3-dimensional models showing chemical point group symmetry

A three-dimensional instructional model used to exhibit and represent specific point groups used in the application of group theory to symmetry operations and molecular structure. A pair of blocks, each have a pair of spaced faces with peripheral side surfaces extending between the faces, the number of side surfaces being equal to the principal rotational axis order of the point group to be represented. The blocks are connectable to one another with a face of each of the blocks opposing. The connection between the blocks can be made using a circular shaft inserted through openings in the blocks, by pins inserted in holes in abutting opposing faces of the blocks, or by adhering the face of one block directly to the face of the other to hold them rigidly in engagement with each other. The interconnection can also be made using a rotational locking mechanism allowing the blocks to rotate with respect to one another. A number of pegs are located in peg openings disposed in the side surfaces of each block, the pegs representing asymmetric units contained in each point group that are repeatable as the point group has the symmetry operations associated with the particular point group performed on it. The pegs can be either removably or permanently contained within the peg openings depending upon the flexibility desired for the model.

FIELD OF THE INVENTION 
The present invention relates generally to instructional aids used in 
teaching concepts relating to organic and inorganic chemistry. More 
specifically, the invention relates to three dimensional models used to 
illustrate specific concepts and specific axial point groups when 
explaining the application of group theory to symmetry operations and 
molecular structure. 
BACKGROUND OF THE INVENTION 
Group theory is a mathematical formalism which has important applications 
in chemistry, physics and geology. In particular, the symmetry operations, 
namely rotations, reflections, inversions, reflection-rotations, and 
identity, inherent in molecular structures, constitute mathematical 
groups. These mathematical groups are known as point groups because one or 
more points at the center of the structure are unshifted by all of the 
operations of the group. Each point group has an overall order, equal to 
the number of times the basic asymmetric unit, or any random point, of a 
symmetrical structure will be repeated. This overall order also equals the 
number of symmetry operations associated with a given point group. Group 
theory and classification schemes for assigning molecules to their point 
groups are taught in advanced undergraduate and/or beginning graduate 
courses in the chemistry and physics curricula of modern universities. 
Many important properties of various organic and inorganic molecules can be 
ascertained through assigning point groups comprised of three dimensional 
symmetry operations to those molecules. For example, depending on the 
point group assigned to a given molecule based on the number of symmetry 
operations found for that molecule, one can determine the presence or 
absence of a dipole moment in the molecule. Whether the molecule is 
optically active can be determined using point group symmetry. Point group 
symmetry may also be used to evaluate whether certain molecules require 
resonance structures for an accurate valence bond representation of 
bonding and also whether the atomic orbitals in particular orientations of 
certain molecules will have a finite or zero net overlap. In short, point 
group symmetry has numerous beneficial applications in evaluate and 
classifying various organic and inorganic molecules, while requiring only 
analysis of certain symmetrical attributes of the structure of the 
molecule. 
The teaching of the principles of point group symmetry is difficult because 
specific molecules belonging to various point groups have very disparate 
structures and their three dimensional structures are often difficult to 
represent in two dimensions. Thus, accurate representations in textbooks 
of various structures and molecules exhibiting certain point groups are 
not readily achievable. 
Various molecular model sets constructed from numerous materials are 
well-known and commonly used teaching tools for illustrating molecular 
structures, mainly the relative positions of the constituent atoms in the 
molecule. Their utility is not readily extended to the systematic 
illustration of the symmetry point groups, because in most molecules, some 
atoms sit on the symmetry axes, planes or centers, and therefore are fewer 
in number than the full order of the group. For example, the molecule 
AuCI.sub.4.sup.- is in the point group D.sub.4 h, that has an overall 
order of 16, but the molecule itself has only four chlorides lying off the 
principal four-fold axis of the point group. 
BRIEF SUMMARY OF THE INVENTION 
It is an object of the present invention to provide three dimensional 
models that accurately and vividly exhibit all symmetries associated with 
various point groups present in certain molecules and present asymmetric 
units equal in number to the overall group order. The models have 
sufficient visual impact to immediately reveal the symmetry operations 
present in the model and to allow students to explore the differences in 
the symmetry of the chemically important point groups. 
It is a further object of the invention to provide three dimensional point 
group models comprised, and assembled from, basic, interchangeable parts 
that may be increased or decreased in size to create either large models 
suitable for classroom and lecture hall use, or smaller models to be 
assembled from inexpensive kits for use by students. For example, in 
analyzing the symmetry of a particular molecule, a model of the point 
group assigned to that molecule could be utilized in a classroom to 
clearly illustrate how the symmetry operations are performed on that 
molecule to assign the molecule that particular point group. 
In particular, models for the axial point groups (C.sub.n , C.sub.nv, 
C.sub.nh, D.sub.n, D.sub.nh, D.sub.nd, and S.sub.2n ; where n is the order 
of the principal rotational axis and can be 2,3,4,5 or 6) can be provided 
by the present invention, thereby illustrating the most important orders 
of rotational axes for molecular symmetry; 2-fold, 3-fold, 4-fold, 5-fold 
and 6-fold. Interchangeable models for the point groups with a 4-fold axis 
can also be used for point groups with 2-fold axes. Models for point 
groups with 6-fold axes can be used to illustrate point groups with 3-fold 
and 2-fold axes. 
In accordance with the invention, a pair of geometric prisms or blocks, 
formed as thick plates of a rigid and durable material, are used to 
construct models representative of specific point groups. The plates 
represent the order of the principal rotational axis of the point group, 
the principal axis passing perpendicularly through the center of the 
plates. Each block has a pair of spaced faces with peripheral side 
surfaces extending between the faces. The faces on each block have a 
geometric shape with a number of sides, allowing the block to simulate a 
point group having a principal rotational axis order equal to the number 
of sides in the geometric shape of the face on the block. In other words, 
the number of times a block can be rotated around its center to a new 
position symmetrical from a previous position equals the principal 
rotational axis order of the block. This number is also equal to the 
number of side surfaces present on the block. For example, triangular, 
square, pentagonal and hexagonal blocks having 3, 4, 5 and 6 sides are 
used to form models representing the corresponding point groups with 
principal rotational axes of order 3, 4, 5 and 6, respectively. 
In a complete model representing a single point group, two geometric blocks 
with the same number of sides are interconnected with faces of each of the 
blocks opposing each other. The opposing faces may abut or be spaced from 
each other. The side surfaces of each block are in either a peripherally 
aligned configuration or a peripherally offset configuration with respect 
to each other. The different configurations are necessary as certain of 
the point groups to be represented have aligned configurations while 
others have offset configurations. The blocks can be interconnected to 
create a rigid, permanent model either by directly connecting adjacent 
faces of the blocks to each other, or by inserting a shaft through a shaft 
opening located in the center of each block, and securing the blocks to 
the exterior of the shaft, leaving the blocks spaced a distance apart from 
one another along the shaft. The shaft may be circular to allow any 
desired peripheral configuration of the side surfaces to be established. 
The shaft also illustrates the direction and position of the principal 
rotational axis of the point group shown by the model. As stated before, 
the order of this principal axis is determined by the number of side 
surfaces present on the blocks. 
In either construction, an appropriate number of pegs representing the 
asymmetrical units for the particular point group to be represented are 
inserted into peg mounting openings in the side surfaces of one or both of 
the blocks to form a model of the desired point group. In performing the 
symmetry operations associated with a particular point group on a model 
representing that point group, it is the number of times the 
configurations of the asymmetric units, represented by the pegs, can be 
repeated that illustrate the overall order of the model and point group. 
The overall order of a particular model or point group is greater than the 
principal rotational axis order of that group for most point groups, 
except for the Cn group, where the orders are equal due to the lack of any 
symmetry operations other than rotations about the principal rotational 
axis. The pegs may be secured in the peg mounting openings, as by an 
adhesive. 
In another embodiment of the invention, the blocks are not permanently 
interconnected to one another. A number of connecting pins are inserted 
into opposed pin openings spaced around the central axis of the blocks on 
the opposing interior faces of the geometric blocks to rigidly, but 
removably, hold the pair of blocks in connection with each other. As the 
pins are removable from the pin openings, the pins may be reused to form 
models of other point groups utilizing different geometric blocks. 
The pegs representing the asymmetrical units of the point groups formed 
using this embodiment may also be attached to the geometric blocks in a 
removable fashion to allow for various point groups with the same order to 
be illustrated using the same pair of geometric blocks. 
In a further embodiment of the invention, the geometric blocks are 
constructed to contain a rotational locking mechanism to allow them to be 
rotatable with respect to one another when interconnected to form the 
point group model. A spring-biased detent is disposed within and extends 
outwardly from the interior face of one geometric block and a series of 
depressions are disposed in an equally spaced, circular pattern on the 
opposing interior face of the second, opposed geometric block. The 
circular pattern of the depressions has a radius equal to the distance of 
the spring biased detent from the center of its respective geometric 
block, to allow the detent to selectively engage each of the depressions. 
The depressions are disposed on the interior face of the second, opposed 
geometric block in such a way so as to allow the blocks to be rotated and 
held in positions wherein the side surfaces of the blocks are peripherally 
aligned or also peripherally offset. Typically, the amount of peripheral 
offset is equal to one-half "turn", i.e. 1/2( 360/ n ) where n is the 
number of side surfaces present on the geometric blocks. The blocks are 
connected to each other with these interior faces opposing each other 
using a threaded bolt inserted through a shaft opening located in the 
center of each geometric block and in the center of the circular pattern. 
As the blocks are rotated with respect to each other, the detent is 
displaced inwardly by the interior face of the opposing block, allowing 
the blocks to rotate, until the detent reaches a depression. The detent 
extends into the depression, locking the blocks in that position until a 
sufficient force is applied to inwardly displace the detent again. The 
necessary rotational forces can easily be generated manually. 
To facilitate rotation of the blocks with respect to one another, before 
connection of the blocks using the bolt, a thin, low friction separating 
disk, having a bolt opening located at its center, may be disposed between 
the two blocks with the bolt opening aligned with the shaft openings of 
the geometric blocks. The bolt is secured on the exterior faces of both 
blocks by a washer and nut threadably mounted on the bolt and tightened 
down onto the exterior faces of the geometric blocks. A pair of rigid 
handles having threaded bolt cavities are threadably mounted onto the ends 
of the bolts that protrude beyond the nuts from the exterior faces of the 
geometric blocks. These rigid handles may also be partially contained 
within handle cavities disposed in the center of the exterior faces of the 
geometric blocks, the handle cavities surrounding the shaft opening and 
extending part way through the blocks. 
In a further embodiment of the present invention, the blocks are replaced 
with a cube to represent both the octahedral (O,O.sub.h ) and tetrahedral 
(T,T.sub.h, T.sub.d) point groups, or the cubic point groups, 
collectively. These point groups have multiple intersecting high order 
rotational axes, namely, fourth-order axes passing through the centers of 
opposite faces of the cube and/or third-order axes passing through 
opposite corners of the cube, and, therefore, high overall orders. To 
illustrate these point groups, a pair of peg mounting openings are located 
in each corner of each face of the cube. The openings are spaced 
equidistant from the diagonal line intersecting the respective corner in 
which the openings are located and also a short distance inward from the 
edges of the faces forming the corner. Each opening extends a short 
distance into the cube so as not to intersect any peg mounting opening on 
an adjacent cube face. The attachment of the pegs within the peg mounting 
openings is achieved in a similar manner as with the previous embodiments 
and a number of pegs utilized also depends, as before, upon the particular 
point group to be represented by the model. 
In addition to the variations in the construction of the blocks described 
above, the construction of the pegs may also be varied. For example, the 
pegs may be formed of a metal mounting portion, that is inserted into peg 
mounting openings located in the side surfaces of the geometric blocks, 
and a wooden knob attached to the free end of the metal mounting portion. 
The pegs, while being removable from the peg mounting openings, are held 
within the openings by magnets disposed in magnet slots extending into the 
block from the interior opposed faces of the blocks. The magnets intersect 
the interior ends of the peg mounting openings. The magnetic attraction 
between the magnet and the metal mounting portion of the peg is sufficient 
to securely hold the peg within the opening, but is weak enough to allow 
the peg to be manually removed.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
FIGS. 1-3 illustrates a 3-dimensional point group symmetry model 10 
constructed according to the present invention. Model 10 is formed of a 
pair of plate-like geometric prisms or blocks 12a, 12b, a shaft 28 and a 
number of pegs 20. Blocks 12a, 12b, shaft 28, and pegs 20 may be formed of 
wood, plastic, metal or other suitable material. Blocks 12a, 12b each have 
an exterior, unopposed face 14 and an interior, opposed face 18. Exterior 
face 14 and interior face 18 of each block are joined by side surfaces 16. 
Depending on the point group to be represented by the model 10, blocks 
12a, 12b will have an appropriate geometric shape and a number of side 
surfaces 16 commensurate with that geometric shape. Specifically, blocks 
12a, 12b have a geometric shape with a number of sides to simulate a point 
group with an axis order equal to the number of sides of the geometric 
shape. FIG. 1 shows four sided blocks 12a, 12b to represent point groups 
with an axis order 4. 
Each side surface 16 has a pair of peg mounting openings 26. With model 10 
oriented as shown in FIG. 1, peg mounting openings 26 are positioned 
equidistant from the vertical center line of each side 16 and along the 
horizontal center line of each side 16. A complete set of openings 26 will 
number 4 times the axis order. With the four sided blocks 12a, 12b shown 
in FIGS. 1 et seq the number of openings will be sixteen, i.e. 4 times the 
axis order 4. This represents all possible symmetry generated positions 
needed to be present in order to generate, in turn, all possible point 
groups of that order. 
Referring now to FIG. 3, pegs 20 are comprised of a peg mounting portion 22 
and a peg knob 24 attached to the free end of peg mounting portion 22. Peg 
mounting portions 22 of pegs 20 are inserted and retained in peg mounting 
openings 26 located in side surfaces 16 of blocks 12a, 12b. While each 
side surface 16 has a pair of peg mounting openings 26, the number of pegs 
20 contained in the peg mounting openings 26 on side surfaces 16 of blocks 
12a, 12b depends upon the particular point symmetry group to be 
represented by the model 10 formed of blocks 12a, 12b. Inserting the 
requisite number of pegs 20 into the proper openings 26 will create models 
for all 7 or 6 point groups for a particular order of rotation. All the 
mounting peg openings 26 need not contain a peg 20 and for permanent 
models only openings 26 that must accept a peg need be formed in blocks 
12a, 12b. 
The blocks 12a, 12b of model 10 are connected to one another by shaft 28 
which is coaxial with central axes of blocks 12a, 12b and therefore 
symbolically represents the axis of the molecule. Shaft 28 is inserted 
through shaft openings 30 located in the center of each block 12a, 12b and 
extending therethrough. 
FIGS. 1 and 3 show the blocks 12a, 12b of model 10 with the side surfaces 
16 of each block in a peripherally aligned position. On attachment of 
blocks 12a, 12b to shaft 28 through shaft openings 30, one of the blocks 
12a or 12b may be rotated about shaft 28 to a position in which the side 
surfaces are peripherally offset and in which corresponding side surfaces 
16 on blocks 12a, 12b are not in alignment with each other to represent 
point groups that contain offset configurations. FIG. 8 illustrates this 
orientation. The amount of peripheral offset is usually equal to one-half 
"turn" i.e. 1/2(360/n) where n is the number of side surfaces of the 
geometric shape of a block 12a, 12b. However, certain point groups may 
have the blocks 12a, 12b at any arbitrary angle of rotation from 0.degree. 
to 90.degree.. Shaft 28 and openings 30 are peripherally circular to 
facilitate such rotation. 
In forming model 10 by connecting blocks 12a, 12b to shaft 28 and 
connecting pegs 20 to blocks 12a, 12b using peg mounting openings 26, an 
adhesive may be applied to shaft 28 and peg mounting portions 22 of pegs 
20 to secure blocks 12a , 12b around shaft 28 and pegs 20 within peg 
mounting openings 26 to form a permanent model 10 of a particular point 
group. 
In summary, FIG. 1 shows a model 10 which can be fitted with sixteen pegs 
20 to represent the point group D.sub.4 h. Center shaft 28 holds the upper 
and lower blocks 12a, 12b together and is deemed collinear with the 4-fold 
axis. 
By appropriate placement of an appropriate number of pegs 20, fixed models 
10 for the point groups with rotational axes of order 4 (C.sub.4 ; 
C.sub.4V ; C.sub.4h ; S.sub.8 ; D.sub.4 ; D.sub.4 h; D.sub.4 d) and 
S.sub.4 can be created. The groups S.sub.8 and D.sub.4 d require the 
blocks 12a, 12b to be offset by 45.degree.; the model for D.sub.4 may have 
the blocks at any arbitrary angle from 0 to 90.degree.. This structure can 
also be used to generate point groups with a C.sub.2 axis (C.sub.2 ; 
C.sub.2v ; C.sub.2h ; S.sub.4 ; D.sub.2 ; D.sub.2h ; D.sub.2d) and the 
trivial point groups C.sub.1 ; C.sub.s ; and C.sub.i. 
An alternative embodiment of the present invention is shown in FIGS. 4-6. 
In this embodiment, shaft 28 and shaft openings 30 in blocks 12a, 12b are 
replaced with pins 32 and pin openings 34. Pin openings 34 are located on 
the interior, opposed faces 18 of blocks 12a, 12b in a square 
configuration about a central, vertical axis of the model 10, as shown in 
FIG. 5. Pin openings 34 are situated on the interior faces 18 of each 
block 12a, 12b so as to be aligned with one another when blocks 12a, 12b 
are connected to form model 10, as seen in FIG. 4. Additional pin openings 
34 may be provided in the opposing faces of blocks 12a, 12b so as to allow 
the blocks to be connected with peripherally offset side surfaces 16. 
FIG. 6 shows pins 32 disposed within pin openings 34 and connecting blocks 
12a, 12b to form model 10. Pins 32 may be secured within pin openings 34 
using an adhesive in a manner similar to that used in the previous 
embodiment for pegs 20 and shaft 28. Or, pins 32 may be removable from pin 
openings 34, allowing pins 32 to be reused to form models 10 employing 
geometric blocks in a different orientation or of a different shape to 
form different point groups. 
FIGS. 7-11 show a third embodiment of the present invention including a 
rotational locking mechanism. In this embodiment, model 10 is constructed 
of a pair of plate-like geometric blocks 12a, 12b, formed of a material, 
such as hard plastic, each having an exterior, unopposed face 14, an 
interior, opposed face 18, and a number of side surfaces 16 with peg 
mounting openings 26 containing a number of pegs 20, similar to the 
previously described embodiments. However, as shown in FIGS. 7 and 8, the 
rotational locking mechanism utilized in this embodiment allows blocks 12 
comprising model 10 to be rotated with respect to one another from a 
peripherally aligned position, as shown in FIG. 7, to a peripherally 
offset position, as shown in FIG. 8. 
Referring now to FIGS. 9-11, model 10 is constructed of a first plate-like 
geometric block 12a located above a similar plate-like second geometric 
block 12b. As shown in FIGS. 9 and 11, blocks 12a and 12b are connected by 
a threaded bolt 38 inserted through shaft openings 30 in the center of 
blocks 12a and 12b. Threaded bolt 38 is secured to the exterior faces 14 
of blocks 12a and 12b by washers 42 and nut 44 threadably attached to each 
end of bolt 38. Washers 42 and nuts 44, when fastened to secure blocks 12a 
and 12b to threaded bolt 38, are contained within handle cavities 46 that 
extend into the exterior faces 14 of blocks 12a and 12b and surround shaft 
openings 30 in blocks 12a and 12b. Threadably mounted onto threaded bolt 
38 above nuts 44 and partially contained within handle cavities 46, are 
handles 36 that contain threaded bolt cavities 37. 
Referring now to FIG. 10, on the interior opposed faces 18 of blocks 12a 
and 12b are located magnet slots 48, with one magnet slot 48 being present 
for each peg mounting opening 26 of blocks 12a and 12b. Magnet slots 48 
extend from interior faces 18 partially through blocks 12a and 12b and 
intersect the interior ends of adjacent peg mounting openings 26. A small 
magnet 50 is disposed within each magnet slot 48. Magnets 50 disposed in 
magnet slots 48 are used to retain pegs 20 within peg mounting openings 26 
as in this embodiment, peg mounting portion 22 of peg 20 is formed from a 
metal and is attracted to magnet 50 when inserted into peg mounting 
opening 26. Peg knob 24 of peg 20 in this embodiment may be formed of a 
wooden sphere attached to the free end of peg mounting portion 22. 
The structure of the rotational locking mechanism included in this 
embodiment is also shown in FIGS. 9-11. On the interior opposed face 18 of 
block 12b is located a spring-biased detent 56. The detent comprises a 
metal housing 58 that encloses and contains detent spring 60 and detent 
head 62, which extends partially out from the end of housing 58. The 
detent 56 is contained within detent opening 59 located on the interior 
face 18 of block 12b. On the interior opposed face 18 of block 12a is a 
series of equally spaced detent depressions 64 arranged in a circular 
pattern about shaft opening 30, as shown in FIG. 10. Detent depressions 64 
enclose a portion of detent head 62 when detent 56 is aligned with a 
detent depression 64 as seen in FIG. 11. A low friction disk 40 having a 
bolt hole 41 located in its center, allowing for insertion of threaded 
bolt 38 therethrough, is placed between blocks 12a and 12b, as seen in 
FIG. 9. Low friction disk 40 disposed between block 12a and block 12b has 
a diameter less than that of the circular pattern of detent depressions 64 
on block 12b to avoid any interference with the engagement of detent 56 
with detent depressions 64. Disk 40 is formed of plastic or other suitable 
material to facilitate rotation of blocks 12a and 12b with respect to each 
other by providing low friction surfaces for blocks 12a and 12b to rotate 
on. 
In utilizing the rotational locking mechanism, a manual rotational force is 
applied to either block 12a or block 12b while holding the remaining block 
stationary. Detent head 62, resting in detent depression 64 is pressed up 
against the edge of detent depression 64, compressing detent head 62 down 
against spring 60, thus pushing detent head 62 within housing 58. With 
detent head 62 compressed within housing 58 by interior opposed face 18 of 
block 12b, second block 12b may be rotated with respect to block 12a until 
another detent depression 64 in the circular pattern rotates over detent 
head 62 allowing detent head 62 to be extended into detent depression 64 
by spring 60. When detent head 62 is partially contained within detent 
depression 64, block 12b is locked in position with respect to block 12a. 
While four sided blocks are shown in FIGS. 1 through 11 in connection with 
the representation of point groups with a axis of order 4, as noted above 
blocks with a lesser or greater number of sides may be used, as shown in 
FIGS. 12a, 12b, and 12c, in accordance with the order of the point group 
to be represented. While blocks having a greater number of sides than six 
may be used, as a practical matter few molecules have rotational orders in 
excess of six-fold. 
A fourth embodiment of the present invention is shown in FIGS. 13a and 13b. 
In this embodiment, the model 10 is formed from a rigid cube 66 that has 
pairs of peg mounting openings 26 located in each corner of each face 68 
of rigid cube 66. Peg mounting openings 26 extend a short distance into 
rigid cube 66 and retain pegs (not shown) similar to those illustrated in 
previous embodiments. In forming these models 10 by inserting pegs (not 
shown) into peg openings 26, an adhesive may be applied to the pegs (not 
shown) to secure them within the peg mounting openings 26. 
By appropriate placement of an appropriate number of pegs (not shown), 
models 10 representing the cubic point groups, namely the octahedral point 
groups (O,O.sub.h) and the tetrahedral point groups (T,T.sub.h, T.sub.d) 
can be created. 
Various modes of carrying out the invention are contemplated as being 
within the scope of the following claims particularly pointing out and 
distinctly claiming the subject matter which is regarded as the invention.