Math Chat: Infinity, Dynamics, the Logistic Map, and Chaos
Details
In this "Math Chat" we will explore some introductory issues about Infinity (including whether it exists) and an introduction to Dynamics, the Logistic Map, and Chaos from Complex Systems. There are 2 hours of highlighted videos and 2 hours of elementary videos (on infinity) for a total of four hours for the ambitious participant. Watching the videos is optional, we will discuss the content of each video by summarizing the content in case you cannot watch them.
Videos on Infinity
• Dr James Grime explains that there are different kinds of infinity in this 8 minute video.
https://www.youtube.com/watch?v=elvOZm0d4H0
• Jeff Dekofsky explains the Hilbert Hotel with a fascinating new (to me) twist at the end of this 6 minute video.
https://www.youtube.com/watch?v=Uj3_KqkI9Zo
• Mark Jago explores four paradoxes of infinity in this 10 minute video.
http://www.youtube.com/watch?v=dDl7g_2x74Q
• James Franklin and N J Wildberger debate the existence of infinity in this 43 minute video:
http://www.youtube.com/watch?v=WabHm1QWVCA
• Elementary "Effective Thinking" videos on infinity with Michael Starbird and his students: 1) Introduction to Infinity: The Cardinality of the Natural and Rational Numbers (23 videos totaling 1 hour 8 minutes) (https://www.youtube.com/playlist?list=PLkCiNL_gZp2cAoO_9ow3XC0Umpg_AGte4) and 2) Introduction to Infinity: Dodgeball and the Cardinality of the Real Numbers (19 videos totaling 49 minutes) (https://www.youtube.com/playlist?list=PLkCiNL_gZp2dA5bERejisfvlziCsnsXHR) Note: these videos provide an excellent but very elementary treatment (advanced members may want to watch the videos speeded up significantly or skip them altogether) of the proof that the cardinality of the natural and rational numbers differs from the cardinality of the real numbers. The videos provide a good exercise for anyone wanting to improve their thinking skills (pause the video often and work out a precise answer to each question before Starbird and his students give hints or solutions, if you get stuck watch a bit more but challenge yourself to answer the questions before the video gives too many clues). They are also worth studying from the perspective of the value of "understanding simple things deeply": how deeply do you understand rational & real numbers and how the properties of those numbers affects "counting" the size of infinite sets? Educators looking to better understand how elementary concepts can be taught from an advanced perspective will also find these videos useful.
Videos on Dynamics, the Logistic Map, and Chaos
• Melanie Mitchell introduces the subject of Dynamical Systems Theory in this 14 minute video.
http://www.youtube.com/watch?v=9KA6PWim2TA
Melanie Mitchell explores population growth rates in this 9 minute video. You can download the NetLogo (https://en.wikipedia.org/wiki/NetLogo) code for the simple population growth model she uses here (http://s3.amazonaws.com/complexityexplorer/IntroToComplexity/SimplePopulationGrowth.nlogo).
http://www.youtube.com/watch?v=haXiK7Ydnp0
• Melanie Mitchell explains Verhulst's logistic model for population growth in this 8 minute video.
http://www.youtube.com/watch?v=rxd1poIvP6s
• Melanie Mitchell explains the Logistic Map in this 11 minute video.
http://www.youtube.com/watch?v=XqlnD5CjIug
• Melanie Mitchell explores this NetLogo model for the Logistic Map (http://s3.amazonaws.com/complexityexplorer/IntroToComplexity/LogisticMap.nlogo) in this 7 minute video.
http://www.youtube.com/watch?v=ri4zMp_mVv4
• Melanie Mitchell explains the sensitive dependence on initial conditions (using this NetLogo model) (http://s3.amazonaws.com/complexityexplorer/IntroToComplexity/SensitiveDependence.nlogo) in this 11 minute video.
http://www.youtube.com/watch?v=U-rjOv3CD3o
• Melanie Mitchell concludes her introduction to dynamics and chaos in complex systems in this 11 minute video. Here is the NetLogo code for the Sine Map.
http://www.youtube.com/watch?v=K9KKAxYbr0g
All of the Dynamics, the Logistic Map, & Chaos videos in one playlist (7 videos totaling 1 hours 10 minutes) (https://www.youtube.com/playlist?list=PLkCiNL_gZp2fqJ1wL40QUWkGicvXJOJ7l).
Additional background resources for NetLogo: Melanie Mitchell's 11 minute video introduction to NetLogo (http://www.youtube.com/watch?v=dKUQ6FuPrwc). Download NetLogo through this link (http://ccl.northwestern.edu/netlogo); Download Mitchell's "Getting Started with NetLogo" (pdf, 863k) (http://s3.amazonaws.com/complexityexplorer/IntroToComplexity/GettingStartedWithNetLogo.pdf), Download the code for AntsNew.nlogo (http://s3.amazonaws.com/complexityexplorer/IntroToComplexity/AntsNew.nlogo).
Notes about related Math Counts events: Past and Future
The discussion on Dynamics, the Logistic Map, and Chaos builds upon our October 2013 discussion "Chaotic math - A look a dynamical systems" which was based on James Gleick's book "Chaos: Making A New Science" (https://www.meetup.com/MathCounts/events/143432202). In addition the July 2014 Math Chat discussed NetLogo in more depth (https://www.meetup.com/MathCounts/events/169050352/) among other topics.
The discussion on Infinity will prepare us to delve into more depth next month when on we discuss Lillian Lieber's book "Infinity: Beyond the Beyond the Beyond" which explores Cantor's theory of transfinites (https://www.meetup.com/MathCounts/events/218584450/) in much more depth.