The idea of building a dodecahedron kite aroused while looking at a book on geometric shapes. We wished to build something new, and the dodecahedron called our attention for its beauty and harmony.

We started searching for drawings or pictures in books and Internet, and although there’s plenty of information about the dodecahedron, we found no reference to kites having this shape. So, we started from zero, designing our own model. Below you’ll find an account of this flying dodecahedron creation.

#### What is a dodecahedron?

A dodecahedron is one of the five –and only five-, regular polyhedrons that may be drawn in three dimensions, together with the tetrahedron, the cube, the octahedron and the icosahedron. It is, then, a closed three-dimensional figure, the sides of which are regular polygons. In the case of the dodecahedron, these polygons are pentagons and their number is 12.

#### Brief Story About The Dodecahedron

It is very interesting to read the story of the dodecahedron and the other platonic solids throughout the centuries.

The origin of the Dodecahedron, the same as for the rest of the regular polyhedrons, is not exactly known, but evidence proves that these solids were already known in Pre-Pythagoras times. It were Pythagoras and his school that systematized and studied these bodies in depth. Pythagoras also affirmed that this shape served as the Universe building plan, while outlining it as well.

The key person in studying the dodecahedron was **Fra Luca Pacioli** (1445-1517), a distinguished mathematics and geometry specialist from the Renaissance. In his masterpiece, De divina proportione, he states the mathematic basics of the divine proportion, considering it a universal principle and beauty goal in Universe. This proportion becomes particularly evident in the Dodecahedron.

Several artists and investigators focused on the regular polyhedrons and on the dodecahedron: Leonardo Da Vince performed beautiful drawings; Alberto Durero used them as models in his studies on perspective. Almost one century afterwards, Johannes Kepler (1571-1630), the German astronomer, finds in the regular polyhedrons an explanation about the planet orbits around the Sun, placing ones inside others. In this Universe pattern introduced in his masterpiece Mysterium cosmographicum, he places the Dodecahedron between the Earth and Mars orbits.

Finally, we find –among several more or less serious articles- the following news: There are scientists who propose that the Universe may not be infinite, but would have a limited size and the dodecahedron shape, according to the last investigations carried out by the North American Space Agency (NASA) and Cape Town University (South Africa)...!

#### Kite Building

In order to join the three edges running in each one of the 20 vertexes from the dodecahedron kite, it is required to know the angle between these edges. This angle is the same as that formed by the adjacent sides in a pentagon, i.e., 108 degrees. The first joints, made of crystal tube, were useless due to their flexibility. They were replaced by joints made of riveted aluminum tube, with a 5 mm inner diameter, using a cardboard template to get the 108-degree angles.

Each edge length determines the largest diameter in the figure. In this case, with a 50 cm edge, you get the largest diameter by approximately 140 centimeters. On a first attempt, we used rods having 50 centimeters long, and the device was big enough to get inside, as you can see in the picture. Considering the properties attributed by the sacred geometry to the dodecahedron, we could have gone through an “inter-dimensional awareness gate” favoring a “quantum leap” in our evolution and other huge life-harmonizing benefits, but… however, we preferred reducing the size to make the kite more handy. Anyhow, while the structure was lying in the living-room, we ensured our visitors that it was a portal (there are plenty in Bariloche area, but none of them has the dodecahedron shape!).

We tried with rods having 30 centimeters long, but it was hard to let it fly due to its weight. Finally, we split the difference and built the third dodecahedron, with rods having 40 centimeters long.

The coating for the prototype was made with extra-large polyethylene bags (the consortium size), joined with transparent adhesive tape. Once we had it ready, we noticed that it wouldn’t go through any door nor window, it was too big to take it outside the house! And we had to dismantle part of it and assemble it again outdoors.

#### The flight

Which sides should be covered, which ones should be left open, where should the hoisting ropes be fastened, how could we make it fly like a kite and not like a wind sleeve? All these questions came up, and fortunately, we found answers for them that worked.

We left two opposite pentagons uncovered, tried several inserting points and flight angles that were not working, until we finally succeeded: a combination fit. As you may see in one of the pictures, the insertion points are three. The dodecahedron went up high and very stable, with medium wind, and it looked quite flashy to us in the sky. Perhaps, after all, that thing about the portal…

We still have to build a version in cloth that can be dismantled, and we’re working on it. In the pages mentioned below, you will find all type of info regarding the dodecahedron. It results particularly interesting a Mexican site dedicated to building balloons under the shape of platonic solids. We hope all this helps as inspiration for creating new models.

**Diana Ross - Roberto Trinchero**

**Images:Links:**

- El Dodecaedro Regular, Martín Pastor - Granado Castro, Departamento de Ingenieria Gráfica,Universidad de Sevilla, España
- El Dodecaedro
- Globos de papel de seda - Méjico
- Dodecahedron Art Project
- Tales of the Dodecahedron
- Sólidos platónicos
- Pitágoras
- Fra Luca Pacioli
- Divina Proporción

We are now transcribing information furnished by our friend Alfonso Pérez Arnal, enlarging the content in this article:

Dear BaToCo fellows, and all of you in general,I hardly ever write, though I read all the messages. On this opportunity, I dare answer because apart from being fond of kites, I’m also fond of mathematics.

The thing is that I don’t consider the regular polyhedron issue as not-for-kitefliers, as I’ve been designing kites based on these geometric bodies for years. Out of them, it is known the tetrahedron-shape kite (4 triangular sides) by Graham Bell, and you can see flying kites in a cube-shape (or hexahedron) (6 square sides), and in an octahedron-shape (8 triangular sides) –I myself have 6 kites in the octahedron-shape. I’ve seen in some book a kite with a rod structure in a icosahedron-shape (20 triangular sides).

From the five regular convex polyhedrons, I still have to build the dodecahedron and the icosahedron. I have ideas and several designs based on these forms, but I need time to build them. Perhaps, one day…

The thing is that there are only five regular convex polyhedrons in the three-dimensional space, known in mathematics as “regular convex polyhedrons”. They are also known as the five platonic solids (as above-mentioned, tetrahedron, octahedron, cube, dodecahedron and icosahedron).

http://es.wikipedia.org/wiki/Poliedro_regular

The other polyhedrons that you mention and that you have, in the shape of special “dices” used in role games, are not all of them regular polyhedrons, according to the strict definition. To be called “regular polyhedron”, all the sides in it should be regular polygons (no rhombus nor trapezoids, nor any other but equilateral triangles, perfect squares, regular pentagons, etc.). Besides, all the sides should be equal (all of them equilateral triangles, or all of them squares, or all of them regular pentagons…). In addition, on each polyhedron vertex, the same number of sides must converge.

I also have a small dice with 10 sides. In fact, all the 10 sides are equal. The little dice shape keeps certain regularity… but the sides are not regular polygons, because they’ve got the shape of a NON-REGULAR quadrilateral, having the aspect of… CERTAIN TYPE OF WELL-KNOWN KITE! A very kiteflying geometric shape, but it isn’t a regular polygon. So you cannot call regular polyhedron to that small dice with 10 sides. It is even specified in this link,

http://es.wikipedia.org/wiki/Dado_de_rol

Another example easy to see: two tetrahedrons stuck by one side would form a polyhedron with 6 sides, all of them equilateral triangles... but there are vexes where 3 sides converge, and others where 4 sides converge. It isn’t considered a “regular polyhedron” either.

The geodetic domes you comment are not regular polyhedrons, either (those called of higher than 1 frequency). The triangles forming its sides are not exactly, exactly equilateral, nor exactly, exactly equal… though they seem so at first sight.

In fact, they keep some sort of regularity in their structure… but not that one required in the “regular polyhedron” definition.

Perhaps, when splitting hairs, one may discover a little mistake in the reference you comment. It should say, “A dodecahedron is one of the five –and only five- regular CONVEX polyhedrons that may be drawn in three dimensions”. There are four polyhedrons called “non-convex regular polyhedrons”, but the matter gets a bit complicated,

http://es.wikipedia.org/wiki/S%C3%B3lidos_de_Kepler-Poinsot

Well, that’s all. I apologize for all the mathematics stuff introduced in this forum… but a enjoy using lots of mathematics and lots of geometry when designing kites. I would really like to build some day my kite design in a dodecahedron shape. Anyhow, there’s one flying somewhere, but I haven’t seen any, yet. Though… may be I’ll start by those I already have, based on the dodecahedron rhombus and the star-shaped dodecahedron rhombus… interesting designs that I also wish to build. So, patience.

Cheers from Spain!

At last I could see a dodecahedron fly! I didn’t notice, when I answered, that the proposal was already in BaToCo page. Congratulations, Diana and Roberto. Besides, the idea you propose leaves the doors open to other quasi-spherical designs with two opposite openings… We could think in using it as a general idea for an icosahedron (20 triangular sides) and other non-regular polyhedrons.

Congratulations, again!