Figure 1. 2011 MAA Study Trip participants in Copan, Honduras (Photo by John Wilkins, used with permission)
Figure 2. Maya ruins in Copan, Honduras (Photo by Cynthia Woodburn, 2011) |
Almost every year since 2003, the MAA has sponsored a study trip to a location of mathematical interest, such as Greece, Egypt, and China. The 2011 study tour had an emphasis on Maya mathematics and included sites in Guatemala and Honduras. Although the timing of the trip coincided with interest in the Maya long count calendar system and the end of a cycle in the year 2012, the group was also exposed to other information which has not received as much publicity and is not as well known. The focus of this article is on some of that information, namely Maya geometry and ways it can be used in the mathematics classroom. (For more information on Maya calendars, check out the Loci: Convergence articles “Maya Calendar Conversions” and “Maya Cycles of Time.”)
The 2011 MAA study tour was organized through the Maya Exploration Center with guide Christopher Powell. Powell (seated at far left in the first row in Figure 1 above) is an archaeologist with over twenty years of field experience. In addition to giving onsite explanations of Maya ruins, Powell presented several lectures to the group. One of these included material from his 2010 Ph.D. dissertation, The Shapes of Sacred Spaces, in which he introduced how the Maya used ratios in their art and architecture, as well as their ingenious way of using a knotted cord to form right angles.
In this article, we start with some background information on the Maya and then explain different ways that the Maya used geometry, going into detail on how the Maya formed right angles and rectangles with sides in various ratios by using knotted cords. Animations and videos created by the first author are included to illustrate the processes. The article concludes with related classroom activities that are tied to the Common Core Standards.
Figure 3. Maya stela in Copan, Honduras (Photo by Cynthia Woodburn, 2011) |
The classic Maya period ran roughly from 250 to 900 CE (or AD) [Coe, p. 26] during which time the Maya constructed hundreds of cities in a contiguous area from what is now southern Mexico across the Yucatan Peninsula to western Honduras and El Salvador, including what is now Guatemala and Belize. Maps of the Maya world can be seen at Latin American Studies: The Maya. The Maya Exploration Center also has maps of several Maya cities. Currently there are about 10 million modern Maya [Coe, p. 11].
The classic Maya made extensive use of geometry in their architecture. There is also evidence that they used geometry in their art. However, efforts to find a measurement system used by the Maya have been mostly unsuccessful. It does appear that they used body parts as units of measure, e.g. a uinic, which is a person’s height or “wingspan” from fingertip to fingertip. There are also several references to the use of a measuring cord in religious ceremonies and when buildings were being laid out. A measuring cord is even mentioned at the beginning of the Maya religious text Popol Vuh in the description of the creation of the universe. Powell, who conducted ethnography interviews of modern Maya shaman priests and master builders, discovered that measuring cords are still in use today.
Figure 1. 2011 MAA Study Trip participants in Copan, Honduras (Photo by John Wilkins, used with permission)
Figure 2. Maya ruins in Copan, Honduras (Photo by Cynthia Woodburn, 2011) |
Almost every year since 2003, the MAA has sponsored a study trip to a location of mathematical interest, such as Greece, Egypt, and China. The 2011 study tour had an emphasis on Maya mathematics and included sites in Guatemala and Honduras. Although the timing of the trip coincided with interest in the Maya long count calendar system and the end of a cycle in the year 2012, the group was also exposed to other information which has not received as much publicity and is not as well known. The focus of this article is on some of that information, namely Maya geometry and ways it can be used in the mathematics classroom. (For more information on Maya calendars, check out the Loci: Convergence articles “Maya Calendar Conversions” and “Maya Cycles of Time.”)
The 2011 MAA study tour was organized through the Maya Exploration Center with guide Christopher Powell. Powell (seated at far left in the first row in Figure 1 above) is an archaeologist with over twenty years of field experience. In addition to giving onsite explanations of Maya ruins, Powell presented several lectures to the group. One of these included material from his 2010 Ph.D. dissertation, The Shapes of Sacred Spaces, in which he introduced how the Maya used ratios in their art and architecture, as well as their ingenious way of using a knotted cord to form right angles.
In this article, we start with some background information on the Maya and then explain different ways that the Maya used geometry, going into detail on how the Maya formed right angles and rectangles with sides in various ratios by using knotted cords. Animations and videos created by the first author are included to illustrate the processes. The article concludes with related classroom activities that are tied to the Common Core Standards.
Figure 3. Maya stela in Copan, Honduras (Photo by Cynthia Woodburn, 2011) |
The classic Maya period ran roughly from 250 to 900 CE (or AD) [Coe, p. 26] during which time the Maya constructed hundreds of cities in a contiguous area from what is now southern Mexico across the Yucatan Peninsula to western Honduras and El Salvador, including what is now Guatemala and Belize. Maps of the Maya world can be seen at Latin American Studies: The Maya. The Maya Exploration Center also has maps of several Maya cities. Currently there are about 10 million modern Maya [Coe, p. 11].
The classic Maya made extensive use of geometry in their architecture. There is also evidence that they used geometry in their art. However, efforts to find a measurement system used by the Maya have been mostly unsuccessful. It does appear that they used body parts as units of measure, e.g. a uinic, which is a person’s height or “wingspan” from fingertip to fingertip. There are also several references to the use of a measuring cord in religious ceremonies and when buildings were being laid out. A measuring cord is even mentioned at the beginning of the Maya religious text Popol Vuh in the description of the creation of the universe. Powell, who conducted ethnography interviews of modern Maya shaman priests and master builders, discovered that measuring cords are still in use today.
According to his doctoral dissertation, Christopher Powell of the Maya Exploration Center observed modern Maya using the geometric fact that a rhombus with equal diagonals must be a square to make sure the corners were square when laying out a building. This fact allows one to easily check whether or not a rhombus is a square by checking if the diagonals are equal with a cord or piece of rope. More generally, a parallelogram with equal diagonals must be a rectangle.
In a lecture during the 2011 MAA Study Tour, Powell explained another clever way that the Maya used a cord to form right angles in laying out a square that he learned from a master builder who had learned it while a shaman apprentice. The method uses a knotted cord and properties of equilateral triangles. The cord has eight knots on it, dividing the cord into seven equal segments with a knot at each end. There are loops at each of the knots for staking to the ground. Since the knots are evenly spaced, when knots 1 and 4 are held together and the cord pulled taut, an equilateral triangle with interior angles of 60° is formed. Then knot 6 is joined with knot 3 and the cord pulled tight resulting in another equilateral triangle formed by knots 4, 5 and 6. Finally knot 8 is joined with knot 5 forming a third equilateral triangle (and all together, half of a hexagon). If one runs a ray (or rope) from knot 1 through knot 2 and another ray from knot 1 through knot 7, the resulting angle is a right (60° + 30°) angle. The animation below (created by the first author) illustrates how the right angle is formed.
The first author also created the following video using student volunteers to demonstrate the Maya way of forming a right angle. See Activity 3 for detailed instructions for recreating this activity with your students.
After analyzing many classic and modern Maya structures and even participating in the construction of some Maya buildings, Powell discovered that several special ratios appear over and over again. These special ratios include square roots of the positive integers up to \(5\) and the golden ratio \[\phi = {\frac{1+\sqrt{5}}{2}}.\]
Figure 4. Flowers for sale outside a church in Chichicastenango, Guatemala (Photo by Cynthia Woodburn, 2011) |
(For more information on the golden ratio, see the “Golden Ratio” section of the Loci: Convergence article “Leonardo of Pisa: Bunny Rabbits to Bull Markets.”) Rectangles whose sides are in these ratios can easily be constructed by use of a measuring cord, which will be explained below. In each case, a square is first constructed. To lay out a typical Maya home, the initial square would have a side length of two uinics [Powell, p. 39].
Powell has theorized that flowers are the answer to why the Maya chose to use ratios of square roots and the golden mean. These particular ratios arise naturally in flowers, which were, and still are, very special to the Maya. The Maya do not cultivate flowers but wildflowers are sacred to them and often used as offerings in religious ceremonies. Also, flowers and shells are displayed prominently in many classic Maya artworks. But the deciding evidence was when a modern Maya shaman told Dr. Powell that his grandfather had said, “The shapes of the flowers are in our houses” [Powell, p. 116]. For details about Powell's tantalizing thesis, we refer you to his dissertation.
To construct a rectangle whose sides are in ratio of \(1\) to \({\sqrt{2}}\), start with a square. Then take a cord and hold one end at one corner, stretching to the opposite corner to form a diagonal. Keeping the length the same as the diagonal and the first end of the cord stationary, swing the other end until it intersects the extension of one of the sides of the square containing the first corner. From this intersection point, create a perpendicular, continuing on to construct a rectangle containing the original square. Since the longer side has length the same as the diagonal of the original square, by the Pythagorean Theorem, the ratio of the sides will be \({\sqrt{2}}\) to \(1\).
If we start with a rectangle with sides in ratio of \({\sqrt{2}}\) to \(1\), and repeat the same process of using a cord to make a rectangle with a side congruent to the diagonal, the result, again by the Pythagorean Theorem, is a rectangle with sides in ratio of \({\sqrt{3}}\) to \(1\).
The pattern continues, allowing one to easily construct rectangles with sides in ratio of \({\sqrt{4}}=2\) to \(1\), as well as \({\sqrt{5}}\) to \(1\).
Another ratio that appeared frequently in Maya architecture is the golden ratio \({\frac{1+\sqrt{5}}{2}}\). To form a golden rectangle using a cord, again start with a square. Measure the length of a side of the square with a cord and fold in half to mark the midpoint of a side of the square. Then stretch the cord from this midpoint to an opposite corner and swing an arc to intersect an extension of the side of the square where the other end of the cord is held. From this intersection point, form a rectangle. The construction is illustrated in the animation below.
To see that the constructed rectangle is indeed a golden rectangle, without loss of generality, suppose the initial square has side length two. Stretching the cord from the midpoint of a side to an opposite corner forms a right triangle with side lengths \(1\) and \(2\). By the Pythagorean Theorem, the hypotenuse has length \({\sqrt{5}}\). So, the resulting rectangle has one pair of sides of length \(1+{\sqrt{5}}\) while the other sides are length 2. Hence, the ratio of sides is the golden mean \({\frac{1+\sqrt{5}}{2}}\).
Many Maya houses, both ancient and modern, have a basic rectangular shape, but with rounded ends. These also can be constructed with the help of a cord. First find the midpoint of one of the sides to be rounded by using the cord, and then stretch the cord from the midpoint to an opposite corner and swing the cord to the other corner. Then repeat the process on the other side of the rectangle.
Figures 5 and 6 below show modern Maya houses built in the traditional style. See Mundo Maya for additional information about the construction of "The Traditional House."
Figure 5. Maya houses in Campeche, Mexico (Photo by Joel Haack, 2011, used with permission)
Figure 6. Maya house in Yucatan, Mexico (Photo by Joel Haack, 2011, used with permission)
An activity for grades 7 and up that applies the Pythagorean Theorem and investigates the special square root ratios that appeared in Maya architecture
This activity can be carried out on a large scale as in the video above or on a smaller scale. For the larger scale activity, using ten to twelve students and about 75 feet of rope, follow the instructions below. For the smaller version, have students work in pairs or groups of three with about 12 inches of string for the knotted cord. If desired, the equilateral triangles can be formed over a sheet of paper with tape used to hold down the knots. Having students use knotted cords to make right angles in the Maya way ties in with Common Core Standard G.CO.12, which recommends making formal geometric constructions with a variety of tools and methods.
The rope or string needs to be “knotted” with eight equally-spaced “knots” which divide the rope or string into seven congruent segments. The “knots” can be actual knots or just “virtual knots” marked on the rope or string. For the video, we used 75 feet of yellow nylon rope with pieces of red tape marking “knots” at ten-foot intervals and with one person assigned to each of the seven knots and others available to help.
The classic Maya utilized geometry extensively in constructing their homes and buildings without the use of modern measuring devices. By using simple knotted measuring cords, they were able to form right angles and rectangles with sides in special ratios which they noticed appearing in nature, especially in flowers. These special ratios included square roots of small integers, as well as the golden mean. Students today can use these same methods to explore and engage with geometrical concepts through hands-on activities, while at the same time gaining an appreciation of some of the mathematical contributions of the classic Maya civilization.
John C. D. Diamantopoulos is an Associate Professor in the Department of Mathematics and Computer Science at Northeastern State University in Tahlequah, Oklahoma. He has been very active in the MAA, both at the section and national level. His mathematical interests include ordinary differential equations, mathematics education, and history of mathematics. Diamantopoulos is also very active in his church, volunteering on computer productions/presentations and any area that needs attention. |
Cynthia J. Woodburn is a University Professor in the Department of Mathematics at Pittsburg State University in Pittsburg, Kansas. She has participated in all of the MAA Study Tours since 2009. Her research areas include computational commutative algebra and history of mathematics. Woodburn is a handbell soloist and has a black belt in Chinese Kenpo karate. |