ed5b00325 1..4

Discovering Symmetry in Everyday Environments: A Creative

Approach to Teaching Symmetry and Point Groups

Kei Fuchigami,* Matthew Schrandt,* and Gary L. Miessler

St. Olaf College, Northﬁeld, Minnesota 55057, United States

Supporting Information

ABSTRACT: A hands-on symmetry project is proposed as an

innovative way of teaching point groups to undergraduate

chemistry students. Traditionally, courses teaching symmetry

require students to identify the point group of a given object.

This project asks the reverse: students are instructed to

identify an object that matches each point group. Doing so

requires students to think about symmetry in their everyday

environment and aids in the development of a more intrinsic

understanding of the assignment of symmetry classiﬁcations.

KEYWORDS: Second-Year Undergraduate, Inorganic Chemistry, Humor/Puzzles/Games, Hands-On Learning/Manipulatives,

Group Theory/Symmetry, Collaborative/Cooperative Learning

he conceptualization and integration of point group

symmetry is a topic of much importance for under-

graduates in chemistry. A strong foundation in symmetry

provides students with a powerful tool for understanding

advanced topics such as group theory, ligand design, spectros-

copy, and crystallography. Therefore, eﬀective teaching

methods and activities for introducing symmetry are valuable

tools for instructors. We propose a new type of project that asks

students to identify objects from their surroundings that match

speciﬁc point groups. This approach diﬀers from typical

textbook problems, which usually ask students to assign point

groups to designated objects described or shown as pictures in

text. This project requires the students to fully understand the

symmetry operation(s) (i.e., rotational symmetry, mirror plane,

etc.) of each point group and challenges them to carefully

analyze and interact with more objects than the limited number

provided in a textbook.

The ﬁnal product of this project is a table reminiscent of the

familiar periodic table of elements, but each element is replaced

by a point group with a corresponding image of an object that

the students found around their campus. The result of our

investigation of objects on the St. Olaf College campus is

shown in Figure 1.

■

CONCEPT

There are many methods for introducing symmetry and point

groups to students. Some of these include online tools,

software, and books with images from around the world to help

visualize point groups.

1−5

In the conventional teaching of

symmetry, students are prompted to assign a point group and/

or symmetry operations to a given object or image.

6−20

Other,

more hands-on approaches, such as those introduced by Flint

and Scalfani et al., encourage students to understand and better

visualize symmetry through 3D models.

21,22

Another valuable

activity is provided on the Virtual Inorganic Pedagogical

Electronic Resource (VIPEr) Web site, called the “Symmetry

Scavenger Hunt.” This activity resembles our project in also

having students seek various point groups around campus.

By participating in our proposed activity, students grasp the

shapes and patterns necessary to analyze molecules and objects

in terms of group theory. Before looking for an object, the

student is required to ﬁnd and understand the symmetry

operations that comprise each point group. The next task is to

visualize the type of object that fulﬁlls each of the symmetry

operations. Some point groups are extremely diﬃcult to ﬁnd or

imagine. Examples of lower order symmetry such as C

or C

may also prove rather diﬃcult to ﬁnd because candidates for

such point groups may turn out on closer observation to have

or other higher order point groups. Many objects have very

similar sets of symmetry operations; therefore, ultimately the

most challenging task is to ﬁnd an object with certain symmetry

operations while excluding others.

Received: May 19, 2015

Revised: February 15, 2016

Published: March 3, 2016

Activity

pubs.acs.org/jchemeduc

Division of Chemical Education, Inc.

1081 DOI: 10.1021/acs.jchemed.5b00325

J. Chem. Educ. 2016, 93, 1081−1084

To understand this project better, the following is a more

detailed explanation of the student’s thought process. For

example, a student is assigned the point group D

.Toﬁnd a

suitable object, the student may ﬁrst decompose the point

group into its corresponding symmetry operations. Under-

standing the operations that are required to fulﬁll the

assignment of this point group will be an important ﬁrst step

to visualizing the type of object that is needed. For example, in

Figure 1 for the D

point group, we have an image of two,

four-legged tables stacked such that the top table is ﬂipped and

turned 45° instead of perfectly mirroring the bottom table. For

this object, the student must ﬁrst understand that if there were

only one table, it would fulﬁll the C

rotational symmetry

operation of D

, but a single table would have no

perpendicular C

axis of rotation required for the D point

group assignment. The student can then further stack two

tables so they are mirror images, but the pair would still have

only D

symmetry; therefore the student must ﬁgure out a way

to remove the horizontal plane of symmetry while adding the

vertical planes of symmetry (4σ

)ofD

. Finally, once the

student rotates the top table by 45°, the student will remove the

horizontal plane of symmetry, and the pair of tables will fulﬁll

the D

point group assignment.

In addition to ﬁnding an object matching a particular point

group, this project requires the students to write a short

explanation of how their object fulﬁlls their assigned point

group. For example, for D

and T

symmetry, we have

assembled toys with the correct shapes, but with faces of

diﬀerent colors; our explanation notes that the diﬀerent colors

on the faces should not be considered for their point group

assignment. In many cases, the objects we have chosen, for

example ﬂowers, require simpliﬁcation to fulﬁll the point group.

This written portion will reveal to the instructor the extent of

the student’s understanding of the point group by explaining

what is required, and what must be omitted in order to fulﬁll

the assigned point group. A complete list of the explanations for

each photo in our periodic table of symmetry is provided in the

Supporting Information.

Traditional teaching methods give students a limited

selection of objects and a speciﬁc number of point groups;

however, this new project gives the students an open-ended

point group challenge.

1,7−9

It is likely to be far more

challenging for students to ﬁnd point groups in objects when

they have the whole world to work with, instead of a limited

selection of objects. We hope that this project allows for the

students to be more engaged and provides them another tool to

learn symmetry from a diﬀerent perspective.

■

DESIGN

In an eﬀort to create a visualization displaying the symmetry in

everyday objects, such as a ﬂower, we captured pictures of

objects observed on our campus with various symmetries. A

photo for each point group was obtained and the collection of

photos was organized into blocks. As a project connected to

chemistry, adopting the form of a periodic table was opportune;

however, to prevent confusion, the instructor should emphasize

that there is no genuine connection between the point group

table the students are constructing and the periodic table of

elements. From the table, students can visualize the hierarchy of

increasing point group symmetry and be reminded of the point

group labels by associating the labels (S, D, or C) with the

blocks in the original periodic table of elements. This table will

not replace any teaching material provided in textbooks such as

Figure 1. Periodic table of symmetry (larger image is provided in the Supporting Information).

Journal of Chemical Education Activity

DOI: 10.1021/acs.jchemed.5b00325

J. Chem. Educ. 2016, 93, 1081−1084

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a point group ﬂowchart, but instead, it can help provide some

clarity to students for whom the current method is not intuitive.

A general guide to the table is provided in Figure 2: within

the C and D blocks, reﬂection plane symmetry is designated by

rows (i.e., the ﬁrst row of the C/D block has no reﬂection plane

symmetry, the second row has vertical or dihedral plane

symmetry, and the third row has horizontal plane symmetry).

However, in the C block the C

symmetry is ignored in this

trend, and the order of rotational symmetry given by column

increases toward the right in each block. Higher order

symmetry is also found near the bottom in each block. The S

block includes C

and C

, despite their labels because their

operation is equivalent to S

and S

, respectively. Much to our

enjoyment, certain connections to the periodic table were kept

such as the S and D blocks along with the general shape.

The photos in Figure 1 were taken primarily in the course of

two days, with revisions made later. As constraints for our

project, only pictures taken on campus were eligible and no

pictures from the Internet were allowed. To make the project

more creative, we made an attempt to allow as few molecular

structures as possible. Molecular models and orbital calculations

were allowed when no other option was present. Future

projects could choose more or less strict interpretations of

these conditions. The ﬁnal product, after revision by faculty and

students, was printed and displayed for all to view. A copy of

the ﬁnal document, in addition to a list of detailed explanation

for each point group assignment, has been made available in the

Supporting Information.

In our selection of each point group object, measures were

taken to ﬁnd objects that represented a deﬁnite symmetry. A

tour of campus led to the discovery of the common point

groups such as D

∞h

∞v

, and C

,in

contrast to the less common higher-order symmetries such as

, and C

, which were more diﬃcult to ﬁnd.

Further exploration and imaginative creations revealed objects

displaying uncommon symmetries (D

, and

). Computational modeling was allowed for two objects, D

and D

Some of the pictures in the table have diﬃcult symmetry

operations to identify. For instance, the bipyramidal structure

of D

symmetry has dots along the corners of a face near the

square plane of the shape. A fun addition to our table, C

,isa

component of St. Olaf College’s Rube Goldberg machine,

which is a four-paddled propeller. As previously mentioned, a

complete list of the explanations for each photo in our periodic

table of symmetry is provided in the Supporting Information

that will discuss these details.

■

CONCLUSION

The proposed Periodic Table of Symmetry project is designed

to help chemistry students understand symmetry by ﬁnding

everyday objects that correspond to given point groups. It will

enhance students’ understanding of how to apply all of the

symmetry operations of a point group and emphasizes the

challenging c omponent of correctly including, and more

importantly, excluding symmetry operations to ensure that

the object has the correct point group. This pro ject is

recommended not as a replacement for traditional approaches

of classifying molecules but rather as a project that will help

enliven students’ interest in symmetry as well as solidify their

understanding after the basic symmetry principles have been

explained.

■

ASSOCIATED CONTENT

Supporting Information

The Supporting Information is available on the ACS

Publications website at DOI: 10.1021/acs.jchemed.5b00325.

Instructor information. (PDF)

Instructor information. (DOCX)

Student handout. The student handout contains the

instructions that will guide students through the activity.

(PDF)

Student handout. The student handout contains the

instructions that will guide students through the activity.

(DOCX)

Example ﬁnal project. In addition, a sample ﬁnal project

is provided for instructors’ use. (PDF)

Example ﬁnal project. In addition, a sample ﬁnal project

is provided for instructors’ use. (DOCX)

Blank periodic table of symmetry. (PDF)

Blank periodic table of symmetry. (DOCX)

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected].

Notes

The authors declare no competing ﬁnancial interest.

■

ACKNOWLEDGMENTS

We would like to thank Dr. Robert Hanson for his revisions of

our project and for the idea of using JMol for molecular

imaging. We would also like to thank all of the various faculty

and students who contributed along the way; their suggestions

and encouragement made this project very enjoyable.

■

REFERENCES

(1) Symmetry @ Otterbein. Symmetry Resources at Otterbein

College. http://symmetry.otterbein.edu (accessed February 2016).

(2) Charistos, N.; Tsipis, C.; Sigalas, M. Teaching Molecular

Symmetry with JCE WebWare. J. Chem. Educ. 2005, 82, 1741.

(3) Kastner, M. E.; Leary, P.; Grieves, J.; DiMarco, K.; Braun, J. Point

Group I, II, and III. J. Chem. Educ. 2000, 77, 1246.

(4) Hargittai, M. Visual Symmetry; World Scientiﬁc Publishing

Company: Singapore, 2009.

(5) Hargittai, M. Symmetry through the Eyes of a Chemist; Springer:

New York, 2009.

(6) Miessler, G.; Fischer, P. J.; Tarr, D. A. Inorganic Chemistry, 5th

ed.; Prentice Hall: Upper Saddle River, NJ, 2013; pp 111−113.

Figure 2. Higher order symmetry is observed at the bottom of the

table. Intrablock symmetry is increased in groups to the right.

Journal of Chemical Education Activity

DOI: 10.1021/acs.jchemed.5b00325

J. Chem. Educ. 2016, 93, 1081−1084

1083

(7) Housecroft, C. E.; Sharpe, A. G. Inorganic Chemistry, 3rd ed.;

Pearson Education Limited: London, 2008; pp 88−114.

(8) Atkins, P.; Overton, T.; Rourke, J.; Weller, M.; Armstrong, F.;

Hagerman, M. Shriver & Atkins’ Inorganic Chemistry, 5th ed.; Oxford

University Press: Oxford, 2010; pp 179−198.

(9)House,J.E.Inorganic Chemistry, 1st ed.; Elsevier Inc.:

Amsterdam, 2008; pp 137−175.

(10) Orchin, M.; Jaffe, H. H. IX − Symmetry, point groups, and

character tables. Part I: Symmetry operations and their importance for

chemical problems. J. Chem. Educ. 1970, 47, 246.

(11) Orchin, M.; Jaffe, H. H. IX − Symmetry, point groups, and

character tables. Part II: Classification of molecules into point groups.

J. Chem. Educ. 1970, 47, 372.

(12) Orchin, M.; Jaffe, H. H. IX − Symmetry, point groups, and

character tables. Part III: Character tables and their significance. J.

Chem. Educ. 1970, 47, 510.

(13) Zeldin, M. An introduction to molecular symmetry and

symmetry point groups. J. Chem. Educ. 1966, 43, 17.

(14) Herman, M.; Lievin, J. Group theory. From common objects to

molecules. J. Chem. Educ. 1977, 54, 596.

(15) Burrows, E. L. Pictures of point groups. J. Chem. Educ. 1974, 51,

87.

(16) Glasser, L. Teaching symmetry: The use of decorations. J. Chem.

Educ. 1967, 44, 502.

(17)Faltynek,R.A.GroupTheoryinAdvancedInorganic

Chemistry: An Introductory Exercise. J. Chem. Educ. 1995, 72, 20.

(18) White, J. E. An introduction to group theory for chemists. J.

Chem. Educ. 1967, 44, 128.

(19) Sein, L. T. Dynamic Paper Constructions for Easier Visual-

ization of Molecular Symmetry. J. Chem. Educ. 2010, 87, 827.

(20) Noggle, J. H. Flow chart for identification of Schoenfliess point

group. J. Chem. Educ. 1976, 53, 190.

(21) Flint, E. Teaching Point-Group Symmetry with Three-

Dimensional Models. J. Chem. Educ. 2011, 88, 907.

(22) Scalfani, V. F.; Vaid, T. P. 3D Printed Molecules and Extended

Solid Models for Teaching Symmetry and Point Groups. J. Chem.

Educ. 2014, 91, 1174.

(23) Symmetry Scavenger Hunt. Virtual Inorganic Pedagogical

Electronic Resource. https://www.ionicviper.org/class-activity/

symmetry-scavenger-hunt (accessed February 2016).

Journal of Chemical Education Activity

DOI: 10.1021/acs.jchemed.5b00325

J. Chem. Educ. 2016, 93, 1081−1084

1084