A. What is a Group.
If we placed an equilateral triangle on a piece of paper and made a tracing of it, we could pick it up and put it back down again and not know if we had changed its orientation. What we would have done is called a rigid motion or an isometry. To be able to tell what happened we could label the corners of the cutout and the corners of the tracing.
One question to ask is “How many ways can we pick up the cutout and put it down?” We could rotate it around its center to move a corner one space in a counterclockwise direction, or two spaces, or even three to come back to the starting position. Some obvious things to note are that we can do one thing and follow it with another and it is as if we had done a third thing in the first place. Two moves of one space each are the same as one move of two spaces. There are a limited number of moves. If we keep moving we end up with previous positions. One move is a waste, if we rotate three spaces it is as if we had not moved at all.
When we have situations like this in math: where things combine to make new things of the same kind we say the system is closed. So our system of three rotations is closed. The move that is the same as doing nothing is called the identity operation. The fact that every operation can be undone to get back to the original position is described by mathematicians by saying each operation has an inverse. When we have a closed system with an identity element and each element has an inverse, we say we have a group. Technically a mathematician would want another additional property: the operations should be associative, a(bc) = (ab)c. Isometries are associative. It took some historical time for mathematicians to recognize that there are non-associative “things”. So we will be historical and not check for associativity. Division is an example of a non-associative operation. Example: 12 Ö (6 Ö 2) makes four, while (12 Ö 6) Ö 2 makes one. Check and see.
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Do the three rotations make the only group for our equilateral triangle? Are there more isometries for the equilateral triangle? Can we move it without rotating it? Flip it along an altitude so the two base angles switch positions but the vertex angle stays fixed. You will have to put labels on the back of your cutout to match the ones on the front. How many flips are there? Do the flips make a group? We have three vertices so we have three flips. Some might say we have four flips, the identity flip of making no flip at all. So, do the four flips make a group? What happens if we flip around the top vertex and then flip around the lower left hand vertex? Do we get a flip? No, we get a rotation, it is as if we rotated two spaces counterclockwise. So the flips are not closed, they do not form a group by themselves. Along with the rotations and the identity the flips will make a group. To verify it make a multiplication table. Let I stand for the identity, R1 stand for a rotation of one space counterclockwise, R2 for a rotation of two spaces counterclockwise, Fl for a flip around the top vertex, F2 a flip around the lower left vertex, and F3 for a flip around the lower right vertex.
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To read the table find the first operation in the first column and then read over to the column with the second operation on top, there is your answer. For example, operation R2 followed by operation Fl is operation F3. Notice that operation Fl followed by R2 is operation F2. Our operations are not commutative. The table helps us see that the system is a group. It is closed because no answer is a new thing, all the answers are operations listed on the edges. There is an identity one column matches the left hand column and one row matches the top row. Every element has an inverse because the identity appears once in each row and column and even when the identity is not on the main diagonal the two operations still commute (R1 followed by R2 = R2 followed by R1 = I).
So we now have a bigger group the identity, the rotations, and the flips. The bigger group contains the group of rotations, so the group of rotations is called a subgroup of the larger group. Are there any other subgroups in the bigger group? Since each flip is its own inverse we could use a flip and the identity to form a group. We also could look at the identity alone and consider it a group. So a group is a set of things that operate on each other always getting themselves for answers, (just saying closure again).
The process of lifting up the figure and putting it down again, an isometry, is also known as a symmetry operation. The symmetries that required us to use both sides of the triangle, the flips, are called improper symmetries.
Here is a problem that might be more obviously relevant to us, where the symmetries tell us something. The queen in chess can move horizontally, vertically, and diagonally. If we had eight different queens could we place them on an eight by eight chess board so that no queen could capture any other queen. You do not need to know anything about chess to solve the question. The question could have been: place eight pieces of some kind on an eight by eight board so that no two pieces are on the same row, the same column, or the same diagonal.
I will not solve the 8 by 8 case for you. Look at the 4 by 4 case. Here is a solution.
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1 2 3 4
2 4 1 3
My symbol means a piece is in the first row second column, another piece is in the second row fourth column and so forth. The number on top gives the row, the number underneath gives the column.
Is this the only solution? Did you find a different one? Are there any different solutions? This is where we apply the symmetries. We can rotate the board ne quarter, one half, or three quarter turns: R1,R2, and R3, we can flip it on its horizontal midline: MH, its vertical midline: MV, or one of its diagonals: D1 and D2.
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What is going on here? Are R1, R2, and R3 the same operation? They give the same result. What about MH, MV, D1, and D2? Let us start with a different configuration, one that is not a solution, and perform the operations and see how many new configurations we get.
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which is not a solution since the first column and the second column pieces are on the same diagonal.
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all distinct results. There were only two answers in the first case because the first arrangement was so symmetrical. The different operations really are different.
Can you find a solution for the eight by eight board? I have one solution that transforms into eight distinct solutions. Is it the only one? I do not know.
Just as we asked how many bricks are possible we also could ask how many groups are possible. The answer to those questions is for another time. To get started on the answers one needs to have names for the groups. Here is one technique.
In our triangle and square examples we had axes of rotation, a three-fold axis for the triangle and a four-fold axis for the square. When we flipped the figures we had two-fold axes of rotation. These operations can be symbolized as 1,2,3,4,5,6, . . . n, for n-fold axes of rotation. 1 means you rotate right back to the starting position. In two dimensions a two-fold rotation has the same result as a reflection in a mirror, which is not the case in three dimensions. The symbol for a mirror is m. This gives names to our two groups: 3m for the triangle and 4mm for the square. The number tells the type of rotation axis and the m indicates a mirror. One might wonder why there is only one m in 3m, while there are two m’s in 4mm. After all, there are three mirrors on the triangle and four mirrors on the square. Either 3m is correct and 4mm has an extra m, or they are both two m’s short. The answer is due to the way the mirrors interact with the rotations and each other. It is logical, ask me to show you with a diagram. The symbols for the groups are known as the Hermann-Mauguin symbols.
I would have liked to have spent more time on Hermann-Mauguin, however I only got it straight in my own head as this project was coming to an end. I even found some useful references in books I had looked at earlier. So I must have learned something.
One special reference for groups and Hermann-Mauguin notation is
Symmetry
by Ivan Bernal, Walter C. Hamilton, and John S. Ricci. The book comes with a stereo viewer, in a pocket on the inside back cover, so one can actually see the three-dimensional crystallographic point groups in three dimensions. Read the book as well, and read it Slowly, with thought.
B. Miller Indices
Crystallographers need ways to describe crystals. One way is to draw pictures, but pictures are hard to copy and write down every time you want to talk about a crystal. Another way is to have numbers associated with the crystal. In fact the numbers will often be found on the pictures so you will know more certainly the orientation of a particular face.
A standard way to attach numbers to figures is to set up a coordinate system. So we are back in Algebra I and Algebra II with graphing. We need a third axis for three dimensional solid objects so we have the z-axis coming out of the page. In Algebra I we learn that the equation of a line whose x-intercept is a and whose y-intercept is b is
x
+ Y =1. a b
In three dimensions the equation
x
+ y + r = 1 a b c
would be a plane that cuts the x-axis at a, the y-axis at b, and the z-axis at c. The three numbers, (a,b, and c) could have been used as indices for the plane. Furthermore, since the crystallographers were only interested in the angles the three numbers could be “reduced” if they had a common factor, giving a parallel plane. Also, since no one likes fractions, the equation was multiplied through by abc to give
bcx + acy + abz = abc.
The numbers bc,ac, and ab are called the Miller indices h,k,1. These indices will be integers and usually single digits. They are written without commas separating them except for the rare occasion when one is more than a single digit. When the intercept is negative the index will also be negative. To show a negative value a bar is placed over the number. Remember, Miller indices will always be whole numbers: positive, negative, or zero.
If you are reading the descriptions of a crystal you will want to translate the Miller indices into intercepts. To change from Miller indices back to intercepts take the reciprocal of each digit and then multiply each by a common denominator to make whole numbers. The numbers will be in the order of the axes,
i.e.
x-intercept, y-intercept and z-intercept.
When saying Miller indices in words, 111 is said as one-one-one, not one hundred eleven.
In Elizabeth A. Wood’s book
Crystals and Light
she has a picture of a pyrite crystal which she shows as a pentagonal dodecahedron.
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She gives the Miller indices of the faces that are visible as 102, 021, 210, 102, 021, 210 and says the point-group symmetry is m3. One is a convenient index since it is its own reciprocal. Zero is not so obvious. The reciprocal of zero is commonly called infinity. So the plane intersects the axis at infinity, or in other words it never intersects the axis, it is parallel to the axis. So we only have to change the indices + 2 and + 1 to their reciprocals + 1/2 and + 1 which become + 1 and + 2 when we multiply by the denominator to get whole numbers. Yes, they are the same values as we started with, but the order is not the same and that means different axes.
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The group symbol, m3, translates into a cubical system with the four diagonals of a cube as axes of three-fold rotational symmetry and the x-y, x-z, and y-z planes as mirror planes. We should be able to cut a model of the crystal out of a cubic 4 by 4 by 4 block. See the plan at the bottom of the previous page.
Here is some explanation. Follow the steps. K is the midpoint of AB, L is the midpoint of FG. Similarly N,M,J, and I are midpoints of their respective segments. Next draw a line through V parallel to JI, a line through T parallel to KL and a line through 0 parallel to NM. Those lines will serve as if they were the ridges of house roofs. Next we need to find the eaves lines and the gable lines. S is the midpoint of BM and R is the midpoint of GO. The story is the same for P and Q. Finally, using the corners as axes of three-fold symmetry rotate each of the ridge lines into the position of KL and mark in the eaves and gables. Now cut off the wedges to get the gable roofs.
How do you cut it out? Make a jig to keep your fingers away from the saw. How do you keep your angles when your guide lines are cut off? I made mine out of clay, cutting it with a wire and putting the cutoffs back to maintain the cubical guide shape. After all the cuts were made then the wedges were peeled off to get the dodecahedron core. The force of the cutting distorted the clay block. I did get a pentagonal dodecahedron, but I can not call it regular nor can I claim that it would not be regular if the process were more precise.
A drawing showing all the lines is shown in figure 8.
I would not expect anyone to follow the drawing unless they drew it too.
C. Projections: Mechanical Drawing
One of the objectives of geometry is to visualize objects in space. How can this be taught? Let the student experience drawing three dimensional objects. Mechanical Drawing is a way to draw even if one believes one is not an artist. Mechanical Drawing is math, it is part of projective geometry. See Morris Kline
Mathematics in Western Cultures
. John Pottage in
Geometrical Investigations
has a number of problems that are solved by mechanical drawing techniques, including a copy of woodcut by Albrecht Durer showing how to construct an ellipse as a conic section by mechanical drawing techniques (page 436).
Look at figure 1. In the upper right we have three pieces of overhead projector transparency film (VWDU, UDST, and WDSR) joined at their edges to form a corner, we place a block in that corner and trace onto the plastic the edges that touch the plastic. Now we unfold the plastic corner and have three views: top view, front view and side view. You have done a parallel projection of each side onto a picture plane namely its piece of plastic. This is all there is to mechanical drawing and blue print reading. So, how can mechanical drawing be a full year course? Easily, the “block” could be much more complicated, needing auxiliary views, section views, shading, maybe even the shadows cast by the “block”. Also time is needed for practice to gain speed, while achieving accuracy and neatness.
Let us look more closely at the figure. How can we improve it? We do not need to show the “pieces of plastic”. We could have some space between the three views so it would be more obvious where each view begins and ends. Notice how D,U, and C appear on both the x and y axes. If you were to draw a line segment from the U on the x-axis to the U on the y-axis you would form a triangle. What kind of a triangle? Notice the dashed lines. The dashed lines stand for invisible edges. Edges HG,FH, and HB are the back edges of the solid block that we would not see in reality.
Let us think some more about this. When we have plans what do we want to get from them? The sizes of the dimensions that make our object, the angles we have to set our saws at to get the pieces to fit. Will these dimensions be on our pieces of plastic? Think about it.
Let us explain “parallel projection” in more detail. If we keep the model in the corner we will be rather cramped, at least we will have one line right on the edge of the plastic sheet, so let us move the model out of the corner. Now we rest our pencil on an edge with its point touching a plastic sheet, slide the pencil along the edge keeping it parallel to the other plastic sheets. You now have a line on your plastic, go completely around the model and you will have one of your three views as before. Let us change our point of view. Look at the pencil; it is always perpendicular to the picture plane. So we could look at the process as keeping the pencil perpendicular to the picture plane and tracing the edges of the model with the “other end”. I say the “other end” because the pencil would have to change its length as the line got closer or farther from the picture plane. Figure 2 is my attempt at illustrating this.
Let us draw a block with some faces not parallel to the picture planes. Turn the model so it “rests” on one of its long edges. See figures 3 and 4. A rectangular brick has been placed in the interior of a plastic box and parallel projected. The subscripts tell what view the point is on, t,f, and s for top, front, and side views. Points with the same capital letter correspond. Figure 4 is the box in figure 3 unfolded. Edge HK is invisible from the front so it becomes a dashed line in the front view. Likewise edge EG is underneath so it is invisible from the top and is a dashed line in the top view. Look at edge CB on the model. Is it equally as long on each of the views? Is it full size on any of the views? If line segment CB on the front view were called x, and if line segment CB on the top view were called y, and line segment CB on the side view were called z, what would be the relationship between all three variables? Why?
So what kind of lines will not be the same length on the plans as they are on the model? Lines that are not parallel to their picture planes.
Let us look at another example. We have a block with one corner knocked off (figures 5 and 6). Knocking the corner off leaves a triangular face FDE. In the three view mechanical drawing there is no full size congruent image of the triangle. In each view only one of the diagonal lines is full size the other two are shortened. If we want to see the true shape of the triangle we have to
develop
it, one of those auxiliary views mentioned earlier. A useful reason to develop all the faces of a figure (other than clarity) is to have a pattern one can cut out of paper to form the object. The paper cutout is a check to see if your drawing is the figure you claim it to be. Figure 7 is the development of the three visible faces in figure 5. It leaves out the back, bottom and left side of the block. When it comes to developing the block, to make a model, we want the matching edges back together again. I chose to put the DC edges together so there would be more room for the triangle with ED as its base. Once the top, front, and side faces are together it is time to make the triangle. With the point of your compass at D set the radius to DF from the top view and draw an arc. Then with the point of your compass at E set the radius to FE from the front view and draw an arc to cross the previous arc. The intersection is F, the vertex of the triangle. The construction is the same as constructing a triangle given three sides, as in geometry class.
