Chemical Education Journal (CEJ), Vol. 12, No. 1 /Registration No. 12-1/Received July 5, 2008.
URL = http://chem.sci.utsunomiya-u.ac.jp/cejrnlE.html


A simple system (named polifacil) for building three-dimensional models of polyhedra starting from drinking straws and raffia.

Aaron  Perez-Benitez*

Facultad de Ciencias Quimicas. Benemerita Universidad Autonoma de Puebla. 14 sur y avenida San Claudio, Col. San Manuel. C. P. 72570. Puebla, Pue. MEXICO. E-mail: <aronpersiu.buap.mx>

 

Rosa Elena Arroyo-Carmona

Departamento de Ciencias de la Salud. Universidad Popular Autonoma del Estado de Puebla. 11 Poniente 1312, Col. Santiago. Puebla, Pue. MEXICO.

 

Enrique Gonzalez-Vergara

Centro de Quimica del Instituto de Ciencias de la BUAP. 14 sur y avenida San Claudio, Col. San Manuel. C. P. 72570. Puebla, Pue. MEXICO.

 

Keywords: Three-dimensional models, Polifacil system, drinking straws, molecular geometry, misconceptions on spatial geometry.


Abstract (In English)
A simple system for building three-dimensional models of polyhedra made of drinking straws and raffia, named Polifacil,a is presented. It is an inexpensive and very suitable tool for being used by children in the learning of the spatial geometry or by undergraduate and graduate chemistry students in the learning of the molecular geometry and group theory. The procedure for building three sets of models is given: a) The trigonal, square, pentagonal and hexagonal pyramids; b) The tetrahedron, trigonal bipyramid, octahedron, square and pentagonal antiprisms; c) The trigonal dodecahedron, monocapped pentagonal antiprism and icosahedron.

The discovery of undergraduate chemistry students' misconceptions on the geometry of usual polyhedra has prompted us to divulgate this system as workshops for children in our "National Weeks of Science and Technology".

Abstract (En este resumen se suprimieron los acentos y los caracteres especiales para mejorar su presentacion en la web)
Se presenta un sistema sencillo, llamado Polifacil, de construccion de modelos tridimensionales de poliedros hechos con popotes y rafia. Es una herramienta economica y muy apropiada para usarse con los ninos en el aprendizaje de la geometria  del espacio o por los estudiantes de quimica de licenciatura y de posgrado en el aprendizaje de la geometria  molecular y la teoria de grupos. Se proporciona el procedimiento para la construccion  de tres series de modelos: a) Las piramides trigonal, cuadrada, pentagonal y hexagonal; b) El tetraedro, la bipiramide trigonal, el octaedro y los antiprismas cuadrado y pentagonal; c) El dodecaedro trigonal, el antiprisma pentagonal monoapuntado y el icosaedro.

El descubrimiento de ideas erroneas en muchos de nuestros estudiantes de licenciatura en quimica, respecto a la geometria  de poliedros comunes, nos impulso a divulgar este sistema en forma de talleres para ninos, en nuestras "Semanas Nacionales de Ciencia y Tecnologia".

Contents.
i) Abstract (In English).
ii) Abstract (In Spanish).
I Introduction.
I.1 Undergraduate chemistry students' misconceptions on spatial geometry.
I.2 The origin of "Polifacil system".
II Construction of 3D models by using Polifacil.
II.1 Construction of trigonal, square, pentagonal and hexagonal pyramids.
II.1.1 Construction, in steps, of a trigonal pyramid.
II.1.2 General procedure for making square, pentagonal and hexagonal pyramids.
II.2 Construction of polyhedra derived from chains of equilateral triangles: Tetrahedron, trigonal bipyramid, octahedron and square and pentagonal antiprisms.
II.3 General procedure for the building of polyhedra derived from a square and pentagonal antiprisms: Trigonal dodecahedron, monocapped pentagonal antiprism and icosahedron.
II.3.1 Construction of a trigonal dodecahedron.
II.3.2 Construction of a monocapped pentagonal antiprism and an icosahedron.
III Application of the Polifacil models in the teaching and learning of the geometry of polyhedral molecules.
IV Conclusion.
V Notes and references.

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I Introduction.
I.1 Undergraduate chemistry students' misconceptions on spatial geometry.
Although geometry is a very important tool in several fields of science, we have found that in our university many of high school and undergraduate students do not have a clear understanding of the basic concepts of spatial geometry. For example, the application of some diagnostic tests at the beginning of an inorganic or organic stereochemistry courses revealed that our undergraduate chemistry students do not match the name of the usual polyhedra with their physical forms. Thus, instead of the tetrahedron, the octahedron and the hexahedron they provided drawings of some polyhedra different to those required and, surprisingly, also... "Polygons!" (Tables 1 - 3: Arroyo-Carmona, 2005; Perez-Benitez, 2008).

Table 1 Alternative drawings for the tetrahedron provided by the tested population (49 undergraduate chemistry students): a) Two triangles sharing and edge (4 %); b) Square (2 %); c and d) Square Pyramid (67 %); e) Tetrahedron (25 %)*. Only 2 % of the students did not respond to the requirement: "Draw a tetrahedron".

 (a)

 (b)

 (c)

 (d)

 (e)


*The left drawing in "e" was considered as correct although they drew a chemical structure that
does not correspond to the requirement made to the students.

Table 2 Alternative drawings for the octahedron provided by the tested population (36 undergraduate chemistry students): a) Octagon (22 %); b) Octagonal pyramid (6 %); c) Octagonal prism (11 %); d) Cube (17 %); e) Octahedron (17 %).* The 27 % of the students did not respond to the requirement: "Draw an octahedron".

 (a)

 (b)

 (c)

 (d)

 (e)


*As in the previous case, the left sketch in "e" was considered as an octahedron (the correct answer)
although this stereographic sketch represents a 6-coordinated central atom surrounded by
ligands in an octahedral fashion.

Table 3 Alternative drawings to the hexahedron provided by the tested population (41 undergraduate students): a) Hexagon (19.6 %); b) Trigonal bipyramid (10.9 %); c) Hexagonal prism (19.6 %); d). Hexagonal pyramid (19.6 %); e) Cube or hexahedron (8.7 %); f) Octahedron (8.7 %).* The 13 % of the students did not respond to the requirement: "Draw a hexahedron".

  (a)

  (b)

  (c)

 (d)

 (e)

 (f)

*The sketch "f" was considered as an octahedron although the student's description was "it has six sides"
("tiene seis lados"), which obviously corresponds to an hexagon and not to an octahedron and much less
to a cube or hexahedron.

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I.2 The origin of "Polifacil" system.
Bearing in mind the problem mentioned above, we have developed a simple system for building 3D models of polyhedra and we have divulgated it as workshops in scientific chemical meetings (Pérez-Benítez, 1991) and also in our "National Weeks of Science and Technology" for children and their parents. This system was derived from the invention of the Finland artist Astrid Gröndlund, who built a geometrical mobile for children's room, made of octahedra. Each octahedron was constructed inserting a thin copper wire into 12 straws arranged as a chain of triangles (Borglund, 1972; Figure 1).

Figure 1 Astrid Gröndlund's octahedron made with straws and a thin copper wire (Borglund, 1972).

Looking for other starting materials, we found that drinking straws and raffia (a commercial plastic ribbon) are well suited: They are resistant enough, accessible, manageable and inexpensive in creating several colorful and lasting 3D models of polyhedra. Thus, they are quite appropriate for the teaching and learning of the geometry of usual polyhedra, from basic to advanced educational levels.

The most important advantage of this system (which we named in Spanish "Polifacil" and means "easy polyhedra") is that, starting from a 2D structure, the angles are automatically determined when the polyhedron is formed. This is strictly true when all of the polyhedron's faces are triangles; but in contrast, when it is shaped only with faces different to 3-side polygons, the resulting polyhedron is unstable, such as in the case of the cube. In the middle are polyhedrons whose topicity is a combination of triangles and other higher polygons, as in the case of square antiprism that is composed of 8 triangular and 2 square faces.

To illustrate the above mentioned, in this article is presented the construction of three sets of models:
1) Pyramids: Trigonal, square, pentagonal and hexagonal pyramids.
2) Polyhedra derived from a chain of equilateral triangles: Tetrahedron (Pérez-Benítez, 1991: pp. 198-200), trigonal bipyramid, octahedron, square and pentagonal antiprisms.
3) Polyhedra derived from square and pentagonal antiprisms: Trigonal dodecahedron, mono- and bicapped pentagonal (icosahedron) antiprisms.

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II The construction of 3D models.
II.1. Trigonal, square, pentagonal and hexagonal pyramids.
Materials:
a) Twenty red drinking straws of 5 cm.
b) Eighteen blue drinking straws of 8 cm.
c) Four pieces of raffiab of 1 m long.

II.1.1 Construction, in steps, of a trigonal pyramid.
1) Insert the raffia into three red straws and tie it firmly to form the trigonal pyramid's bottom (Figure 2a-b). One end of the raffia should be left short (SE) and the other large one (LE) (Figure 2b)c.
2) Insert the LE into two blue drinking straws and pass it by the inner part of the triangle at the point 2 (Figure 2c).
3) Pull it firmly and then make a simple knot as it is illustrated in Figure 2d.
4) Insert the LE by the inside of the straw 2-3 and make a simple knot at the point 3 (Figure 2e). It is necessary to make a knot at each vertex that the LE arrives; if it is not well done the polyhedron losses rigidity.
5) Insert the LE into one blue straw (Figure 2f) and tie it by joining the point 4 with the apex 5 (Figure 2g).
6) Cut the remaining raffia of the LE or retain it if you wish to hang the polyhedron.

Figure 2 Detailed procedure to build a trigonal pyramid starting from three red and three blue drinking straws and a piece of raffia (in black lines).

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II.1.2 General procedure for making square, pentagonal and hexagonal pyramids.
The extend structures for building these pyramids are illustrated on the left of the Figure 3. They can be constructed as follows: After the elaboration of the pyramids' bottoms with four, five or six red straws, followed by the insertion of two blue straws for the construction of their first triangles on the edges 1-2, the LEs must be inserted inside of the straws 2-3 and tied it at their vertices number 3. Then, a couple of blue straws have to be added for constructing the second triangle and the LEs are tied at vertices number 4.

In the case of square pyramid, the LE is going back to the vertex number 6 and tied with the vertex number 5 to conclude the model (Figure 3a right), while in the case of pentagonal and hexagonal pyramids their LEs must be inserted into the straws 4-5 and tied them at the vertices number 5. Then, one or two blue straws have to be added and joined to the vertex number 6 and 7 to conclude the pentagonal pyramid (Figure 3b right) or the LE has to be tied at the vertex 6 for making the third triangle of the hexagonal pyramid (Figure 3c left). Then, the LE is going back to the vertex number 9 and joined with the vertices number 7 and 8 to conclude the model (Figure 3c right). The terminated models are presented in Figure 4.

Figure 3 Extended structures (left) for making square, pentagonal and hexagonal pyramids (right) starting from red (5 cm) and blue (8 cm) drinking straws and a piece of raffia.

 

Figure 4 3D models of a trigonal, square, pentagonal and hexagonal pyramids made of drinking straws and raffia.

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II.2 Construction of polyhedra derived from chains of equilateral triangles: Tetrahedron, trigonal bipyramid, octahedron and square and pentagonal antiprisms.
All of these models can be constructed from their extended structures which are chains of two, four, five, seven and nine equilateral triangles (Figure 5 left). With exception of the trigonal bipyramid, one straw must be added after the construction of those chains of triangles.

Note that we use only straws of the same size (red color code) for building these models even if the length of the straws can be conveniently changed for increasing or decreasing the height of the trigonal bipyramid and the square and pentagonal antiprisms (Figures 5d, 5h and 5j). The reason of this fact is that the construction of trigonal dodecahedron and icosahedron (other regular polyhedron) is based on the square and pentagonal antiprisms (see next section).
Even though this time the detailed instructions for building these models are not provided to avoid repetitions (compare for example the extended structure of trigonal pyramid (Figure 2f) with that of tetrahedron (Figure 5a)), we have noted that some children have difficulties to make, for example, the trigonal bipyramid, because they carry out the step illustrated in the Figure 2e instead of introducing two straws to make the triangle 2,5,4 of the trigonal bipyramid's extended structure (Figure 5c). So, be sure of following the sequence: three straws - knot - two straws -knot - two straws - knot, etc., which gives rise to: first triangle - second triangle - third triangle, etc.

The terminated models are presented in Figure 6.
Materials:
a) 43 red drinking straws of 5 cm.
b) 5 pieces of raffia of 1 m long.

Figure 5 Extended structures (left) for building a tetrahedron, trigonal bipyramid, octahedron and square and pentagonal antiprisms (right), starting from drinking straws and raffia.

Figure 6 3D models of a tetrahedron, trigonal bipyramid, octahedron and square and pentagonal antiprisms made of drinking straws and raffia.

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II.3 General procedure for building polyhedra derived from the square and pentagonal antiprisms: Trigonal dodecahedron, monocapped pentagonal antiprism and icosahedron.
Even thought trigonal dodecahedron and icosahedron can be constructed by closing their respective extended structures (not showed here), we have found that their construction is more understandable if they are built in steps, being the first one the construction of the square and pentagonal antiprisms (see above).

In fact, other advantage of this system is that a polyhedron can be modified very easily. For example, the square antiprism can be converted in a trigonal dodecahedron if their square faces are transformed in triangles by the addition of one straw in each face; thus, the blue and green lines of the Figure 7 accomplish this purpose.

In the same way, the pentagonal antiprism is converted in an icosahedron if their two pentagonal faces are capped, which signifies that the two pentagonal faces have to be transformed into pentagonal pyramids, following the procedure described in the section II.1.2 (Note the resemblance among the Figures 3b and 8a). The semi-extended structure for building the monocapped pentagonal antiprism is presented in the Figure 8a, while the Figure 8c schematizes the semi-extended structure for making the bicapped pentagonal antiprism or icosahedron (Figure 8d).

The added straws (green and blue) for building these three models (sections II.3.1-2; Figures 7 and 8) are of different color than the initial polyhedron (red) only for didactical purposes but all of them must be of the same size. The terminated models are presented in Figure 9.

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II.3.1 Construction of a trigonal dodecahedron.
Materials:
a) The square antiprism built in the section II.2 (Figure 5g-h).
b) 1 green drinking straw of 5 cm.
c) 1 blue drinking straw of 5 cm.
d) 0.5 m of raffia.

Figure 7 Procedure to build a trigonal dodecahedron starting from a square antiprism and two additional straws (green and blue) of the same length as the red ones.

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II.3.2 Construction of a monocapped pentagonal antiprism and icosahedron.
Materials:
a) The pentagonal antiprism built in the section II.2 (Figure 5i-j).
b) 5 green drinking straws of 5 cm.
c) 5 blue drinking straws of 5 cm.
d) 1 m of raffia.

Figure 8 Procedure to build a mono- and bicapped pentagonal antiprisms (b and d, respectively) starting from a pentagonal antiprism (a). The bicapped pentagonal antiprism will be a regular icosahedron (d) if all of the straws are of the same length.

 

Figure 9 3D models of a trigonal dodecahedron and icosahedron made of drinking straws and raffia.

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III Application of the Polifacil models in the teaching and learning of the geometry of polyhedral molecules.
We had been using these models in our inorganic stereochemistry courses (Figure 10a) for the teaching and learning of the geometry of polyhedral molecules such as those described in the table 4. In the case of molecules which posses a central atom, the Polifacil models can be conveniently adapted simply by tying pieces of raffia from the vertices to the center of the polyhedron and modeling balls of plasticine at its sites and with the colors you consider necessary (Figure 10b).

At this point is very useful to introduce the IUPAC's nomenclature of the polyhedra which are used often for the representation of chemical structures you could be dealing in a class. It is also possible to include other topics related with the molecular structure such as the Valence Shell Electron Pair Repulsion Theory (VSEPR) and Coordination Chemistry.

 (a)

 (b)

Figure 10 a) Building Polifacil 3D models in the classroom for teaching-learning molecular geometry in an inorganic chemistry course; b) Polifacil model of a tetrahedral molecule possessing a central atom (e.g. CH4).

 

Table 4 IUPAC's nomenclature and symmetry point group of the polyhedra constructed in this work (Hartshorn, 2007).
 Polyhedron  IUPAC symbol  Symmetry point group  Related chemical structure
 Trigonal pyramid  TPY-3  C3v  NH3
 Square pyramid  SPY-5  C4v  BrF5
 Pentagonal pyramid  PPY-6  C5v  XF6
 Hexagonal pyramid  HPY-7*  C6v  MoF82-
 Tetrahedron  T-4  Td  CH4
 Trigonal bipyramid  TBPY-5  D3h  PCl5
 Octahedron  OC-6  Oh  SF6
 Square antiprism  SAPR-8  D4d  ZrF82-
 Pentagonal antiprism  PAPR-10  D5d  Fe(C5H5)2 (staggered)
 Trigonal dodecahedron  DD-8  D2d  Mo(CN)84-
 Monocapped pentagonal antiprism  PPRP-11  C5v  B11H112-
 Icosahedron  IC-12  Ih  B12H122-

* We assigned arbitrarily this polyhedron's symbol.

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IV Conclusion.
Polifacil models
have received good acceptance as a teaching and learning material of the spatial geometry, from basic school to undergraduate and graduate levels. For example, during several "National Weeks of Science and Technology" promoted in our country by the Council of Science and Technology of Mexico (CONACyT) and in our city by Council of Science and Technology of Puebla (CONCyTEP), many teachers of mathematics invited us to make workshops for teaching the Polifacil system to their children (Figure 11).

On the other hand, we have developed static (e. g. models of carbon's hybrids: Fuentes-López, 2005) and dynamic models (e. g. model for illustrate the fluxional behavior in 5-coordinated atoms; Pérez-Benitez, 1992) based on the Polifacil system which will be presented later. In all of these cases the simplicity and versatility of this system, besides the low cost of the starting materials make to the Polifacil models a very attractive and useful way of learning and teaching the spatial geometry.

 (a)

 (b)

 (c)

Figure 11 Teaching Polifacil system for building 3D models of polyhedra in primary schools of Puebla, México, during a "National Week of Science and Technology".

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V Notes and references.
a)  We coined the word "Polifacil" for naming in Spanish to this construction system of polyhedra; it is composed by two words: Poli and facil coming respectively from poliedro (polyhedron) and facil (easy).  The orthographic accent in the letter "a" was suppressed for the html presentation.

b)  Raffia is a plastic ribbon based on rayon or polypropylene. It can be bought in any haberdashery.

c)  In the Figures 2, 3, 5, 7 and 8, the straws are represented by red, blue or green straight lines while raffia is in black curved lines. Moreover, the ends of the straws or the vertices are numbered for a better understanding of the procedure.

1) Arroyo-Carmona, R. E.; Fuentes-López, H.; Méndez-Rojas, M. Á.; Pérez-Benitez, A. "La geometría: ¡Un pie que cojea en la enseñanza de la estereoquimica". Educ. Quim. 16 (Núm. Extraord.), 2005, pp. 184-190.

2) Borglund, E. "Avec de la paille". Ed. Sélection J. Jacobs. Paris, 1972. pp. 36-37.

3) Fuentes-López, H. "La construcción de modelos tridimensionales como estrategia didáctica para la enseñanza-aprendizaje de las hibridaciones del carbono a nivel licenciatura". Thesis of master of education in sciences. Faculty of Chemical Sciences. Benemérita Universidad Autónoma de Puebla. January, 2005.

4) Hartshorn, R.M.; Hey-Hawkins, E.; Kalio, R.; Leigh, G. "Representation of configuration in coordination polyhedra and the extension of current methodology to coordination numbers greater than six". Pure Appl. Chem. 79 (10), 2007, pp. 1779 -1799.

5) Pérez-Benítez, A.; González-Vergara, E. "Un tetraedro (o un tetraedro alargado) a partir de un popote y un cordel". Educ. Quím. 1991, 2, p.p. 198-200.

6) Pérez-Benítez, A.; González-Vergara, E. "Taller de construcción de poliedros moleculares". III Congreso Iberoamericano, X nacional de química inorgánica y II simposio de química del silicio. Zacatecas, Zac. April, 1991.

7) Pérez-Benítez, A.; Arroyo-Carmona, R. E. "¿Sabes qué es un hexaedro? La respuesta y un modelo plegable elaborado con materiales de desecho". Saberes Compartidos. 2008, 2, pp. 50-54.

8) Pérez-Benítez, A.; González M. R.; González-Vergara, E. "Un modelo dinámico sencillo para ilustrar la pseudorrotación de Berry". Rev. Soc. Quím. Méx. 36, 1992, p. 96. Abstract C/31.

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