The scottish scientist Alexander Graham Bell (1847-1922) is known for his contribution to the advent of the telephone. His patent for this invention (revoked in 2002 by the United States Congress in favor of Antonio Santi Giuseppe Meucci) assured him a fortune. Without financial worries, Alexander Graham Bell was able to devote himself to other studies. Among them was aviation.

One of the technological issues confroning scientists in the early 20th century was the possibility of building large, aerodynamically stable flying devices. One of the arguments against this possibility was given by the astronomer and mathematician Simon Newcomb (1835-1909):

     “ Let us make two flying machines exactly alike, only make one on double the scale of the other in all its dimensions. We all know that the volume, and therefore the weight of two similar bodies are proportional to the cubes of their dimensions. The cube of two is eight. Hence the large machine will have eight times the weight of the other. But surfaces are as the squares of the dimensions. The square of two is four. The heavier machine will therefore expose only four times the wing surface to the air, and so will have a distinct disadvantage in the ratio of efficiency to weight.”

Alexander Graham Bell proposed a model of aerodynamically stable kite and whose size can be increased keeping constant the efficiency ratio by weight. The idea of Bell: to use regular tetrahedrons as cells. In this activity you will find a step by step to build the one of the tetrahedral kite developed by Alexander Graham Bell. To understand why this kite isn't a violation of Newcomb's argue, just follow the orientation of the student accompaniment form.



Click on the figure bellow to exhibit 3D interactive schemes of tetrahedral kites with 1, 4, 16, 64 and 256 tetrahedron cells, respectively. Na janela que se abrirá, para ampliar ou reduzir o esquema, mantenha o botão direito do mouse pressionado e, então, arraste-o.

1 célula tetraédrica 4 células tetraédricas 16 células tetraédricas 64 células tetraédricas 256 células tetraédricas


In the photo of Figure 2, in addition to Alexander Graham Bell, also appears his wife Mabel Gardiner Hubbard. Figure 3 is a reproduction of one of the pages of the tetrahedron kite patent made by Alexander Graham Bell (click on Figure 3 to enlarge it).

Alexander Graham Bell
Figure 1
    Alexander Graham Bell e Mabel Gardiner Hubbard
Figure 2
    Uma das páginas da patente da pipa tetraédrica feita por Bell
Figure 3


The argument given by Simon Newcomb for the impossibility of building large flying machines is a rereading of the Principle of Similitude given by Galileu Galilei in his piece Discorsi e Dimostrazioni Mathematische from 1638. According to this principle, if a biological organism increases its size, it will have to change its own structure. CConsider, for example, the situation of two similar animals, where one of them is twice the scale of the other. The bone “thickness” of the bigger animal wil be 4 times larger than the “thickness” of the corresponding bone of the smaller one, but this bone will tolerate a weight 8 times higher. Therefore, the bone structure of the larger animal will be much more fragile when compared to that of the smaller animal. By Principle of Similitude, a “bigger version” of the small animal will prefer to change its structure (for example, increasing more than 4 times the bone “thickness”) to insure some robustness. It is for this reason that those giant spiders from horros movies, can't exist. The poster of the movie “Tarantula!” is available here.


The following section was extracted from the novel “Gulliver's Travels” written by Jonathan Swift (1667-1745):

     “The reader may please to observe, that, in the last article of the recovery of my liberty, the emperor stipulates to allow me a quantity of meat and drink sufficient for the support of 1728 Lilliputians. Some time after, asking a friend at court how they came to fix on that determinate number, he told me that his majesty’s mathematicians, having taken the height of my body by the help of a quadrant, and finding it to exceed theirs in the proportion of twelve to one, they concluded from the similarity of their bodies, that mine must contain at least 1728 of theirs, and consequently would require as much food as was necessary to support that number of Lilliputians. By which the reader may conceive an idea of the ingenuity of that people, as well as the prudent and exact economy of so great a prince.”, 2018.

The calculation of the volume made by the Lilliputian mathematicians is correct: if Gulliver is 12 times taller than a Lilliputian, so his volume is 123 = 1728 times greater (assuming that Gulliver and the Lilliputians are similar). However, if the metabolism of Lilliputians is equal to Gulliver's metabolism, it is not correct to state that, for having a volume 1728 times greater, Gulliver has to receive 1728 times more food than a Lilliputian would receive. The energy supplied by food is mostly transformed into heat and the rate of heat loss is proportional to the surface area of the body and not to its volume. Note that the surface area of the body of a Lilliputian is 144 times smaller than the surface area of Gulliver's body, whereas the heat generated by his body is 1728 times smaller. So either the body temperature of a Lilliputian is much smaller (he would not have warm blood) or he would have to eat more (in comparison to its size) to generate more energy (like a mouse that is constantly nibbling).

Illustrations from the novel “Gulliver's Travels” by Jonathan Swift no Project Gutenberg.

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Responsible: Humberto José Bortolossi.
Idealization: João Júlio Dias Bastos Queiroz and Humberto José Bortolossi.
Construction: João Júlio Dias Bastos Queiroz and Mayara Andrade Viana.
Revision: Carlos Eduardo Castaño Ferreira, João Júlio Dias Bastos Queiroz and Humberto José Bortolossi.
English version: João Júlio Dias Bastos Queiroz.

A Pipa Tetraédria de Graham Bell Versão 29/05/2009
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