Triangle problem

The site Five Triangles has lots of good mathematical problems for middle schoolers. I really liked a recent one:

I wanted to see what configurations with this property looked like, so I made this brief animation:

Braid groups for nine year olds

Homeschooling my daughter in math has been a very interesting experience so far. As a mathematician, I try to re-imagine the topics in our curriculum with the goal of introducing deeper concepts as soon as I can.

Recently thanks to Allen Knutson, on Google+, I found out about a great video series by Ester Dalvit on braids and knots. My daughter has made it though about half of it, which already covers quite a lot of mathematical concepts (especially from a nine-year old's perspective). So I am trying to use it as a springboard to do some algebraic concepts that most K-12 students never see, like noncommutative groups.

So far I've just made a worksheet on braids along with a reference sheet, which went over pretty well today. I'm not sure how much farther I should push it now.

MOOCs - the hype and the pleasant reality

MOOCs - massive online open courses - are all the rage right now, and many people are wondering what their eventual impact will be.  For example, an opinon piece in the New York Times quotes someone as saying that students might be told to "take the following online courses over the summer or over a certain period, and then, when you’re done, you will come to campus and that’s when our job will begin.”

I don't think that's going to happen to a great extent.  Before explaining why, I'd like to make it clear that I am a big fan of MOOCs.  I've signed up for many courses on Coursera, EdX, and Udacity over the last two years, and learned a lot.  I use some of them to supplement my homeschooling, and I plan on incorporating some of their material into courses I teach.  I wish there had been MOOCs when I was in middle school and high school - I'm certain I would have eaten them up.

But the current hype can be pretty unrealistic.  Its interesting to compare the expectations of MOOCs to a very similar phenomenon of 100 years ago, the correspondence course.  Here's the beginning of an article in the journal Science (volume 24 from the year 1906):

There was tremendous interest in the general population in correspondence courses.  Here are some enrollment numbers for one group of correspondence schools (I'm sure there were many others in addition):

Consider that the population of the United States was about a quarter of what it is now, so this is comparable to enrolling a total of almost 4 million people today.  Its not hard to imagine that people thought this would radically change the nature of higher education.

There are many things that make colleges and universities attractive and useful and important to students that are not easily captured by a correspondence course or MOOC.  To be fair, the social and collaborative aspects of MOOCs make them far superior to a correspondence course.  But even for the limited goal of learning a well-defined list of concepts and skills, I think the shared commitment of physically showing up to class is psychologically crucial for most people and it will continue to be so. 

More math worksheets for the adventurous

These are still a work in progress, but here is another collection of worksheets I made for my daughter as part of her homeschooling in mathematics.

These include the ones I posted about previously.

I am planning on polishing up the complete collection once I stop making them for the year, sometime in June.  One thing that is missing is acknowledgements - I have taken many ideas from many sources, such as Fawn Nguyen, Dan Meyer, Kate Nowak, Alfred Posamentier, and many others including state math tests and standards from around the country.

Top mathematics videos

These are all worth watching.  The first two are especially good for kids:

Vi Hart has many, many awesome math videos.  Here's just one to get you started.  OK, I can't resist linking to this one too.  Very hard to pick a favorite.

Donald Duck in Mathmagicland.  This is from 1959, so parts of it are a bit dated but it had a significant influence on me when I saw it sixth grade.  I know many other friends and family who remember it fondly as well - a real classic.

A video from 1988 from Caltech, very well done: http://archive.org/details/theorem_of_pythagoras.

Möbius Transformations.  Short but very well done.

Niles Johnson's Hopf Fibration video.  Hopefully Niles will continue making more videos like this.

Not Knot part I and part II.  This is a window onto some really advanced mathematics, but I think its worth showing to just about everyone.  Perhaps more than any other video I know, it visually conveys one of the many amazing structures of modern mathematics.  Its a good antidote to anyone who thinks that math=arithmetic.

Dimensions is a 9-part series of short movies that have excellent animations.  Of course some episodes are better than others.

There are some other series which are more educationally oriented (and generally less visually stunning) but worth checking out: the Mathalicious project, the TED-Ed math section,  and the Numberphile youtube series.

Finally, two movies which are good are "Between the Folds" (on origami and mathematics), and the BBC Horizons documentary "The Proof" which is about Wiles' solution to Fermat's Last Theorem.  These are both about an hour, so they require a longer attention span than the ones listed above.

If you have any favorites which aren't listed here, let me know.

The Singular Value Decomposition and Congressional Voting

I am teaching a class about the SVD (Singular Value Decomposition) of a matrix this week.  I was inspired by a nice article of Carla Martin and Mason Porter, "The extraordinary SVD", to compute the SVD of the voting record of the 112th Congress (House of Representatives) to show to my class.

If you're interested in how this is done, here is the Sage code I used as a Sage worksheet. 

Here is the projection onto the first two singular vectors:

Third grade math worksheets

These are some (about 76 I think) worksheets that I have used in homeschooling my daughter this year:

http://www.d.umn.edu/~mhampton/ThirdGradeWorksheets_1_to_76.pdf

She was 8 and 9 during this time.  Some of them were too hard, and she required quite a bit of assistance.  Also, since she is learning German, there are a handful of questions that use German number names. 

They would require significant editing to be appropriate to a broad audience, but I am proud of some of them.  Its hard to find good workbooks that include more than the bare minimum material that kids in the US are over-tested on.

Motion sensitive LED belt

Recently I built a motion sensitive LED belt for a dance performance by my wife. In case it is of interest to others, this post will briefly describe how I made it.

I started with the LED belt kit by Adafruit:
http://www.adafruit.com/products/332,

for which they supply an excellent tutorial:
learn.adafruit.com/digital-led-belt 

To this I added a 3-axis accelerometer, the ADXL335 from Adafruit:
http://www.adafruit.com/products/163

Adding the accelerometer is very simple.  The 3V in and ground are soldered with short wires to the corresponding pins on the Atmega32u4 breakout board, and wires from the X,Y, and Z outputs can be soldered to the F4, F5, and F6 pins (see http://github.com/adafruit/Atmega32u4-Breakout-Board for more details or http://www.adafruit.com/datasheets/af_at32u4bb_pinout.pdf for a good diagram).

After a lot of tweaking, here is the code we actually used for the performance:
http://www.d.umn.edu/~mhampton/LED_belt_accelerometer.pde

Here is a short video (with a special guest star - our dog - in the background):

Sorry about the graininess - hopefully I can get a somewhat better version soon.  I'm also hoping for good closeup pictures of the soldered connections.

Tuition myths

There have been many, many articles written over the past few years bemoaning the rapid rise of college tuition. Often there is an implication that the budgets of colleges and universities have ballooned. While it is true that personnel budgets have grown faster than inflation, this is primarily due to the increased cost of healthcare. The real culprit behind most tuition increases at public colleges and universities is a massive decrease in state spending per student. In this post, I will try to illustrate the reality of this for my own institution, the University of Minnesota Duluth.

It is currently Minnesota law that "the state must provide at least 67 percent of the estimated expenditures" for resident undergraduate students, including those from other states such as Wisconsin with which we have a reciprocity agreement (link to statute). There is a cutoff for people who rack up too many credits without graduating, but the law would cover almost all of the current students at UMD.

The actual state contribution is much, much smaller than the law requires. Currently the state provides around 20% of the "O & M" (operations and maintainance) costs for UMD students.

The plots above show, in 2012 inflation-adjusted dollars:

• The nominal tuition at UMD over the last five years (in blue).
• The actual average tuition (in green). This is the actual tuition taken in by UMD divided by the number of students (data is from fall enrollment).
• The black line is what the tuition would have been if the state had maintained its 2007 spending levels for UMD, in inflation-adjusted dollars. This is based on the total dollar amount from 2007, so it does not account for the increase of enrollment at UMD.
• The red line is what tuition would be if the state obeyed its own laws and funded 2/3 of the costs of undergraduate education (for residents of MN and its reciprocity partners). This would have kept tuition constant in real terms at about $5000/year.

How much would that last scenario cost? For UMD, it would cost about $6,750 per student per year. If we use this amount for the whole state, we can estimate the impact on the state budget. Every year about 25,000 MN high school students enter a public university or community college. Let's assume they all eventually get four-year degrees (the "worst"-case scenario from a budgeting point of view, since many do not finish). Then this program would cost approximately $675,000,000 per year. That's a lot, but it isn't a crazy number, considering the current MN higher education budget, excluding loans, is around $200 million.

Thoughtful comments are welcome.

Some worksheets for the curious eight year old

My increased use of Google+ and to a lesser extent Facebook has meant this blog, never very active, has fallen on hard times. But it might get revived a bit by a new project: my wife and I decided to try homeschooling our daughter.

While I am all in favor of a certain amount of drill problems to learn arithmetic, I think usually in schools the exercises are too narrowly focused on one topic. So I have started making some worksheets with more of a mix of subjects. In the slight chance that they are of interest to other parents or kids, I will try to post them here. Here are the first eight, as PDFs and the source LaTeX:

Worksheet PDFs


LaTeX sources

In case you are worried that I am reinventing the wheel, I am very actively looking at other material. We already have some more standard curricular materials, and we are also using the Khan Academy quite a bit - a really fantastic resource for basic mathematics.

Review of "Sage: Beginner's Guide" by Craig Finch

I was asked by the publisher to review "Sage: Beginner's Guide" by Craig Finch. They sent me a free copy; I have no other conflicts of interest. I am generally biased towards Sage itself, as an avid user and minor developer.

On Amazon you can browse the table of contents, which gives a pretty good idea of the strengths of the book, namely basic computation and plotting, numerical calculations, and data analysis. The focus was an excellent choice considering what is already available. The current free Sage Tutorial is oriented much more towards pure mathematicians. There is a Numerical Computing With Sage as part of the standard documentation, but at the moment its quite short and nowhere near as helpful as Finch's book.

I liked the style of the book a lot. There are many code examples that illustrate how to accomplish concrete tasks, along with good explanations of what they are doing. Many of these are things that are unfortunately far from obvious to a beginner (or even intermediate) Sage user. Despite using Sage heavily for the last five years, I learned some new things. The book is particularly strong in showing how to use Numpy, Scipy, and Matplotlib. Sage wraps a lot of the functionality of these projects, but if you want to do something that isn't included in the standard interfaces it can be quite mystifying.

Chapter 9, "Learning Advanced Python Programming", might have been a little ambitious. There's nothing wrong with it, but its too short to provide enough. Fortunately there are a lot of good books, some of them free, that cover Python programming in much more depth. I would have preferred some of this space and effort to be devoted to using Cython and the @interact command, which are covered very briefly in Chapter 10.

I teach several classes using Sage and I will definitely advertise this text as a useful optional supplement (I consider it a little too expensive to add on as a mandatory second text). It would be nice if some institutions considered using Sage instead of its commercial competitors such as Maple, Matlab, and Mathematica - you could probably give every student a copy of this book for the money saved from license fees!

The only thing I disliked about the book was the quality of the illustrations. Sage output that was in LaTeX was not typeset, but instead looks as if a PNG was copied from a screenshot. Some of the examples would have benefited from being in color. The quality of the plots is also somewhat poor. This is not too big a deal if one is following along with Sage, since you can reproduce the figures. None of them are bad enough to obscure the content.

Overall this is a very impressive and useful introduction to Sage that should help any beginning user a great deal.

Top Ten Talk Titles at the International Congress of Industrial and Applied Mathematics

10. The Most Likely Path to Systemic Failure

9. Exploding Rocks

8. Moving Mucus from the Outside In

7. Ducks in Array: Inferring Individual Rules from Collective Behaviour

6. The Nonlinear Dynamics of Jellyfish Swimming

5. The Neuromechanics of Insect Locomotion: How Cockroaches Run Fast and Stably Without Much Thought

4. Transformational Acoustics: Acoustic Cloaks, Carpets and Wormholes

3. A Semi-Implicit Blob Projection Method for Tiny Insect Flight

2. Mathematical Model for Contemplative Amoeboid Locomotion

1. Warping Peirce Quincuncial Panoramas

Ultrafilters

One of the challenges in learning mathematics is the vocabulary. Its hard not just because of the sheer number of words. Some words make an old word strangely precise, like "continuous". But many others are naming concepts which have no common equivalent, and each of these requires wholly new paths of thought.

Recently I was reading a book on the positivity of multivariate polynomials, and I had to recall what ultrafilters are. Even having learned that before, its a bit of a struggle to internalize. It did inspire me to make the following "ad":

A family of Mobius transformations

I was trying to sketch the behavior of a Mobius transformation in my complex analysis course today. Its hard to convey on the blackboard, so I tried making a video which shows a homotopy of the image of the unit disk, from the identity to (1+z)/(1-z) and back again.

Sci-fi history in a painting

This is pretty fantastic - Ward Shelley's "The History of Sci­ence Fiction":



(The crop above is just the tip of the iceberg.)

Plotting the zeta function

I lectured a tiny bit on the Riemann zeta function for the first time in my complex analysis course, which inspired me to make the following plot. Colors are the argument of the Riemann zeta function, brightness is proportional to magnitude. The default brightness map gives very low contrast, so I modified the magnitudes. To help with seeing the magnitudes a contour map with exponentially spaced contours is overlaid.

def xy_to_zeta_size(x,y):
    return abs(zeta(N(x+I*y)))
cvals = [e^i for i in srange(-7,1,.25)]
cp = contour_plot(xy_to_zeta_size,(-6,3),(-3,3),contours = cvals, fill=False, plot_points = 201)
rzeta(z) = zeta(z)/norm(zeta(z))^(.25)
rzf = fast_callable(rzeta,domain=CDF)
cparg = complex_plot(rzf,(-6,3),(-3,3))
show(cparg+cp,figsize=[18,12])

Youtube videos of n-body choreographies

For a presentation I'm doing tomorrow, I made some animations of n-body choreographies (n equal masses which share a common trajectory). I put these up on youtube since that seems to make more sense than having them mangled by Blogger or hosting them myself:

Twenty-one bodies
Eight bodies
Three bodies on a figure eight

The videos are done using Sage, the Tachyon raytracer, and ffmpeg. The orbit data is from Carlos Simo.

Here's a 2D picture of the twenty-body choreography.

Are you listening, pig?

I'd just like to share some recommendations that are all connected by the theme of visual and concrete computing. In the last few years I've become more and more convinced that (in the United States at least) our educational system is almost completely failing to provide adequate skills in computing - both the theory and practice of computer science. This is sad because I think there are many fun and interesting ways to learn about programming, logic, and algorithms. Here are some examples:

1. LOGO. When I was in elementary school I was exposed to Logo, a very visual platform for learning programming. If you have a child, make sure you check out an implementation and show your kid. No matter what computer you have, its almost certain that there are several free Logo programs for it.

2. Similar in spirit to Logo, but maybe more fun and less deep is the game Light-Bot at Armor Games. You can drag and drop visual program elements to write a program that instructs a robot to complete a variety of puzzles. Very fun, well done.

3. There are novels that glamorize hackers and computing to some extent, but nothing I've seen compares to the plot in Souls in the Great Machine by Sean McMullen. The plot is brilliant and complicated, but all you need to know is that they enslave all the mathematicians to create a giant human computer.

4. I was reminded of Eyes of the Calculor by this fantastic video on youtube. The creator has implemented a gigantic arithmetic unit within the virtual Minecraft environment. He mentions he was inspired by the book The Elements of Computing Systems: Building a Modern Computer from First Principles. I hadn't heard of that before but I will definitely take a look. If you watch the video you'll get the title of this post.

Torus knots in Sage with Tachyon and @parallel

After working a bit more on a patch improving the Tachyon raytracer in Sage, I made a movie of a torus knot. The code below was parallelized to use 20 cores at a time on one of the Sage Foundation's lovely Sun Fire X4450s. Frames were animated with ffmpeg. Blogger might degrade the video, so I recommend downloading the original here.

pk = 3
qk = 23
res = 180
trad = .5
lab = 'tk_' + str(pk) + '_' + str(qk)
def tknot(theta):
    return [(2 + trad*cos(qk*theta/pk))*cos(theta), \\
(2 + trad*cos(qk*theta/pk))*sin(theta), trad*sin(qk*theta/pk)]
@parallel(ncpus = 20)
def tknotter(anum):
    th = anum*2.0*pi/res
    T = Tachyon(xres = 800, yres = 600, camera_center = (4,4,2.5), \\
raydepth=12, antialiasing=2)
    loc = tknot(th)
    nloc = [abs(x)/norm(vector(loc)) for x in loc]
    T.texture('s1',color=nloc,specular=1)
    T.texture('p1',color=(.5,.5,.5))
    T.sphere(loc,.05,'s1')
    ci = 0
    for sth in srange(0,th,2*pi/res):
        e1 = tknot(sth)
        e2 = tknot(sth+2*pi/res)
        ts = [abs(x)/norm(vector(e1)) for x in e1]
        T.texture('t'+str(ci), color=ts)
        T.fcylinder(e1, e2, .025,'t'+str(ci))
        ci += 1
    T.plane((0,0,-4),(0,0,1),'p1')
    T.light((6,6,11),.1,(1,1,1))
    T.light((5.9,5.9,11),.1,(1,1,1))
    T.save(DATA+lab+'%03d.png'%anum)
    return 1

qin = range(pk*res)
s = list(tknotter(qin))

for i in range(40):
    r2 = os.system('cp '+DATA+lab+'%03d.png '%(pk*res-1) + DATA+lab+'%03d.png'%(pk*res+i))

anum = qin[-1] + 41
th = anum*2*pi/res
T = Tachyon(xres = 800, yres = 600, camera_center = (5,5,3))
T.save(DATA+lab+'%03d.png'%anum)

The animation was then done with the following code; if you want to do something similar you would have to use the appropriate path to your ffmpeg executable.

import subprocess
subprocess.call('/home/mhampton/ffmpeg/ffmpeg -qmax 2 -i ' + \\
DATA + 'v%3d.png ./t2vest.mp4', shell=True, stdout=file('/dev/null','w'),stderr=file('/dev/null','w'))

Flight in Curved Spaces

I was recently reminiscing about the movie Not Knot, and began thinking about recreating some of the hyperbolic scenes. But then today I stumbled upon a really nice application that does exactly what I was thinking about and more:

http://www.geometrygames.org/CurvedSpaces/index.html

Its a pretty small download, works on most platforms, very fun.