Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Throughout our learning, and in fact throughout our lives, we add knowledge to our toolkit that we can later reference as needed. The tools in our toolkit can then be used to solve problems, to further build our understanding of related concepts, and to support the validity of an argument. Most likely, you are already familiar with some of these tools from previous coursework. You know how to perform computations, have used geometric formulas, and can recognize many shapes.

Although you already know some of the facts and formulas, we will develop a toolbox from the ground up in this course, starting with basic principles and assumptions that are intuitively obvious. With geometric formulas and theorems, we will want to explore why these are true before adding them to our toolbox. When we need to assume a statement without proving it, we will acknowledge the assumption and identify it as a principle rather than a theorem. Theorems will be justified logically from the principles and previously proved theorems. Definitions are developed to give us helpful terminology and state the characteristics needed to identify the object. We typically do not include every property of an object in the definition; some properties can be proved from the initial definition. For example, we define a rectangle to be a quadrilateral with four right angles. The property that opposite sides of a rectangle are congruent is not included in the definition because it can be proved from the definition.

Depending on the level of a geometry textbook and the author’s goals, you will see some differences between the statements of geometric definitions and principles (also called axioms). For this reason, a proof you find elsewhere may not be sufficient here due to its reliance on different assumptions. Students are encouraged to use the resources in this text for their work. In addition, you should be mindful of the order in which theorems are introduced as you cannot use a theorem that comes later in the textbook as an argument in a proof. On the other hand, this textbook is not as rigorous as some geometry textbooks. In order to maintain the exploratory and intuitive nature of this book, the author will include some principles here that might appear as proven theorems in other texts. At times, proofs may be included as exercises leaving the choice of rigor up to the course instructor.

Our course toolbox currently contains the definition of area 1.1.1, the area formula for a rectangle 1.1.2, and Additivity of Area Principle 1.1.3. In other words, we can think about the area of a region as the number of \(1\times1\) blocks that fill the interior, we can use the area formula for a rectangle, and we know that the area of a region is equal to the sum of the areas of its parts. Hopefully, you will agree that length should also be additive; namely, that if we extend a segment of length \(n\) by a segment of length \(m\text{,}\) the resulting segment has length \(n+m\text{.}\) We restate this below:

Since we are interested in discovering facts and formulas that will hold for a whole class of geometric shapes, we will often use variables to represent numerical quantities such as length, area, and angle measure. We include in our toolbox the algebraic rules and procedures in A.2 that allow us to perform calculations, simplify expressions, and solve equations. There is a section A in this book’s appendix (back matter) on algebra for your review and reference.

Eventually, we will add more tools to our collection including the Pythagorean Theorem, area formulas for other shapes, and other geometrical theorems and formulas. The reasons behind these will be explored in future sections and based on definitions and the initial principles.

As we encounter and state new definitions, these too will be added to our toolbox. As you work through the next exploration, you will encounter terms like trapezoid and isosceles. Definitions for these and other terms are provided at the end of this section in case you would like to refresh your memory. You can also use the index or textbook’s search tool to find definitions.

In this exercise, we practiced using our initial principles and identified some tools, namely the Pythagorean Theorem and the area formula for a triangle, that will be developed later. We also noted that there are multiple ways to solve a problem. Solutions do not need to use formulas and should evolve from what we know and what we have in our toolbox. Still, we want to grow our toolbox to increase our ability to solve problems and justify claims.

Perhaps you moved figures around and lined up sides to determine whether the lengths were the same.

In the first part of this course, we will work primarily with polygons. What is a polygon? Giving a clear, clean definition of the word "polygon" is difficult so we will describe it verbally and give several examples. If you search for the definition in books or on the web, you will find different attempts to give a clear definition. We will describe a polygon as being a closed plane figure formed by a sequence of straight sides. Thus, a polygon is a shape that lies in the plane or on a flat surface. By closed, we mean that a polygon will have a well-defined interior; the polygon forms a solid boundary between this interior and the rest of the plane.

As seen in Figure 1.2.3, we distinguish between convex and concave polygons.

Not all flat shapes are polygons as illustrated in Figure 1.2.5.

Some specific types of polygons are defined below. We add these to our toolbox. An easy way to find definitions later is to use the index at the end of the textbook.

Some polygons are defined by the types of angles they have.

Quadrilaterals may also be described by the existence of parallel sides.

Triangles and other polygons may also be classified by whether some of their sides have the same length. Two sides of a polygon are said to be congruent if they are the same length.

While area measures the interior region bounded by a polygon or other closed figure in the plane, we can also measure the boundary itself.