Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

This class may be different from many of the mathematics classes you have taken in the past. You will be encouraged to explore geometrical ideas, solve puzzles, make conjectures, discuss with classmates, make connections between ideas, explain your solutions, and justify your reasoning. You should not expect to be shown how to solve problems by the instructions. In fact, there may not be one single right way to complete some of the problems but rather many acceptable ways of coming to a solution. Your teacher will encourage you to share these alternative approaches with the class.

Perhaps this course would be easier for you if you were simply told how to solve the problems, but the objective of this course is not to learn facts and methods, but rather to develop your mathematical thinking and communication skills. In this age of technology, computers can do many of the algorithms and processes that were at one time the core of mathematics education. However, they have not replaced man’s role of developing strategies to solve problems and recognizing when mathematics can be applied to real-world situations. Developing these reasoning skills will help you more in your future than simply being able to quote the Theorem. On the other hand, it is still useful to know the Pythagorean Theorem so that you are able to recognize when it can (and cannot) be used to solve both elementary and complex problems. This theorem along with some other basic formulas you have learned in the past (and can look up online) form a toolbox for our work, but we will give more emphasis to the creative process that is mathematics.

In A Mathematician’s Lament, Paul Lockhart describes a nightmare in which music was taught in the same way that mathematics is currently taught. He says,

So I encourage you to embrace the experience that is mathematics. Approach each puzzle with an open mind, make guesses as to what is happening, and experiment to determine whether your ideas are correct. Play with the materials and the concepts and ask new questions. In this course, you might even be invited to explore a question you posed instead of the one the teacher originally posed! Mathematics is not just about finding the answer; even when there is an answer, our work should evoke new questions: Why? What else?

The purpose of Exploration 1.1.1 is two-fold. It introduces the exploratory nature of the course and it also suggests a few important geometric facts and definitions that we will use as our initial geometrical foundation. These facts are chosen because they are intuitively obvious, but they give us some building blocks we will need to justify more complex ideas. We state them here for clarity and future reference beginning with a definition for area.

We noted that each tetromino was made up of four unit squares and each pentomino consisted of five unit squares; thus the area of a tetromino is four square units and the area of a pentomino is five square units. We also recalled the area formula for a rectangle.

Another observation is that if you can fill a closed region with smaller shapes so that none of the smaller shapes overlap and no part of the region is left uncovered, then the area of the region is equal to the sum of the areas of the smaller shapes. While this may seem like a simple idea, we will see that it allows us to determine the area formulas of other geometric shapes and to prove the Pythagorean Theorem. We can dissect a shape into smaller shapes, rearrange the pieces into another shape, and argue that the two regions have equal area because they both consist of the same collection of smaller shapes.

We applied these two principles in Task 1.1.1.e where we attempted to construct rectangles using pentominoes. Since pentominoes have an area of five square units, we could only construct rectangles with areas divisible by 5, namely 5, 15, 20, 25, 30, 35, 40, 45, 50, 55, and 60.

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To construct a rectangle of area 10, the same pentomino must be used twice.

In the same way, any shape made up entirely of tetrominoes must have an area divisible by 4.

Finally, we observed that when the lengths of all sides of a two-dimensional figure are doubled (or tripled), the area of the figure is multiplied by 4 (or 9). This is easy to observe in a rectangle with length \(\ell\) and width \(w\text{.}\) When we create a new rectangle with length \(L=2\ell\) and width \(W=2w\text{,}\) the resulting area is \(LW=(2\ell)(2w)=(2\cdot 2)(\ell w)=4\ell w\) square units.

Hopefully as you worked through these tasks, you allowed yourself to experiment with different arrangements of squares and polyominoes. This openness to creative exploration is essential to mathematical thinking. Phrases like "what if" or "I wonder" arise often as mathematicians create and make sense of new ideas. If you find yourself wanting to explore a question arising from a task but not explicitly asked, jot down your question and share it with your instructor. Your instructor might encourage you to investigate your question or even invite your classmates to consider it.

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Some questions may be addressed in a later section, too tangential, beyond the scope of this course, or even unsolved by the mathematical community. Thus your instructor will make the decision on which questions to pursue and when.

You may have noticed that you and your peers discovered different ways of completing the same task. Multiple approaches and multiple solutions are often welcome and even encouraged! If you have a strategy that is different than your group members, try both separately and then compare. You have the ability to do the mathematics, but it will be more meaningful if you attack tasks in a way that makes sense to you. This will require you and your peers to take some risks, to challenge each other to speak up, and to be nonjudgmental of others.

In fact, doing mathematics is a social activity. When we notice patterns and make conjectures, we share our ideas with others and use logic to make convincing arguments. For over 2500 years, mathematicians have been adding to the body of mathematical knowledge; some of which has been shared with you during your studies. The field of mathematics continues to grow.

Unfortunately, most students have had little opportunity to creatively explore in their mathematics classes. You have probably memorized formulas and performed calculations in many of your mathematics classes. While formulas and calculations are useful tools in mathematics, they are not the essence of mathematics and can often be done via a computer. In this course, we will develop the creative thinking, reasoning, and problem-solving practices that set humans apart from computers and are valued by employers. We will think outside the box, make predictions, and wonder why. Different approaches and strategies are encouraged. Relax and have fun exploring mathematics!