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Mathematics and Architecture of the Incas in Peru

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Mathematics and Architecture of the Incas in Peru AC 2011-2240: MATHEMATICS AND ARCHITECTURE OF THE INCAS IN PERU Cheri Shakiban, University of St. Thomas I am a professor of mathematics at the University of St. Thomas in St. Paul, Minnesota, where I have been a faculty member since 1983. I received my Ph.D. in 1979 from Brown University in Formal Cal- culus of Variations. My recent area of research is mostly in computer vision, with applications to object recognition. My publications are in diverse areas of mathematics and engineering. I love to work with undergraduate students, in particular, underrepresented students, to get them involved in doing research in mathematics and encourage them to give conference presentations/posters and submit their work for publication. In addition to teaching regular math courses, I also like to create and teach innovative courses such as ”Mathematical symmetry of Southern Spain” and ”Mathematics and Architecture of the Incas in Peru”, which I have taught as study abroad courses several times. Michael P. Hennessey, University of St. Thomas Michael P. Hennessey (Mike) joined the full-time faculty as an Assistant Professor fall semester 2000. He is an expert in machine design, computer-aided-engineering, and in the kinematics, dynamics, and control of mechanical systems, along with related areas of applied mathematics. Presently, he has published 41 technical papers (published or accepted), in journals (9), conferences (31), or magazines (1). In 2006 he was tenured and promoted to the rank of Associate Professor. Mike gained 10 years of industrial and academic research lab experience at 3M, FMC, and the University of Minnesota prior to embarking on an academic career at Rochester Institute of Technology (3 years) and Minnesota State University, Mankato (2 years). Mike holds a Bachelor of Mathematics from the University of Minnesota (with distinction), an MS in Mechanical Engineering from MIT, and a Ph.D. in Mechanical Engineering from the University of Minnesota. He is also a member of ASME, SIAM, and ASEE. c American Society for Engineering Education, 2011 Mathematics and Architecture of the Incas in Peru Abstract This novel study-abroad January (J-term) course co-taught by the authors (from mathematics and mechanical engineering) is described in detail. It evolves as a journey, both academically and geographically, with most topics being directly related to the Incas and / or symmetry. Students learned about quipus (a complex system of knot-tying for recording numerical and other data), la chakana symbol, la yupana (an abacus-like calculating device), geometrical symmetry of frieze and wallpaper patterns commonly depicted on buildings and in textiles, and the stability of structures, important because of the common occurrence of earthquakes. In addition, the famous enigmatic Nazca lines and the steamship, Yavari, on the highest elevation commercial lake in the world, (Lake Titicaca) were studied onsite. This adventurous course, whether by land, air, or water, traversed through much of Peru (0-14,000 ft), including the cities of Lima, Nazca, Arequipa, Puno and Lake Titicaca (islands of Uros and Taquile), Cusco, Aquas Calientes, and the beach community of Huanchaco Trujillo. The highlight of the course was visiting the famous lost city of the Incas, Machu Picchu. Samples of student work, assessment, and lessons learned are provided that offer practical advice for others considering offering similar courses. Keywords: Peru, Incas, study-abroad, mathematics, architecture, engineering 1. Introduction Study-abroad courses at many US universities and colleges are commonplace these days, especially on traditional liberal arts topics like languages or history, in disciplines where field work is highly valued such as geology or biology, and for courses that by definition possess attributes with an international flavor to them (e.g. international business). That’s great, but what about mathematics or engineering? Historically speaking, there haven’t been that many study- abroad courses offered in mathematics or engineering and participation rates have been low.1,2 Clearly, one major source of motivation (beyond an individual faculty member’s professional interests) for offering our course described below is to appeal to the needs of this underserved population of mathematics and engineering students, as well as students from other technical disciplines that value analytical thinking (e.g. physical sciences and economics), with regard to meaningful, discipline specific study-abroad course offerings. One well-established, popular, and effective mechanism for offering study-abroad J-term courses at the University of St. Thomas is through a regional 5 school consortium, UMAIE3 (Upper Midwest Association for Intercultural Education). On a yearly basis UMAIE typically offers between 20-30 courses. Each school contributes faculty and students to create an annual January curriculum, with the support of a coordinating organization for enrollment, marketing, logistics and financial management. In our case, no other courses (of the courses offered in J-term 2010) were mathematics or engineering related. The niche market for this course is for students (26 total in our case) who have completed 3 semesters of Calculus (a strict requirement), which narrowed it down to mostly upper classman in engineering and / or mathematics, although other technical majors, such as chemistry were represented. The course can be used as either a 4 semester-credit mathematics or engineering course, appealing to many who are seeking a minor, typically in mathematics.4 Course description (from J-term 2010 UMAIE brochure) The ancient Incas had elaborate mathematics, astronomy, architecture, and communication systems. The purpose of this course is to study their mathematics and see how they incorporated it in their art, architecture, calendar and communication systems. They devised an amazing tool, the quipu, to record numerical data for accounting and for transmitting complex messages with astonishing mathematics which is very similar to the binary codes used in the early versions of computers. Another aspect of their culture filled with mathematics is their architecture that was also enhanced with use of symmetry in their art form. The frieze patterns that occurred on buildings and found on their pottery represent all seven basic types of mathematical frieze patterns. We will visit many of the buildings that were built by the Incas all throughout Peru including the Inca city of Machu Picchu. The gigantic stones used in this city were carved and put together with such precision and mathematical accuracy that to this day one cannot fit a thin knife between stones. The stones are held together not due to mortar but because of pure craftsmanship. We will also visit and study the geometry of the mysterious Nazca Lines, which are a set of zoomorphic, phytomorphic, and geometric figures that appear engraved in the surface of the Nazca desert, a high arid plateau that stretches for 37 miles. They are the most outstanding group of drawings on the ground in the world. From a pedagogy point of view, the course is structured such that there are different elements that work together synergistically, including on-site visits (possibly with accompanying instruction), travel, readings, lectures, student presentations (both technical & tourist oriented) to promote ownership and engagement in the course, and traditional homework in a 300+ page custom bound booklet & workbook with over 90 traditional mathematics problems5 (also posted to Blackboard, Bb). The itinerary is shown below in Table 1. Table 1 Abbreviated itinerary Date (1 / X / 2010) City Sites (excluding class and tourism) Saturday, 2 Twin Cities / Lima Sunday, 3 Lima Pachacamac, Archaeological and Anthropological Museum Monday, 4 Lima Tuesday, 5 Lima / Nazca Wednesday, 6 Nazca Nazca lines Thursday, 7 Nazca / Arequipa Friday, 8 Arequipa Monastery of Santa Catalina Saturday, 9 Arequipa Sunday, 10 Arequipa / Puno Monday, 11 Puno Chulpas of Sillustany, Yavari steamship Tuesday, 12 Puno Full-day motor boat to floating islands of Uros and Taquile Wednesday, 13 Puno / Cusco Puka Pukara (Red Fortress), La Raya, Raqchi (Temple of Wiracocha God), and church of Andahuaylillas Thursday, 14 Cusco Sacsayguaman, Kenko, Pucapucara and Tom-Bomachay Saturday, 16 Cusco Sacred Valley of the Incas, Pisac Indian market, Urubamba, Ollantaytambo fortress, and Chincheros Sunday, 17 Cusco / Aguas Calientes Ollanataytambo train station Monday, 18 Aguas Calientes / Cusco Machu Picchu Tuesday, 19 Cusco / Lima Cusco airport Wednesday, 20 Lima / Huanchaco Trujillo Thursday, 21 Huanchaco Trujillo Temple of the Moon and the Sun, Chan Chan with visits to the Temple del Arco Iris, the site museum, and the Tshudi Palace Friday, 22 Huanchaco Trujillo Saturday, 23 Huanchaco Trujillo / Lima Sunday, 24 Lima / Twin Cities The lead author and course originator (Shakiban, Mathematics Professor) is an expert on the topic of mathematical symmetry and fractals, in addition to being a seasoned international traveler (having travelled to over 50 countries), often to sites that exhibit interesting symmetry or fractal attributes. Symmetry and fractals appear in many places including manmade forms, such as textiles and impressively in architecture, which, to maximize appreciation, makes sense to be physically on-site. The second author (Hennessey, Mechanical Engineering Associate Professor) specializes in kinematics and dynamics of mechanical systems and this is his first involvement
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