 23
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Figure 2: Step 21
Figure 4: Step 24
Send us pictures with your Origami Five Intersecting
Tetrahedra!!!
References:
1. L. L. Foster, On the Symmetry Group of the Dodecahedron, Mathematics Magazine, Vol. 63,
No. 2 (Apr., 1990), pp. 106-107
2. J. A. Gallian, Contemporary Abstract Algebra, (Houghton Mifflin, 2006)
3. T. Hull, Project Origami-Activities for Exploring Mathematics, (A K Peters, Ltd, 2006)
4. F. Ow, Modular Origami (60◦ unit), British Origami, No. 121, 1986, pp. 30-33.
5. Tom Hull’s origami page: http://kahuna.merrimack.edu/∼thull/fit.html
6. Video ”How to fold a Five Intersecting Tetrahedra”
http://www.instructables.com/id/How to fold a Five Intersecting Tetrahedra Dodecah/