Project 9.1
In 1928 Frank Ramsey, an English mathematician, showed that there are patterns implicit in any large structure. The accompanying diagram (which resembles the aestheometric design of Projects 7.2 and 7.3) typifies the problems that Ramsey theory addresses.
“How many people does it take to form a group that always contains either four mutual acquaintances or four mutual strangers?”
In the diagram, points represent people. A red edge connects people who are mutual acquaintances, and a blue edge joins people who are mutual strangers. In the group of 17 points shown, there are not four points whose network of edges are either completely red or completely blue. Therefore, it takes more than 17 people to guarantee that there will always be four people who are either acquaintances or strangers. Write a report on Ramsey theory.
What is Ramsey Tic-Tac-Toe?
Reference:
Ronald L. Graham and Joel H. Spencer, “Ramsey Theory,” Scientific American, July 1990, pp. 112-117
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Project 9.2 Historical Quest
The Garden House of Ostia was constructed in the 2nd century, in the city of Ostia, whose population reached 50,000 at its peak. (See the “What in the World” comment at the beginning of Chapter 7.) This city was a major port of Rome, which was about 25 km away. The Garden Houses are of interest because of the geometry used in its construction. The key to its construction, according to archaeologists Donald and Carol Watts, is a “sacred cut.” In searching the records of the architect Vitruvius they found that the basic pattern begins with a square (called the reference square) and its diagonals. Next quarter circles centered on the corners of the square are drawn, each with a radius equal to half of the diagonal. The arcs pass through the center of the square and intersect two adjoining sides; together they cut the sides into three segments, with the central segment being larger than the other two. By connecting the intersection points, you can divide the reference square into nine parts, as described in the article. At the center of the grid is another square that can serve as the foundation for the next sacred cut. Experiment by drawing or quilting some “sacred cut” designs.
References:
“A Roman Apartment Complex,” by Donald J. Watts and Carol Martin Watts. Scientific American, December 1986, pp. 132-139.
An historical reference, along with a very interesting site on the Garden Houses of Ostia is found at this site:
http://www.ostia-antica.org/indexes.htm