Monday - Friday, July 12-16, 2004
After discussing the purpose of the group, and discussing how we would decide on three projects for our group to work on, we set about working on a spatial ability activity How Many Cubes, one of the Technology Problems (tPoW) in Math Tools which is the newest area of the Math Forum. The tPoWs are modeled on the Math Forum's Problems of the Week, with the additional feature that these problems take advantage of online, interactive mathematics tools. How Many Cubes uses a Java applet that allowed us to "test" out conjectures by manipulating and stacking virtual cubes. The applet written by Joel Duffin specifically for this problem is linked from the tPoW or can be found directly at http://matti.usu.edu/mathforum/spaceblocks.html. A similar version can also be found at the National Library of Virtual Manipulatives for Interactive Mathematics.
After this activity, Art (with Steve running the software) led an introduction to the Geometer's Sketchpad. We may schedule a GSP session one afternoon.
Roger Howe led us on an exploration of polyominos. We began by determining the number of properly equivalent (no flipping) and non-properly (flipping allowed) equivalent trominos, then the tetrominos, followed by pentominos and for some, hexominos. We next explored ways of classifying them, starting with perimeter. We ended up exploring the concept of an enclosing rectangle, and used them to classify the polyominos (for example, those of you familiar with pentominos, the enclosing rectangle for the "F" pentomino is a 3x3 rectangle, but for the "N" pentomino it is a 4x2 rectangle). We conjectured and proved some preliminary results concerning the areas and perimeters of the class of polyominos and the enclosing rectangles. We were left to explore the number of polyominos that have the smallest possible area for a given enclosing rectangle, and for a give enclosing rectangle, determine the largest possible polyomino perimeter.
Roger continued to lead us on our exploration of the mathematics behind the polyominos. We began by continuing our discussion/formulation of a proof that the minimum area of a polyomino for a given m x n enclosing rectangle is m + n – 1. A number of proofs were offered, many based on sliding pentominos to an edge.
We next explored the number of minimal area polyominos that could connect the lower left corner of an enclosing rectangle to the upper right corner. We quickly realized that this problem was the same as the "counting paths" problem from the morning session. We were left with the challenge of how to "weed out" the non-properly equivalent polyominos. On of the strategies that evolved was to look at the polyominos as a graph consisting of vertices and edges as opposed to a number of squares sharing edges.
Next we explored the number of ways a pentomino could be covered by dominos. We then explored how the number of dominos, the perimeter, and the area of the pentominos were related. Two equivalent formulas were suggested:
2) D = 2A – 0.5P
D:= the number of dominos
P:= the perimeter of the polyomino
A:= the area of the polyomino
Jerry shared with the group an induction proof of the second formula.
We began by deciding on the working group projects. We have decided on 3 projects in the group, falling into 3 broad themes: Symmetry and Geometer's Sketchpad, Non-Euclidean Geometry and Planetary Orbits, Polyominos in Algebra and Geometry.
Roger finished up with polyominos, tying up loose ends and proposing questions for further study. Three proofs of significant results of the past three days were shared by participants. A combinatorial proof that the relationship between dominoes, perimeter, and the area is given by 2D + P = 4A was given by Darryl. An induction proof of D>= A-1 was given by Steve. Mary shared a proof that the minimum area of a spanning polyomino of a m x n rectangle is given by m + n – 1.
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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.