Communicating Math- Euler's Line and Euler's Circle

         This week in class we discussed Leonhard Euler and all he has contributed to mathematics. He lived from 1707-1783 and is known as the last mathematician to know all of mathematics. One of my favorite areas of math is geometry. When I heard about Euler's Line and Euler's Circle, I thought they were incredible discoveries. The Euler line is a straight line through the orthocenter, triangle centroid, circumcenter, de Longchamps point, the center of Euler's Circle, and other centers of a triangle.
         When I studied Euclidean Geometry, I was unaware and often confused by all of the different centers of a triangle. So first, I am going to explain how to find each of these centers, and then explain Euler's line that he discovered.
          First, we will look at the orthocenter. The orthocenter, point D, is where the three altitudes of the triangle intersect as shown below.

         Next, we have the triangle centroid, point G below. This center is where the three medians intersect. The median is found by finding the midpoint of each side of the triangle and connecting the midpoint to the opposite vertex. 

             The circumcenter, point H below, is the point where the perpendicular bisectors of each side intersect.

           Another center of circle, de Longchamps point, point K below, can be found. De Longchamps point is found by reflecting the orthocenter over the circumcenter as shown below.

        

          Finally, we will look at the center of Euler's circle which is also known as the nine-point circle. This circle is found by connecting nine points together. You connect: the midpoint of each side of the triangle, the foot of each altitude, and the midpoint of the three segments that connect the orthocenter to the vertices. The Euler circle is shown below with center at point S.

           Once Euler has found all of these centers of a triangle, he realized you could draw a straight then through them.

            The pink line above is the Euler line passing through the orthocenter(point D), the triangle centroid (point G), the circumcenter (point H), the de Longchamps point (point K), and the center of Euler's circle (point S). Since this image is quite complex with all the lines you need to find these centers, I simplified it below.

           The only triangle that does not contain Euler's line is an equilateral triangle. For equilateral triangles, all the centers of the triangle fall on exactly the same point. Thus, you cannot create a line connecting all these points when there is only one point.

           I think that realizing that Euler's line holds true in all triangles except equilateral triangles is quite incredible. When I first learned about it I was amazed. It made me wonder how many of these types of discoveries in geometry I don't know about yet or have not yet been discovered. 

          Also, if I ever become a geometry teacher, I would definitely introduce my class to Euler and his line and circle. I think these concepts would grasp students interest and make them more excited to learn about geometry.  

The Joy of X A Guided Tour of Math, from One to Infinity by Steven Strogatz

             I read the book The Joy of X A Guided Tour of Math, from One to Infinity by Steven Strogatz. All topics of math are addressed in this book in a way that anyone can understand. You do not need to be a math major or a mathematician to understand and enjoy this novel.

            The book is split up into six parts: Numbers, Algebra, Shapes, Change, Data, and Frontiers. Numbers addresses negative numbers, multiplying versus addition, division, and representing numbers. The second part about algebra discusses how algebra involves variables, imaginary numbers, word problems, and functions. Next, the book talks about shapes, geometry, proofs, and pi. After Shapes, the fourth part is about change which includes calculus, differential equations, and vectors. The fifth part is Data which talks about statistics, probability, and linear algebra. The last part is Frontiers which covers number theory, group theory, topology, and spherical geometry.

            Although this may sound like too many topics to address in one book, the author does a nice job. First, the book starts with easier topics and then slowly moves to more challenging mathematics. Throughout each chapter, Steven Strogatz does an excellent job of giving examples for every topic. These examples explain how these complicated math concepts apply to everyone’s day to day life. Additionally, he gives many diagrams and visuals to help you further understand what he is explaining. In part six, for example, he describes how group theory can be explained through how you flip your mattress to make sure it wears evenly by giving visuals and easy solutions.

            The Joy of X is a great read for many audiences. Teachers could find examples that would help them explain to their students how the math they are learning in class applies to their real lives. Math majors receive an overview of many of the topics they have learned throughout their studies and they may find an explanation they had never heard before to find further clarification. Anyone who dislikes math can read this and learn how math is all around them. I think everyone will find more joy in math when they read this book.

            Personally, right from the beginning, the book captured my attention with the following quote, “math always involves both invention and discovery: we invent the concepts but discover the consequences” (5). I don’t think I have ever thought about math like this before. Mathematicians defined what numbers mean and what addition means, but have no way of controlling the results of what they have defined.

            I highly recommend you read this book. It is an easy read and will help you see math in a whole new way. You will finally understand the importance of what your teacher was teaching you in high school, or you will find the beauty of math and how it is found all around you. 

Nature of Mathematics- Is math a science?

          

            At first glance at the question, “Is math a science?’, I would say no. All throughout school I loved math class and disliked most science classes. So from that I do not want to say math is a science. Additionally, in every school math class and science class are separate subjects you must learn. However, that does not answer the question; that is just my gut reaction.

            Thinking about it further, I have still concluded that math is not a science. Math is used in science, but science does not have to be used in math. Science involves investigating and studying the physical and natural world while math uses procedures and operations to explore possible worlds.

            Some instances that science uses math include when scientists have to quantify an object’s mass or length. Scientists sometimes use mathematical data charts to organize their information and find patterns. Additionally, chemists use algebra to figure out the proportions of compounds to use to make a reaction. However, weighing and measuring objects, organizing information, and algebra are not strictly used for scientific information. Science borrows these tools and ideas from math. Without these tools, scientists would not be able to discover many things they now know.

            Math on the other hand does not ever need to use science. Math does not need to use real world examples to make discoveries. It can use possible worlds and be very abstract.

            I am not trying to say that there are no relationships between math and science. Both are very structured in the way new information is found. Mathematicians use proofs and scientists use the scientific method. However, proofs and the scientific method are not the same, but both are systematic. For proofs, definitions and axioms are used to create a logical argument that shows a statement is true. For the scientific method, you ask a question, do research, create a hypothesis, experiment, draw a conclusion, and then report results. These differences help me to think that math is not a science.

            I think that mathematics follows more rules and is more logical. Once something in math is proven, it will always be true. Scientists use experiments which can give varying results. It is not as black and white for scientists to be certain if the results of their experiments are true one hundred percent of the time.   
            In conclusion, I do not think math is a science. Math and science have many connections but I do not believe these connections are enough to say math is a science. I think math is used as a tool for science, but science does not need to be used in math. Math is so much more than the ways it is used in science which makes be believe it is not a science.

           Even though I think math is not a science, it is important to hear others points of view of the question and challenge what I think. Additionally, it is possible my students one day will ask me this and I will be open to hear their opinions and share mine with them.

Citation

http://mathforum.org/library/drmath/view/52363.html