Enjoy Math! 2 -- Tower (English) / ART TOWER MITO

1-6-8 Goken-cho, Mito-shi, Ibaraki-ken, 310-0063 Japan
Mail to: webstaff@arttowermito.or.jp
Phone: (029)227-8111 / Fax: (029)227-8110

Make Your Own Tower from a Single Sheet of Paper!
Experience of ATM's Web staff

First, in order to make a diagram of the tower, let's observe the real tower itself. See how stands up so straight?

You can see how three ribbons, folded inward and outward in a triple helix, connect a series of beautifully intertwined equilateral triangles. As long as you have a diagram, it should be no problem at all to make it! Since we learned from Mr. Otsu's class that there are a total of 58 faces of equilateral triangles on the tower, it should be possible to make a model using two ribbons of 19 equilateral triangles and another of 20 (19+19+20=58), assembling them to form the body of the tower. (That is the method that the student in Mr. Otsu's class tried to use).

Mr. Otsu commented that he was too clumsy to make a 3-D figure from a flat piece of paper. ATM's Web staff, however, was confident enough (!) in its own dexterity to make the figure. Well, then, although it is quite difficult to paste the three ribbons together, they should be easy to connect as long as the parts pasted together are kept to a minimum. By putting together the three ribbons on the diagram, making sure that the inward and outward folds are made properly, you should be able to make a fine-looking model.

First, the parts that need to be folded outward are the areas where the ribbons join together vertically. That is the important ridgeline of the clockwise triple helix that climbs upward. You can tell from the photograph that the line that starts from the upper right, coming down along the ridgeline, consists of upward folds, while the line that goes down toward the bottom right consists of inward folds.

We felt that we were capable of building the model of the tower since we knew how to make Web pages (!), and you see the photo at the bottom left of the completed model that we made from a simple diagram. (Click on the simple diagram to make it larger).

Trying to emulate the effect of paper balloons made from origami, we printed out the diagram on regular printer paper (A4-sized), cutting it out so as to make the model.

At first, we made the outward folds, followed by the inward folds. Starting with the point of the tetrahedron at the very top, along with the bottom, we taped the figure all around, fixing the model in place using double-sided adhesive tape. Then twisting the model gently just as if we were squeezing a towel lightly, we put the same double-sided tape along the center ridgeline, one part at a time.

We then formed the base, and attempted to connect it with the tower itself (hope it goes OK!).

We did it!
Next, we pasted the three supporting columns on, adjusting the length while correcting the differences in the folds of the tower and the base. Despite being so flimsy, it did stand up quite sturdily. (It took only around 20 minutes to make from the time we cut the figure out. The photo shows us looking proudly at our product).
* When pasting on the supporting columns, after connecting the tower body firmly to the base, make sure to lay the tower on its side and carefully adjust the length as you go along. The center will be pushed along as you go.

Also, if you are using thick paper to construct the model, it helps to inscribe a slight incision along the fold with a cutter knife (on top in the case of outward folds, and underneath in the case of inward folds). That will let you make cleaner folds.

Numerical Data about the ATM Tower

28 tetrahedrons (9.6m on each side), joined together vertically (Maximum height of 99.4736m, with a height of 94.0736m from the top of the tower to the bottom of the base)
Inscribing an internal cylinder with a radius of 4.9883m, the length of each equilateral-triangle face of the vertical base is 17.0361m (both top and bottom).

(As it forms a certain thickness, the upper equilateral triangle is hidden in the diagram. It is the triangle formed by joining 1, 3, and 5 in the figure below.)

Nodal point axes for joining the tower base and body
* In order to finish it along the outward folds, the completed figure is reversed left and right. The numbers given for the axes are expressed in terms of the distance from the center in meters (in the actual standing tower).

	X	Y	Z
1	0.00000	9.76541	0.00000
2	0.00000	9.76541	-3.00000
3	8.51806	-4.98831	0.00000
4	8.51806	-4.98831	-3.00000
5	-8.51806	-4.98831	0.00000
6	-8.51806	-4.98831	-3.00000
7	3.71806	3.32554	0.00000
8	0.00000	-4.98831	3.03679
9	-3.71806	3.32554	6.07157
Upper node of supporting column #1	-1.10165	4.86514	15.17893
Upper node of supporting column #3	4.95742	0.55426	9.10736
Upper node of supporting column #5	-2.89183	-4.06455	12.14315

Architect, Arata Isozaki (Go to "Designing Mito" page)

Go to Tower Measurements page

Return to Class page

You can purchase a cute miniature version of the Tower, the ATM Tower Key Ring, in the ATM shop.

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