Beautiful graphs and beautiful monsters

Linkdump from today's mathematical explorations

Beautifully graphs with clear explanations and interpretations are one of my favorite things.  This blog post by the 538 blog is a prime example of just that.

http://fivethirtyeight.com/features/four-strikes-and-youre-out/

While discussing counterexamples with a student today I mentioned the Weierstrass function. Here is an great blog post which gives some backstory to the development of and reaction to the function that is everywhere continuous and nowhere differentiable. A thumb on the nose to Newton.

http://nautil.us/issue/11/light/maths-beautiful-monsters
http://www.math.washington.edu/~conroy/general/weierstrass/weier.htm

The Weierstrass function is just one of many "Pathological Functions": http://en.wikipedia.org/wiki/Pathological_%28mathematics%29

There are many here http://math.stackexchange.com/questions/740/useful-examples-of-pathological-functions and here http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function.  There is also an excellent book titled Counterexamples in Analysis which covers even more of these pathological functions in great detail. http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753

Old Math Books

Sometimeshttp://www.reddit.com/r/math turns has some wonderful curiosities on its frontpage. Several users have math texts from the mid to late 1800s and posted a few images. As it turns out, each text has been digitally archived and can be viewed for free by anyone with an internet connection and a PDF reader. The discussions about typesetting, mathematical notation, and mathematical exposition (writing style), was interesting.  Below are some highlights.

Plane and solid geometry by George Wentworth (1899)
imgur gallery
discussion thread
full text on Internet Archive

A treatise on differential equations by George Boole & Isaac Todhunter (1865)
imgur gallery
discussion thread
full text on Internet Archive

Elements of Algebra By George Roberts Perkins (1852)
imgur gallery
discussion thread
full text on Google Books

BBC -- Beautiful Equations

 

Artist Matt Collings chats openly with scientists as he takes a tour of some of the most important equations in physics. Equations are explained to be an unambiguous generalized shorthand for expressing relationships about physical quantities. "A little increase in mass means a gigantic increase in energy." How much, exactly? Well, the details are actually worked out in a way that anyone can follow.

This documentary is well worth watching as it gives an approachable and concrete explanation of what all the symbols on the page actually mean and how truly remarkable it is that such universal truths, which literally apply to all objects in the universe, can be expressed in such a compact manner. 

The concept of "Mathematical Beauty" is explored and (spoiler alert) it basically means that a single, compact equation can very simply and very generally express the relationship between quantities (mass, energy, time, velocity, force, etc.).

BBC Page for "Beautiful Equations" http://www.bbc.co.uk/programmes/b00wltbm

Matt Collings's website: http://www.emmabiggsandmatthewcollings.net/recent

The five equations presented: 

1) The relationship between mass and energy: E = mc^2 
(E = energy, m = mass of an object, c = the speed of light)

2) The force of gravity between two bodies: F = (G * m1*m2)/(r^2)
(G = the gravitational constant, m1 and m2 = "the first object having mass m1, the second object has mass 2" and r = the distance between the two objects)

3) Time dilation: t' = t*sqrt[ 1 - (v/c)^2 ]
 t = time in your frame of reference, as you move at velocity v.
t' = time according to someone watching you from another frame of reference at rest to the first. c = the speed of light. 

4) Dirac Equation: (i*gamma*del - m)*phi = 0 
Difficult to summarize without getting into complicated mathematics. Essentially it predicted that all matter has anti-matter components and it described how the most elementary of particles behaved. More here.

 "What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating." Paul Dirac 1969 full speech here

5) Hawking Equation:  S = A/4
S = entropy (disorder) , A = the area of the black hole

It is worth mentioning another blog post which goes nicely with this theme.

"Math doesn't use fancy words and weird notations because mathematicians are trying to confuse non-mathematicians. It’s not done out of spite, or out of some desire to exclude non-mathematicians from the club. It’s about precision."

http://www.goodmath.org/blog/2014/03/24/yet-another-cantor-crank-size-vs-cardinality/

As a particularly nice example, I offer up the equation for the Fourier Transformation.



At first, it can seem intimidating and overwhelming -- so many symbols, fractions within superscripts, what does it all mean? Well, each symbol has a very specific and unambiguous meaning, quite a lot of punch is packed into this array of symbols!


The image above was originally seen here and explained for the lay reader in this blog.

Mathematical Beauty

This week has been especially beautiful, mathematically speaking.

n-body problems are interesting and challenging.  By using the laws of gravitation and a computer and some very clever symmetry, many beautiful (and in a perfect world, stable) orbits are possible!

This set of simulations is one of the most amazing and beautiful I have ever seen. Toggle the trajectories, particularly on the last two in the list.

http://www.maths.manchester.ac.uk/~jm/Choreographies/

Biofabric is a creative way to re-imagine network diagrams. What is particularly impressive is the short animation going from the old view to the new. See the blog detailing the process here.

before

after

Visualizing sports stats is a great way to engage students with the study of mathematics! Here are some of my favorites.  

http://statmilk.com/NFL/0/? and http://www.statmilk.com/NCAAF/0/? let you visualize just about anything you would want to know about the NFL and NCAA.

http://www.chartball.com/football/ offers a creative and interesting way to visualize football games. 

I hope you enjoy these and are able to work them into your classes.  I imagine that many interesting discussions can arise from having students explore statistics with the help of these websites. 

Math & Juggling Talks & Workshops

I had the pleasure of giving a talk about the mathematics of juggling at the University of Maryland for their annual Maryland Day celebration on 27 April 2013.

Here is a short news clip.



Math and Juggling (Recut) from Matt Meiselman on Vimeo.

And here is the promo JHU CTY Online did about my work and teaching philosophy:

I will be giving several other talks and workshops this year.  A tentative schedule is below.

Workshop, Demos & Appearances ==> Summer 2013

• Sat 08 June 2013 -- Montgomery County Be Active Wellness initative 10am-2pm at the South Germantown Recreation Park next to the soccer plex.
• Mon 10 June 2013 -- Maryland School for the Deaf (Columbia) workshop + demo for 4th & 5th graders
• Sat 15 June 2013 -- Ripley's Believe it or Not + World Juggling Day the Baltimore Inner Harbor
• Sat 20 & Sun 21 July -- Artscape! 
• Sun 25 August 2013 -- Deaf Senior Association afternoon workshop + demos

If you are interested in having me give a talk and/or workshop about mathematics, juggling, or both, please feel free to contact me via twitter: @mathteacher1729 or email: mathteacher1729@yahoo.com

New blog series -- 'Little known free math gems"

 "Little known free math gems"

These are tiny, free, and surprisingly powerful programs for mathematics students from algebra through college-level which can be used on virtually any laptop or desktop computer, even old ones.   

All of my screenshots will be from a laptop running Windows Vista Pro with 1.88 GHz on 1GB of RAM.

I'll be reviewing the following: I hope to do one review each week. 

1. DPGraph
2. GrafEq
3. MVT
4. Winplot
5. dfield & pplane
6. Linear

Also I plan to include:

1. screenshot techniques (cropper, mspaint)
2. useful & interesting wolframalpha commands
3. desmos
4. Geogebra resources
5. Touch Trigonometry

If you have any specific requests or know of a program I have not mentioned, please feel free to contact me.