# See details. - Math Awareness Month

## Transcription

See details. - Math Awareness Month

EXPLANATION OF FIGURES for MATH AWARENESS MONTH Fern Y. Hunt October 28, 2014 Given an equilateral triangle and any interior point π, let π1 , π2 , π3 be the distances (length of the perpendicular) to each side of the triangle (see Figure 1). Then the sum of the distances π1 + π2 + π3 = β, where β is the length of the altitude of the triangle. This fact (theorem) was discovered by Vincenzo Viviani, a 17th century Italian mathematician from Florence. He was also Galileoβs last pupil! Letβs assume that β = 1. Then we have π1 + π2 + π3 = 1. By allowing values ππ to be zero, points on the boundary of the triangle can also be included. Thus every point in the triangle can be associated with a triple of non-negative numbers that sum to 1. Although these numbers seem mysterious, in fact they can be calculated if we know the ordinary (π₯, π¦) coordinates of the point π.The formulas that one would use can be derived using some planar geometry. Consequently we can think of functions of 3 variables as functions that are defined on the equilateral triangle, when the variables are non-negative numbers that sum to 1. This situation arises frequently in many areas of science and engineering- chemistry, metallurgy, genetics and economics just to name a few. The translation from mathematics to applications often comes from understanding how a function changes when variables change. Visualization of the functionβs surface is an important tool for gaining this understanding. This poster shows an example of one such function that arose in the authorβs own work. Imagine two towns with traffic routes joining them along 3 possible routes (in the original problem the towns were locations in a network) . Individual cars or trucks choose route 1 at random with probability π1 , route 2 with probability π2 and route 3 with probability π3 . Since probabilities are non-negative and since the only travel between the towns occurs on the 3 routes, the probabilities must sum to 1. The function of interest is the entropy (not the thermodynamic quantity) , π»(π1 , π2 , π3 ) = β(π1 log(π1 ) + π2 log(π2 ) + π3 log(π3 )). Figure 2 shows the surface height and level curves of π» plotted over the equilateral triangle with altitude 1. If π is a point in the triangle , the height of the surface above the point is π»(π1 , π2 , π3 ) the value of π» 1 1 1 corresponding to the coordinates of π. The point with coordinates οΏ½3 , 3 , 3οΏ½ is at the 1 1 1 center of the triangle (see the pink dot). Its value π» οΏ½3 , 3 , 3οΏ½ = log(3) is the highest point on the surface and represents the maximum value of the function. . As π» decreases, the surface heights decrease and the size of the corresponding level curves are ovals of increasing size until a critical value π» = log(2) is reached (the pink oval), the maximum value of π» if there are just 2 roads (i.e. when one of the roads is eliminated). The probability values where this happens are the coordinates of the points where the log(2) level curve is tangent to the sides of the triangle. Note that in this case one of the coordinates is zero. For π» < log (2), for each level, there are 3 disconnected curves each one intersecting a side of two triangles. This is reflected in the discontinuity in the curve in the triangle and the surface itself. More discussion of this can be found at the website. Finally at π» = 0, the level curve reduces to 3 points, the vertices of the triangle, corresponding to the selection of one route and the exclusion of the other two. Credits for images: Created by Terence Griffin with data supplied by Fern Hunt Acknowledgement: The author is grateful to Dan Kalman of American University who provided helpful and very useful comments and advice. REFERENCES: Posamentier, Alfred S, and Ingmar Lehmann. The Secrets of Triangles: A Mathematical Journey. Amherst, N.Y: Prometheus Books, 2012. pp 20-21, 77-78. For a proof without words of Vivianiβs theorem see: Wolf, Samuel, "(No title)", Mathematics Magazine, Vol. 62, no. 3 (1989), p. 190 There are interesting visual enhancements of this proof on the MAA website: http://www.maa.org/publications/periodicals/convergence/proofs-without-words-andbeyond-proofs-without-words-20 accessed October 29, 2014 Information about the theorem and its applications can be found in the cited Wikipedia articles. The articles and the references they point to were helpful in the preparation of this article. http://en.wikipedia.org/wiki/Viviani%27s_theorem accessed October 29, 2014, 12:10pm http://en.wikipedia.org/wiki/Vincenzo_Viviani accessed October 29, 2014 , 12:11pm P Figure 1 Figure 2