Rectangular States and Kinky Borders

On many maps of the United States, the states of Colorado and Wyoming appear to be rectangles.


Indeed, the enabling legislation creating the two states specifies their extent strictly in terms of lines of latitude and longitude, rather than rivers, mountain divides, or other geographical features. Wyoming stretches from 41°N to 45°N latitude and from 104° 3'W to 111° 3'W longitude. Colorado's borders lie between 37°N and 41°N latitude and between 102° 3'W and 109° 3'W longitude. Originally, the lines of longitude were measured west from Washington, D.C.

However, on the surface of a sphere, although lines of latitude are parallel, lines of longitude converge as you go north or south away from the equator. So, the northern border of each state is a little shorter than its southern border. The difference for Colorado is about 21 miles.

Hence, to a first approximation, both states appear to be trapezoids rather than rectangles, though on a curved surface. A trapezoid has one set of parallel sides.

But there's an additional wrinkle that complicates the picture.

When the states were created, surveyors mapped their boundaries using transit and compass, chronometer, and astronomical readings. They also relied on data from previous surveys and interviews with residents of the affected areas. Following the appropriate lines of latitude and longitude as best as they could, the surveyors established the borders, marking them from milepost to milepost over distances stretching hundreds of miles.

The boundary between Utah and Colorado runs 276 miles from Four Corners (the only place in the United States where four states share a point) to the Wyoming border, for example. On its northward trek, the original survey ended up about 1 mile west of where the surveyors had expected to intersect the Wyoming line, indicating that the surveyed border had at least one kink in it. Indeed, subsequent surveys revealed a discrepancy between mileposts 81 and 89 (northward from Four Corners) and another between mileposts 100 and 110. The errors put kinks into what should have been a straight line.

There were similar surveying errors along other borders, including those that define Wyoming. Interestingly, once a border is defined on the ground and accepted by the interested parties, it becomes official, even if it doesn't follow the written description.

So, perhaps it's best to describe Colorado and Wyoming as polygons. And Utah can then join the group of states that are polygons. The trickier question is determining exactly how many sides these polygons have.

Colorado's legal border, for example, "is a polygon formed by a series of line segments that run between physical monuments that were put in place by . . . survey parties," Stan Wagon and John J. Watkins comment in the September College Mathematics Journal. "This polygon has hundreds of sides."

It's time to take a closer look at the borders and start counting. The main kink in the Utah-Colorado border is visible with a few clicks on a Google map, as is an abrupt jog in the border between Colorado and New Mexico.

References:

2007. Which states are polygons? College Mathematics Journal 38(September):259.

Case, W.F. Why does the eastern border of Utah have a kink in it? Utah Geological Survey.

Van Zandt, F.K. 1976. Boundaries of the United States and the several states. U.S. Geological Survey Professional Paper 909.

The Number Pad Game

The number pad of a computer keyboard or calculator presents a three-by-three array of the digits from 1 to 9. (We exclude the digit 0, which is usually in a row by itself.) This array of numbers can serve as the basis of a simple, two-player game.


In the game, the first player turns on (or clears) the calculator, presses a digit key, then presses the + key. The second player responds by pressing a digit key in the same row or column as the last digit key pressed by the first player, except the key pressed by the first player. So, if player A presses 5, player B can respond with 4, 8, 6, or 2, but not with 5.

The first player responds to the second player's move in the same way. The two players take turns alternately until a player reaches a sum greater than a specified amount, say 30. That player loses.

Is there a winning strategy for this game? How does it depend on the specified "fatal" total?

"You can visualize the game as being played by stacking wooden blocks on top of one another," Alex Fink and Richard Guy write in the September College Mathematics Journal. "When the height exceeds 30 blocks, the stack topples and the player who was responsible loses."

For a tower of "tolerable" height 30, player A can win by touching 9 on the first move. No matter how B moves, A can always ensure that B exceeds 30. A can also win by starting with 3, but B can prolong the game by touching 1 at each turn.

Fink and Guy also work out the winning moves for any "tallest tolerable tower." For example, as noted above, the winning initial moves for 30 are 3 and 9. Interestingly, there is always a winning first move for the first player unless the tallest tolerable towers are 27, 43, or 64 in height. For 27, 43, or 64, there is no such strategy for the first player.

For shorter towers, you have to pay attention and pick your opening move carefully. However, that changes for heights greater than 107. "For towers taller than 107, you can always win by playing 3, 5, or 7—your opponent can never reply with one of these numbers, and whatever is played, you can reply with either of two of them," Fink and Guy report.

You can also reverse the rules and play the game so that the first player to exceed the total wins (rather than loses). Under this rule, there are no winning moves for towers of 12, 42, 76, 97, and 40k + 114. Curiously, beyond 124, the set of winning moves repeats itself with a period of 80. So, the same winning initial moves work for towers of 125 and 205, and so on!

Other variants of the game are possible. Fink and Guy consider configurations that include the 0 key in a separate row and look at how what happens depends on whether the key is in the middle, left, or right column—or even straddles two columns.


Any of these game variants can be a pleasant pastime. In some cases, the patterns of winning moves are simple enough to master that you could have a distinct advantage over any friends who dare to take on the challenge.

References:

Fink, A., and R. Guy. 2007. The number-pad game. College Mathematics Journal 38(September):260-264.

Symmetries of Beaded Beads

To create a beaded bead, you need a supply of beads, a needle with plenty of thread, and a lot of patience and care. It also helps to consult the innovative designs and patterns of Gwen Fisher and Florence Turnour. The result is an elegant form of beadwork—sparkling, colorful clusters of beads typically woven around one or more large holes.


Fisher is a mathematician at California Polytechnic State University in San Luis Obispo, so it's natural for her to look at these creations mathematically. Indeed, many beaded beads can be viewed as polyhedra, where the hole through the middle of each bead corresponds to a polyhedron's edge.

Different weaving patterns bring different numbers of these "edges" together to form the vertices of a polyhedron, Fisher says. The holes of 12 beads, for example, can be strung together to form the 12 edges, eight vertices, and six faces of a cubic bead cluster.

Such polyhedral beaded beads also have particular symmetries, classified by the three-dimensional finite point groups. Earlier this month at MathFest 2007 in San Jose, Calif., Fisher described weaving techniques that allow the realization of beaded beads with all the possible symmetries of polyhedra.


"Any polyhedron can be modeled as a beaded bead," Fisher and Blake Mellor of Loyola Marymount University write in a recent paper published in the Journal of Mathematics and the Arts. In other words, given any polyhedron, it's possible to weave a beaded bead with the same set of symmetries.

"The challenge is to create the patterns to accomplish this so that we are also creating beaded bead that are objects of beauty in their own right," the authors note. "In meeting this challenge, we developed many new designs that may not have been created otherwise." Fisher displayed many of the resulting bead clusters at MathFest.

A symmetry of a geometric object is a rigid transformation, such as a reflection or rotation, that leaves the appearance of the object unchanged. Any rotation or reflection of a sphere leaves the object unchanged, so the symmetry group of a sphere is the infinite collection of these motions, commonly designated O(3). Any given polyhedron shares some finite collection of these symmetries, so its symmetry group is a finite subgroup of O(3).

These infinitely many subgroups divide naturally into 14 classes: the prismatic groups, which correspond to the seven infinite frieze groups (border patterns); and seven additional groups related to the symmetric groups of the Platonic solids.

Fisher notes that beaded beads with any of the prismatic symmetries can be created using a weaving technique, known as the fringe method, that she and Turnour developed.


Other techniques allow the realization of such exotic objects as the beaded Sierpinski tetrahedron.

"Bead weaving is a phenomenally rich medium for creating mathematical art in three dimensions," Fisher and Mellor conclude. And the construction of beaded beads leads to additional mathematical questions. For example, what is the minimum number of beads required to realize a given point group? What is the minimal length of thread required?

References:

Fisher, G.L., and B. Mellor. 2007. Three-dimensional finite point groups and the symmetry of beaded beads. Journal of Mathematics and the Arts 1(No. 2):85-96.

Peterson, I. 2007. Knitting network. MAA Online (Jan. 29).