Snow White and the Seven Pixels
        I began the semester just playing with the different mapping software while trying to find where my math and me fit into the class.  I was very unsure of how math could be related to mapping and how mapping could and would be interesting to me, a science geek.  As I worked with the first software introduced, Arcview, I found myself drawn to the color scheme for the map.  As I changed the colors manually, I noticed that the color had numbers related to them.  These numbers were broken down into three categories, with the final color depending on the setting of each category.  I found it intriguing that you could actually choose your color by putting numbers into the computer.  It reminded me of painting by number as a child.
     I also had a side thought about mapping ski resorts, since that is a hobby of mine.  I worked with this idea, first using Atlas to plot the points of several ski resorts in Colorado, but then changed to digital maps of Colorado and plotted the ski resorts using their latitudes and longitudes.  In this part of my project I learned how to maneuver and work my way through a new program, testing different buttons to find what I was looking for.  I also learned how to edit my web page, make points in the map clickable using Map Edit, and put text into my map using Adobe Photo Shop.  I became more comfortable with a computer and authoring tools.
     Now I needed to somehow link together or relate my two interests of colors as numbers (the math) and skiing (the mapping).  With Sandiís help, I decided to try and relate these interests using group theory from Modern Algebra.  For the remainder of the semester I have been researching and reading about group theory.
Group Theory
Terms
Group: a set of 1:1 functions on finite sets that could be grouped together to form a closed set satisfying the following properties:
            associativity- the operation is associative; (ab)c=a(bc) for all a,b,c in the set.
             identity- there is an element, e (the identity), in the set such that ae=ea=a for all a in the set.
             inverses- for every element a in the set, there is an element b in the set (the inverse) such that ab=ba=e.
             closure- any pair of elements in the set can be combined without going outside the set, which means that the final product is already a part of the set.

*note: the group is abelian if it also is commutative, ab=ba, but not all groups satisfy this property.

Subgroup: a subset, H, of the original group satisfying the group conditions under the operation of the original group, G.
     Proper subgroup: if H is a subgroup of G, but does not equal G, totally contained within the group.
     Trivial subgroup: the identity alone.
     Nontrivial subgroup: any subgroup that is not the identity.

Order of a Group: the number of elements the group contains.

Order of an Element: the smallest positive integer n such that g^n=e, g is the element.

     The abstract approach to determining if a set is a group is that the set has only one identity element, the right and left cancellation laws hold, ba=ca implies b=c, and ab=ac implies b=c, and the inverses are unique.  Since I was hopefully going to relate these mathematical concepts to skiing, Sandi and I decided to work with the symmetries of a square to explain the mathematical concepts in hopes that it could then be related to the various snow colors of the slopes.  The symmetry group of a plane figure includes the set of all symmetries of the figure, which are rotations and reflections.  First one must imagine a square with the four corners numbered from 1 to 4.

Next you look at all the different rotations and reflections that can happen.
 
From here you write down where the number moved to during the rotation or reflection.  For example:  when you don't rotate the square at all you indicate this by writing (1)(2)(3)(4), which means that 1 goes to 1, 2 goes to 2, 3 goes to 3, and 4 goes to 4.
    When you rotate the square clockwise 180 degrees:  (13)(24), this means 1 goes where the 3 was and the 3 goes where the 1 was, the 2 goes where the 4 was and the 4 goes where the 2 was.  When you reflect it over the x-axis (12)(34), 1 goes to the 2 place, 2 goes to the 1 place, 3 goes to the 4 place, and 4 goes to the 3 place.  The following table gives the eight possible symmetries of the square.  No matter what combination of rotations and/or reflections you do, you will always end up with one of these eight products listed in the first column and row of the table.  Another way of saying this is, the table is filled in without introducing any new elements.  This shows that the set I am working with is closed.
 
 Table 1:  Multiplication table of the eight possible outcomes from various movements of the square.
8 Outcomes 
(1)(2)(3)(4)
(1234)
(1432)
(13)(24)
(12)(34)
(14)(23)
(1)(3)(24)
 (2)(4)(13)
(1)(2)(3)(4)
(1)(2)(3)(4)
(1234)
(1432)
(13)(24)
(12)(34)
(14)(23)
(1)(3)(24)
 (2)(4)(13)
(1234)
(1234)
(13)(24)
(1)(2)(3)(4)
(1432)
(1)(3)(24)
(2)(4)(13) 
(14)(23)
(12)(34)
(1432)
(1432)
(1)(2)(3)(4)
(13)(24)
(1234)
 (2)(4)(13)
(1)(3)(24)
(12)(34)
(14)(23)
(13)(24)
(13)(24)
(1432)
(1234)
(1)(2)(3)(4)
(14)(23)
(12)(34)
 (2)(4)(13)
(1)(3)(24) 
(12)(34)
(12)(34)
 (2)(4)(13)
(1)(3)(24)
(14)(23)
(1)(2)(3)(4)
(13)(24)
(1432)
(1234)
(14)(23)
(14)(23)
(1)(3)(24)
(2)(4)(13) 
(12)(34)
(13)(24)
(1)(2)(3)(4)
(1234)
(1432)
(1)(3)(24)
(1)(3)(24)
(12)(34)
(14)(23)
(2)(4)(13) 
(1234)
(1432)
(1)(2)(3)(4)
(13)(24)
(2)(4)(13)
 (2)(4)(13)
(14)(23)
(12)(34)
(1)(3)(24)
(1432)
(1234)
(13)(24)
(1)(2)(3)(4)
 
     So far I have shown that the set is closed.  Next the set has a unique identity of (1)(2)(3)(4).  If you notice from the table that when any element in the set is multiplied by (1)(2)(3)(4) the product is the original element.  Therefore by definition of identity, (1)(2)(3)(4) is the identity.  For example when you multiple (1234) x (1)(2)(3)(4) you get (1234).  The next property is a unique inverse for each element.  This means that for each of the 8 elements in the set there is another element in the set that when multiplied together gives the identity as the product.  If you look at table 1 you will see that for every element there is another element, or itself, that when multiplied together gives the identity as the product.  For example: when (1234) x (1432)=(1)(2)(3)(4).  The last property is associativity and one cannot see from the table, but if 3 elements are multiplied together, the order in which they are multiplied does not matter, a(bc)=(ab)c.  For example: b=(1234), c=(1432),a=(13)(24)  do (bc) first which gives (1)(2)(3)(4) then multiply by (a) which gives (13)(24).  Now we do the other side: do (ab) first which gives (1432) multiply by c=(1432) which gives, as table 1 shows, (13)(24).  Therefore the set is associative, closed, has a unique identity, and has unique inverses.  This set is a group.  This group is not abelian because it does not satisfy the commutative property  with all the elements of ab=ba: (12)(34) x (1234)=(13)(2)(4), whereas (1234)x(12)(34) = (1)(3)(24).  These are not the same and therefore the property is not satisfied.  Since the group has eight elements its order is eight, by the definition noted earlier.
        The way these elements are multiplied together is a bit confusing, so I will try to explain it here.  Take for example (1234) x (1234)
1. Look at the 1 in the first element, it goes to 2, now look at the 2 in the second element, it goes to the 3
                You can write: (13
2. Look at the 3 in the first element it goes to the 4, look at the 4 in the second element it goes to the 1, ( the parenthesis show that it is closed and therefore almost like a circular pattern within, so the number before the parenthesis, goes back to the first number of the element in the same parenthesis)
                You can write :(13)
3. Look at 2 in the first element, it goes to the 3, look at the 3 in the second element, it goes to the 4
                You can write: (13)(24
4. Look at the 4 in the first element, it goes to the 1, look at the 1 in the second element it goes to the 2
                You can write: (13)(24)  and you are done.

        Within this group there is the possibility that there are subgroups.  Using the definition from above, I was able to find eight subgroups and the identity.  Five are of the order 2 and three are of the order 4 because of how many elements are in the subgroup. I=(1)(2)(3)(4) which is the trivial subgroup
    Order 2: (12)(34), I
                  (13)(24), I
                  (14)(23), I                    We know these are subgroups because they are their own inverse. (Proper)
                  (1)(3)(24), I
                  (2)(4)(13), I
 
    Order 4:  (1234),(1432),(13)(24), I
                   (12)(34),(13)(24),(14)(23),I            These subgroups are all closed, have inverses, an identity, and are associative. (Non-trivial)
                   (1)(3)(24),(13)(24),(13)(2)(4),I

        The symmetries of a square are a dihedral group which appear frequently in art and nature, which is why we chose to analyze it.  Now I had done some math and could try and relate it to the snow covered mountains.  At first I labeled the corners of the square using four different shades of white to represent the different colors of snow a skier might see depending on the shadow being made, due to placement of the sun and trees.  This was quite simple because now instead of numbers as the group, I had specific colors where the number previously existed.  I decided to look at it from a shadow perspective and try to define each time a day by one of the eight elements.  I was trying to prove that with any movement, no matter how complicated, the outcome was equivalent to one of the eight elements of the Dihedral group above, numbers or colors.  The part I had a problem with, in this model, was the fact that the ski slope is not a square and therefore does not have the symmetry I needed it to.  I had tried many ways to relate the shadows from the sun hitting the ski slope at different angles, or with different terrain, but because the symmetry is different I couldn't go any farther.  Many times I would get to a conclusion and realize that I couldn't put trees in an area for one shadow, and not the other.  The other aspect that was hard to put in was when an element was closed.  At first I was unsure how to show this, but then I realized I might be able to illustrate the parenthesis as leveling out along the slope.  I again ran into trouble modeling all eight elements according to this group.  So instead of dwelling on this issue we decided to look at the mapping in another way.  We noticed that we could look at the elevations of a mountain using Visual Terrain.  I labeled parts of Steamboat Colorado on the map below and then the program interpolated the rest.
 
Map: Elevation of Steamboat Colorado, (700 means 7000)
.

From this map I was able to use Visual Animator and make a 3-D map of the terrain, as shown below.
 
 Map: 3-D Steamboat Colorado

 
        From this map you can see some of the shadowing I was talking about.  Depending on such factors as where you were on the mountain, what time of day, and the terrain,  different shadows could be seen.  In this program I was able to change the view by moving the target and the camera, so I could shift the shadows that are illustrated in this map.  To remember the elevations I had to choose colors for the contour lines that were visible and definable, so I went with as many basic colors as I could.  Then when I was choosing to color the 3-D map I chose the summer color schema.  Can I relate this all back to what I originally was thinking about, colors?
        Yes, I can.  I went back to my group of the square symmetry and made a lattice.  A lattice is another way to show relationships between the various elements or subgroups in a group.

 
        The lattice shows that as you proceed downwards those elements are subgroups of the ones connected above them.  Since I was able to make a lattice for this group, why not try it for the way colors are chosen.  First I hold one aspect of the color at 0 so I am only dealing with two colors, then I make a lattice by choosing the max number between the x component or y component of the ones I am combining.  This is best understood through an illustration.

This illustration looks like something we have all seen before, but probably don't remember, and that is Pascal's Triangle.

        If I were to overlay Pascal's triangle over my lattice of two colors it would fit perfectly because they are mathematically related through binomials.  Pascal's triangle gives the coefficients while the color lattice gives the powers for the x and y coordinates.  For example:  the third row of Pascal's triangle is 1  2  1, and the third row of the 2-color lattice is (2,0)  (1,1)  (0,2).  Now using a form of the equation x^2 + 2xy + y^2, where the powers change according to the elements in the 2-color lattice.  (2,0) is the 1 in Pascal's triangle, and is therefore x^2,  (1,1) is the 2 in Pascal's triangle, and is therefore 2x^1y^1, and (0,2) is the 1 on the right side of Pascal's triangle, therefore it is y^2.  If we skip down and look at row 4, the equation above will work its way in from x^4 to y^4.  The 2-color lattice is a way of tracking the coefficients.  This shows that we could use a Binomial Color guide where numbers are listed rather than colors, like on the color formula guide.  The binomial equation used to determine an element in Pascal's triangle is n!/(r!(n-r)!) where n is the row number starting at 0 and r is the position of the element in each row starting at 0, which is now related to the 2-color lattice I have made.  Since the color schemes are split into 3 main categories, in some programs it is blue, green, and red, while in others it is hue, saturation, and value, each of which can range from 0 to 255, the example above is a simplification of this where one category is held at 0 and the other two can range from 0 to 255.  At the point 000 the color is black and as long as one stays at 0 the other two have to change by almost 100 to get a significant color change.  Adobe Photo shop uses Pantone colors, which I was able to obtain a guide for.  Through a function one could choose a four color mode or a three color mode.  Since the color guide I had was four colors I chose to the CYMK mode, where C is cyan, Y is yellow, M is magenta, and K is black.  I then tested out this theory of choosing a color by numbers.  I chose a simple one that holds in this case two colors constant, but I still had on the screen R,G,B so I assumed since I no longer had green that I could replace it with the yellow number.  Pantone gave the percentage of yellow as 62.5 and Blue as 37.5.  I had to round up or down because the computer would only take integers, but I put in for R =0, G=159, and B=96, and I came up with a match for the color of Pantone 361U.
        I found this project to be very interesting and helpful.  I had fun doing the project so I spent more time learning about the tools, making mistakes, and learning the concepts that I previously had trouble with.  My confidence working with computers has greatly increased.  I would have never thought that through the semester I would have integrated skiing, math, mapping, and colors, but I did.
 
Text References

Childs, L. (1995).  A Concrete Introduction to Higher Algebra. (2nd Ed.)  New York, NY: Springer.
Gallian, J. (1998).  Contemporary Abstract Algebra. (4th Ed.)  Boston: Houghton Mifflin Co.
Mackiw, G. (1985).  Applications of Abstract Algebra.  New York, NY: John Wiley & Sons.

Computer Programs

Digital Mapping software
Map Edit
Adobe Photo shop
Visual Terrain and Animator
Netscape Communicator