Chemical Education Journal (CEJ), Vol. 17/Registration No. 17-102/Received March 5, 2015.
URL = http://chem.sci.utsunomiya-u.ac.jp/cejrnlE.html

Teaching Helical Chirality with a Paper Helicopter

ARROYO-CARMONA Rosa Elena¹, REYES-LUCAS Pablo¹, RAMIREZ-GUTIERREZ Ramses Elias¹, RIVERA-ORTEGA Jose Angel¹, MELENDEZ-BUSTAMANTE Francisco J.¹, PEREZ-BENITEZ Aaron¹*, GUERRERO-SANCHEZ Alfredo Jibran², CUENCA-LARA Humberto², RANGEL-SANCHEZ Armando² and COCA-SANTILLANA Flor²

*E-mail: aaron.perezcorreo.buap.mx

Abstract (Resumen)
The construction and usage of a paper helicopter in the teaching-learning of P/M-chiral descriptors are presented. These descriptors are used to characterize helical molecules such as helicenes and liquid crystals.
(Se presentan la construccion y el uso de un helicoptero de papel en la enseñanza-aprendizaje de los descriptores quirales P/M (Plus/Minus). Estos descriptores se usan en la caracterizacion de moleculas helicoidales tales como los helicenos y los cristales liquidos.)

Keywords: Paper helicopter, helical chirality, P/M-chiral descriptors, hands-on activity, teaching stereochemistry

Contents

Introduction

Helical chirality

The construction of a paper helicopter

The physics of a paper helicopter

Conclusion

Acknowledgement

References

Introduction

Chirality is the physical property of an object to be non-superimposable with its mirror image. This word is related with the ancient Greek kheir, which means hand. It was introduced in 1904 by Lord Kelvin in his Baltimore lectures on "Molecular Dynamics and the Wave Theory of Light", as follows [1, 2]:

"I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself"

In this context is very important, of course, to mention the work of Louis Pasteur, the father of modern microbiology and a pioneer of stereochemistry. He was capable manually to separate, left and right handed crystals of sodium ammonium tartrate (Figure 1), the so called "racemic mixture" (1:1 mixture of dextro- and levo-rotatory samples of ammonium sodium tartrate) [1, 3].

Figure 1 Louis Pasteur and a schematic representation of enantiomorphic crystals of (+)- and (-)-ammonium sodium tartrate, which were manually separated by himself [4].

Actually, among chemists, physicians and people related with the synthesis and production of drugs, the term chiral is very popular, and necessary, because more than one-half of marketed drugs are chiral [5]. This fact acquired great relevance since 1962, when ca. 12,000 babies suffered phocomelia due to the consumption of racemic R/S-Thalidomide by their mothers being pregnant ("Thalidomide tragedy"). Although initially was believed and demonstrated that R-enantiomer was sedative for pregnant mothers and S-enantiomer was teratogenic for their babies [6-9]; actually is known that Thalidomide racemices in-vitro as well as in-vivo [10].

Helical chirality

There are three types of chirality elements: Chirality center, chirality plane and chirality axis. The existence of at least one of these elements in a molecule makes it chiral, meaning non-superimposable with its plane-mirror image.

Many types of molecules present a chirality axis [11]; however, there are different kinds of descriptors to name each one; for example, chiral descriptors P/M, Ra/Sa and Δ/Λ are mentioned in IUPAC's Gold Book [12]. In particular, in the case of molecular (e.g. helicenes) and supramolecular (e.g. biological) helices is more common the use of P/M-chiral descriptors, whereas in helical coordination compounds (e.g. helicates) Δ/Λ-chiral descriptors predominate.

M (or minus) and P (or plus) chiral descriptors are used respectively, to describe left- and right-handed helices (Figure 2). Since a geometrical point of view, if we take a small part of the left-handed helix then we will observe it possesses a negative slope (m<0); by contrast, a right-handed helix can be described with a positive slope (m>0).

At this point, students must remember that line's slope is defined as:

m = Δy/Δx = (y2-y1)/x2-x1 = rise/run or tan(α),

where α is the corresponding angle in standard position (An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the terminal side. The angle is measured in a counter-clockwise direction (Figure 2, bottom) [13]).

Figure 2 Left- and right-handed helices are described with M- and P-chiral descriptors and possess negative (m<0) and positive slopes (m>0), respectively. Mr. Slope and the definition of an angle in standard position are at bottom.

On the other hand, classical examples of molecular helices are helicenes, which are aromatic ortho condensed rings. When the number of rings is higher than four, the system cannot already be planar and adopts a helical structure to liberate the steric congestion. In the case of [5]-helicene the racemization barrier at 27 °C is 24.1 kJ/mol and it increases to 36.2 kJ/mol for [6]-helicene (Figure 3) [14].

Figure 3 Line formulas and ball and sticks representations of ortho condensed rings. When the number of ortho condensed rings are higher than four, M- and P-helices are produced (i.e. [5]- and [6]-helicenes).

The physics of a paper helicopter

Although there are several forces acting on a paper helicopter during its flight, they can be reduced to (Figure 4): a) A downward force due to earth gravitation (g); b) A couple of forces, perpendicular to its body, that point in opposite directions; c) An upward force due to minor pressure on the top of blades (Bernoullie's Theorem [15]).

Once the helicopter's descending begins due to gravity, airflow is produced between its blades. Because blades are not in straight line, forces do not cancel but produce a whirlpool effect, whose clockwise or counter clockwise direction depends on their initial orientation (Figure 4c). Such orientation does not change during the flight and blades rise up to an equilibrium position.

For a better observation of spin direction, we designed three helicopters: A "standard helicopter" (Figure 5); a helicopter possessing broad blades (Figure 6) to reduce its spin velocity and a helicopter showing inclined blades since the beginning (Figure 7). In all cases, enantiomorphic helicopters are provided.

Figure 4 a) Perpendicular view to helicopter's body illustrating the main forces acting on a paper helicopter; b) Top view showing the direction of airflow between blades; c) The whirlpool direction produced by opposed and unaligned forces of airflow.

Building paper helicopter models

a) Cut out the templates of Figures 5-7.
b) Crumple the circles in small balls as compact as possible.
c) Observe at this moment that both structures do not fly but drop in the same manner.
d) Cut along central line a-b and bend the wings in opposite directions following the instructions into templates labeled as M and P.
e) Drop (or throw catching the ball of) M-helicopter and observe the direction it flies. Repeat the same procedure for P-helicopter. Check that once the planarity breaks (after bending their blades up to the indicated positions) they become chiral and enantiomorphic objects from one other (Figure 8).
f) If you look at M-helicopter parallel to paper´s plane, you will observe its wings have negative slope. This helicopter flights in counter-clockwise direction describing an M-helix. By contrast, P-helicopter's wings have positive slopes and flights in clockwise direction describing a P-helix. This fact can be correlated with pictures of Figure 2.
g) Terminated models are presented in Figure 8. See video for paper helicopters' flights in slow motion (download as mp4, time 1:56 to 2:40 minutes) or use models possessing wide blades (Figure 6) for verifying by yourself, the directions they flight.

Figure 5 Templates for building standard paper helicopters capable to flight in counter-clockwise (M-helicopter) and clockwise (P-helicopter) directions.

Figure 6 Templates for building paper helicopters possessing wide blades. This fact reduces their velocity and helps to visualize the direction of spin.

Figure 7 Templates for building paper helicopters with inclined blades. This feature helps to stablish the slope's sign of the whirlpool produced by M- and P-helicopters while dropping.

Figure 8 Terminated models of left (M = minus) and right (P = plus) standard helicopters. Note they are enantiomorphic (mirror image each other).

Conclusion

In our stereochemistry and didactic of chemistry courses, we take apart the spatial configuration of helicenes and related them with the spatial configuration of paper helicopters, as a funny hands-on activity for the teaching-learning of P/M-chiral descriptors (Figure 9a-b).
Although paper helicopter is known since the beginnings of 90's [16] and it has been used in the teaching-learning of statistics (Figure 9c) [17, 18], nature has anticipated to man with their own helicopter seeds, which disperse kilometers far from tree to ensure its reproduction (e.g. Gyrocarpus americanus (Figure 9d), an endemic tree of Oaxaca, Mexico) [19].

Acknowledgement
Special thanks are expressed to Dirección General de Planeación Institucional, BUAP, for the financial support to our research group BUAP-CA-263 (Investigación Experimental y Teórica de Nuevos Materiales y Educación en Ciencias).

(a)	(b)
(c)	(d)

Figure 9 Building paper helicopters in our classroom for the teaching-learning of helicity: (a) left- and (b) right-helicopter; (c) Physical parameters of a paper helicopter for statistical purposes; (d) A gyrocarpus americanus seed from Mexico.

References

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[5] Millership, J.; Fitzpatrick, A. "Commonly used chiral drugs: a survey". Chirality, 5, 573-576 (1993).
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[13] Roberts, Donna. "Standard Position and Reference Angles". Retrieved from: <http://www.regentsprep.org/regents/math/algtrig/ATT3/referenceAngles.htm>
[14] Rajca, A.; Miyazaka, M. in: "Functional Organic Materials: Syntheses, Strategies and Applications" by Müller T. J. J., Bunz Uwe H. F., John Wiley & Sons, Germany (2007), p. 567.
[15] Dingle, L.; Tooley, M. H. "Aircraft Engineering Principles". Routledge, New York (2013), p. 481.
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[17] Erhardt, E. (2006). Retrieved from: <http://statacumen.com/pub/Erhardt_slides_RSM_paper_helicopter_20061104.pdf>.
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[19] Rivera-Hernández, J. E. "Notas sobre Hernandiaceae: Primer registro de Gyrocarpus americanus JACQ para Mexico y de Sparattanthelium Amazonum Mart para Oaxaca". Acta Botanica, enero, numero 078, 67-76 (2007). Retrieved from: <http://www.redalyc.org/articulo.oa?id=57407804>.