Polygons

Summary: Polygons have properties making them important in engineering, architecture, and elsewhere.

Shapes and figures define how people view the world. Polygons are special figures whose properties and relationships are prevalent in nature and are used extensively by architects, engineers, scientists, landscapers, and artists. Specifically, polygons are traditionally planar (two-dimensional) figures that are closed and comprised of line segments that do not cross. These line segments are called “edges” or “sides,” and the points where the edges meet are called “vertices.” Planar polygons are very important in engineering, computer graphics, and analysis because they are rigid, they work well with functions, and they are easy to transform. Other types of polygons are also useful, such as spherical, hyperbolic, complex, or near polygons.

Properties of Polygons

Polygons are named by the number of their sides. Typically, polygons with more than 10 sides are called n-gons.

Calculating angle sums, areas, and perimeters of polygons is important in architecture, landscaping, and interior design. Understanding properties of triangles and parallelograms facilitates these kinds of calculations. For instance, the sum of the measures of the interior angles of a polygon can be determined by realizing that a polygon with n sides can be divided into n-2 triangles, and that the sum of the measures of the interior angles of any triangle is 180 degrees. Using these ideas, a carpenter could easily determine the angles at which, for example, the sides of a hexagonal window frame should meet. Furthermore, the ability to create polygons from triangles and the ability to rearrange or duplicate some polygons to form parallelograms allow the derivation of area formulas. Michael Serra describes in his 2008 book, Discovering Geometry: An Investigative Approach, how the area of a parallelogram can be derived from a rectangle, and the area of a triangle can be derived from a parallelogram.

Real World Examples

Polygons are prevalent in the world. Even traffic signs come in the shapes of triangles, rectangles, squares, kites, and octagons. The properties of polygons make them useful in many areas including architecture, structural engineering, nature, and art.

Polygons are sometimes used in architecture for their structural benefits. Trusses formed from triangles provide support for bridges and roofs because, unlike other polygons, triangles do not tend to deform when force is exerted on a vertex. Fences are often formed into polygons because they can be built by linking together straight segments of material that are of equal size and shape. The buildings that comprise the Pentagon building in Washington, D.C., are arranged in a pentagonal shape because, according to Stephen Vogel, walking distances between buildings are less than in a rectangle, straight sides are easier to build, and the symmetrical shape is appealing. In the 1850s, Orsen Fowler popularized octagonal-shaped houses because octagons have larger areas than rectangles with the same perimeter. Thus, octagonal houses provided maximal living space while keeping heating, cooling, and building costs similar to that of the smaller rectangular house with the same outer wall space.

Properties of quadrilaterals and triangles facilitate the creation of squares and right angles. For example, using the properties of a square’s diagonals, an approximate baseball diamond could be constructed by cutting diagonals of equal length from string or rope. To form the square, the diagonals would be positioned to bisect (halve) each other at right angles. The ends of each string would then mark the square’s four corners. The same format could be used to create a rectangular play area, except the diagonals would not be perpendicular. According to Sidney Kolpas, although unaware of the Pythagorean theorem, ancient Egyptians used right triangles to reconstruct property boundaries after the annual flooding of the Nile River. To create a 90 degree angle, Egyptians would create a 3-4-5 right triangle by tying 13 equally spaced knots in a rope, placing stakes at knots 4 and 8, then drawing the ends of the rope at knots 1 and 13 to meet.

Polygons are prevalent in nature. Mineral crystals often have faces that are triangular, square, or hexagonal. The cross section of the Starfruit is shaped like a pentagonal star. Katrena Wells describes practical applications of hexagons, such as the often hexagonal shape of snowflakes and the hexagonal markings on many turtles’ backs.

Tessellations of polygons are arrangements of polygons on a plane with no gaps or overlaps. These are also seen frequently in nature. Marvin Harrell and Linda Fosnaugh discuss many examples, including the facts that bees use a hexagonal tessellation for their honeycomb, some plant cell structures form hexagonal tessellations, and cooling lava may have formed the tessellating hexagonal columns of basalt rock at the Giant’s Causeway in Ireland. Interestingly, a giraffe’s skin is covered with a tessellation of various approximate polygons.

When creating sketches of objects or animals, artists often use polygons as the basis of their work by breaking the figure down into polygons and circles, then smoothing and filling in the details of the drawing after the rough polygonal sketch is created. Michael Serra explains how artist M.C. Escher used tessellations of triangles, squares, and hexagons as a framework, then rotated or translated various drawings along the sides of each polygon in the tessellation to create marvelous patterns of reptiles, birds, and fish. Islamic artists covered their buildings with ornate tessellations of polygons. A prime example is the Alhambra Palace in Grenada, Spain.

Investigating polygons as they exist in the world is one method of introducing geometry and instilling a value of geometry to people of all ages. Examining polygons with hands-on learning activities and real-world examples provides students with opportunities to investigate the characteristics and properties among polynomial shapes and helps them grasp an understanding of geometry at a higher level.

Development of Polygons

Planar polygons have been important since ancient times. Up until the seventeenth century, polygons that inscribed and circumscribed a circle were used by Archimedes and many others to estimate values of π. In 1796, at the age of 19, Carl Friedrich Gauss constructed a 17-sided polygon using a compass and straight edge. A year earlier, he had described the area of a polygon, which is often referred to as the “Surveyor’s formula,” although this concept also is attributed to A. L. F. Meister in 1769. The concept of a tiling or tessellation also requires polygons, and these have a long history of representation in art, weaving, architecture, and mathematics. Johannes Kepler studied the coverings of a plane with regular polygons, and in 1891, crystallographer E. S. “Yevgraf” Fedorov proved that there are 17 different types of symmetries that can be used to tile the plane. Planar polygons also star as main characters in Edwin Abbott’s 1884 novel Flatland and the subsequent twenty-first-century movies. In the early twenty-first century, young children investigate the mathematical properties of planar polygons in primary school.

Other types of polygons are also interesting and useful. Non-convex polygons like a star polygon, where line segments connecting pairs of points no longer have to remain inside the polygon, were studied systematically by Thomas Bredwardine in the fourteenth century. Generalized polygons in the twentieth century include complex polygons investigated by Geoffrey Shephard and H. S. M “Donald” Coxeter; Moufang polygons, named after Ruth Moufang; and near polygons. In 1797, Norwegian surveyor Caspar Wessel explored planar and spherical polygons in his theoretical investigation of geodesy. M. C. Escher represented hyperbolic polygons in his tessellated artwork. Some twenty-first-century college geometry texts contain spherical and hyperbolic polygons.

Bibliography

Bass, Laurie E., Basia R. Hall, Art Johnson, and Dorothy F. Wood. Geometry: Tools for a Changing World. Upper Saddle River, NJ: Prentice Hall, 1998.

Botsch, Mario, Leif Kobbelt, Mark Pauly, Pierre Alliez, and Bruno Levy. Polygon Mesh Processing. Natick, MA: A K Peters, 2010.

Cohen, Marina. Polygons. New York: Crabtree Publishing, 2010.

Fowler, Orson. The Octagon House: A Home for All. New York: Dover Publications, 1973.

Guttmann, A. J. Polygons, Polyominoes and Polycubes. Berlin: Springer, 2009.

Harrell, Marvin E., and Linda S. Fosnaugh. “Allium to Zircon: Mathematics.” Mathematics Teaching in the Middle School 2, no. 6 (1997).

Icon Group International. Polygons: Webster’s Timeline History, 260 B.C.–2007. San Diego, CA: ICON Group International, 2009.

Kolpas, Sidney J. The Pythagorean Theorem: Eight Classic Proofs. Palo Alto, CA: Dale Seymour, 1992.

Serra, Michael. Discovering Geometry an Investigative Approach. Emeryville, CA: Key Curriculum, 2008.

van Maldeghem, Hendrik. Generalized Polygons. Basel, Switzerland: Birkhäuser, 1998.

Vogel, Stephen. “How the Pentagon Got Its Shape.” May 2007. http://www.washingtonpost.com/wp-dyn/content/article/2007/05/23/AR2007052301296‗4.html.

Wells, Katrena. “Hexagons in Nature—6-Sided Shapes Made Fun: Teach Practical Application of Natural Beauty of the Hexagon.” http://primaryschool.suite101.com/article.cfm/hexagons-in-nature--6-sided-shapes-made-fun.