Posts Tagged ‘geometry’

December 14, 2013 – Dr. Vladimir Dragovic – “Unusual Billiards”
We will present an attractive crossroad of mechanics and mathematics. We will talk about interesting problems of billiards on triangular and elliptical tables. Video clips illustrating the subject will be demonstrated. We will provide basic definitions and some historical background.

About the lecturer: Vladimir Dragovic is a professor of Mathematics at UT Dallas. He has been a full research professor and the Head of the Department of Mechanics of the Mathematical Institute of the Serbian Academy of Sciences and Arts. He served as the Director of the Mathematical High School in Belgrade (2004-2008), known as one of the world’s most successful schools in the IMO competitions ( http://en.wikipedia.org/wiki/Matemati%C4%8Dka_gimnazija).

**Please note:  Math Circle will be on winter break from December 21, 2013 through January 4, 2014.  We will resume with a talk by Mathew Crawford on January 11, 2014 (see schedule).

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220px-Straight_Square_Inscribed_in_a_Circle_240pxLearn all about the classic theory of geometric objects with only straight edge and compass. What can you construct, and what is impossible? Which regular polygons can you construct? What if you have help by being given a fixed parabola? Come with your pencils and be ready to draw (compasses provided, or bring your own)!

To perform well on geometry problems on math competitions it is necessary to have a deep understanding.  This understanding can be achieved by retracing the footprints of the very first mathematicians whose only tools were a straight edge and compass.

Dr. Kane is one of our most popular lecturers and is recently retired from his position as a professor at the University of Wisconsin – Whitewater.  Along with Dr. Andreescu he is the co-founder and coordinator of the Purple Comet! Math Meet.  Dr. Kane is also the co-chair of the AIME committee and a faculty member at the AwesomeMath summer camp.

Along with his wife, Jane E. Mertz, Jonathan Kane is the author of several important research papers on the role of culture and gender in mathematical achievement including “Debunking Myths about Gender and Mathematics Performance.”

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Cosmin Pohoata will return to the Metroplex Math Circle this week to give a talk on symmedians.  This will be an excellent lecture particularly for some of our more advanced students and those preparing for mathematical contests where symmedians can be a powerful part of their tool kits.

Abstract: We will introduce symmedians from scratch and prove the entire collection of interconnected results that characterize them.  Applications from contests around the world will also be presented if time permits.

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Dr. Yotov sends his appreciation for the large and attentive group we had last Saturday.  He also forwarded this link to more resourcess:

Here is a wonderful link to materials on Geometry and Arts:  <http://myweb.cwpost.liu.edu/aburns/sigmaa-arts/index.html>

Dr. Yotov’s talk had all of the great elements of a Math Circle lecture.  He began with a very visual and accessible discussion of one point perspective.  The lecture then built upon this concept to introduce the additional complexities of two-point, three-point and even four-point perspective!  By the end of the talk Dr. Yotov bridged from these ideas to the very challenging but useful concepts of projective geometry.

One of the highlights at the end of the lecture was a look at forced perspective art.  The examples by Julian Beever are only possible by applying the mathematics discovered during the Renaissance and explained to us by Dr. Yotov.

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For our next Math Circle we are very pleased to have Dr. Mirroslav Yotov who will speak about “Linear perspective or how Geometry helps in drawing realistic pictures

Have you ever had troubles in drawing a picture and making the painting look natural? Well, I have! And now I know that simple rules from Geometry can help resolve this problem. It was actually the great painters of Renaissances who developed the part of Geometry, called Projective Geometry, in order to put an order in the painting process! Many masterpieces in visual arts, including movies, were made following these rules. I will share with you my excitement of what I know from Geometry, and how it is applied to the visual arts. The presentation will be accessible to people who know about straight lines, circles, planes, and spheres.

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Our September 18th speaker, Dr. Branislav Kisacanin,  is an accomplished author.  His book, Mathematical Problems and Proofs, has enjoyed very favorable reviews on Amazon:

This is a very interesting book. If you have mastered the bare essentials of set theory (through the upside down A for “for all” and the backwards E for “there exists”), AND either 1) the bare essentials of combinatorics (through Pascal’s triangle), or 2) the bare essentials of number theory (through the definition of the Moebius function and the statement of the Chinese Remainder theorem), or 3) the bare essentials of geometry (through the law of cosines), AND if you are very talented in mathematics, then this book is a “MUST READ”. It matters not whether you are a high school student or a professional mathematician. You will find new and fruitful insights and quite a few interesting problems in this book. For the beginner, there are several tantalizing (but somewhat oversimplified) references to advanced topics such as Paul Cohen’s proof of the independence of the continuum hypotheses and Wiles’ proof of Fermat’s last theorem. For the professional there are footnotes with references to little known and suprising results obtained in the 20th century. But, unlike the claims in the “editorial review”, this book neither prepares you to read the literaure nor is it a store house of exercises which will help you take your problem solving abilities to the next level. The “editorial review” is “off”, but the book is “right on”. Only the title is unfortunate. Where it will help the career mathematician is in the “beer hall” or “coffee house” or “tea time” discussions with other mathematicians. It is just chock full of beautiful little “gems” which can be shared with one’s friends. This is the kind of beautiful stuff that makes mathematics truly interesting and exciting and though I have searched for a book like this for the last 35 years, this is by far the best in its class which I have found. Dr. John Aiken, Jan 5, 2003.

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Last week’s presentation was well attended and so good that Dr. Andreescu has agreed to continue his talk this coming week!

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This weekend the director of Metroplex Math Circle, Dr. Titu Andreescu, will give the lecture.  For those of you who are not fortunate to have heard Dr. Andreescu speak, here is some information about him:

Dr. Titu Andreescu, University of Texas at Dallas

Titu Andreescu received his Ph.D. from the West University of Timisoara, Romania. The topic of his dissertation was “Research on Diophantine Analysis and Applications.” Professor Andreescu currently teaches at The University of Texas at Dallas. He is past chairman of the USA Mathematical Olympiad, served as director of the MAA American Mathematics Competitions (1998–2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993–2002), director of the Mathematical Olympiad Summer Program (1995–2002), and leader of the USA IMO Team (1995–2002). In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the world’s most prestigious mathematics competition. Titu co-founded in 2006 and continues as director of the AwesomeMath Summer Program (AMSP). He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a “Certificate of Appreciation” from the president of the MAA in 1995 for his outstanding service as coach of the Mathematical Olympiad Summer Program in preparing the US team for its perfect performance in Hong Kong at the 1994 IMO. Titu’s contributions to numerous textbooks and problem books are recognized worldwide. Related articles on this site.

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290px-Triangle_inequality.svgThis Saturday we are very pleased to have as our returning speaker, Dr. Dorin Andrica.   He will share with us many applications of the Triangle Inequality which students will find useful on upcoming math contests.

Dorin Andrica received his Ph.D. in 1992 from “Babes-Bolyai” University in Cluj-Napoca, Romania; his thesis treated critical points and applications to the geometry of differentiable submanifolds. Professor Andrica has been chairman of the Department of Geometry at “Babes-Bolyai” since 1995. He has written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. He is an invited lecturer at university conferences around the world: Austria, Bulgaria, Czech Republic, Egypt, France, Germany, Greece, Italy, the Netherlands, Portugal, Serbia, Turkey, and the USA. Dorin is a member of the Romanian Committee for the Mathematics Olympiad and is a member on the editorial boards of several international journals. Also, he is well known for his conjecture about consecutive primes called “Andrica’s Conjecture.” He has been a regular faculty member at the Canada–USA Mathcamps between 2001–2005 and at the AwesomeMath Summer Program (AMSP) since 2006. Related articles on this site.

Please note that this meeting will be held in ECSS 2.311.

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In anticipation for our speaker this Saturday I thought I would post what Wikipedia has to say about Thebault’s Theorem.   No preparation is required for each Math Circle lecture, but then again it can never hurt:

thebault_theoremThébault’s problem I

Given any parallelogram, construct on its sides four squares external to the parallelogram. The quadrilateral formed by joining the centers of those four squares is a square.

It is a special case of van Aubel’s theorem.

Thébault’s problem II

Given a square, construct equilateral triangles on two adjacent edges, either both inside or both outside the square. Then the triangle formed by joining the vertex of the square distant from both triangles and the vertices of the triangles distant from the square is equilateral.

Thébault’s problem III

Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle. Then construct two additional circles, each tangent to AM, BC, and to the circumcircle. Then their centers and the center of the incircle are colinear.

Until 2003, acadamia thought this third problem of Thébault the most difficult to prove. It was published in the American Mathematical Monthly in 1938, and proved by Dutch mathematician H. Streefkerk in 1973. However, in 2003, Jean-Louis Ayme discovered that Y. Sawayama, an instructor at The Central Military School of Tokyo, independently proposed and solved this problem in 1905.[1]

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