Zometool and 3d Geometry

We gave my youngest son a Zometool set for Christmas and we’ve all been having a great time with it.  He’s built some basic shapes following their guide as well as an impressive army of “Zome bots”  that have occupied the play room waiting to attack my feet without warning.  We’ve also had some fun playing with some 3D geometry.

One interesting problem to revisit was one from the NY Times Wordplay column in September:


(and also from Cut the Knot here:  http://www.cut-the-knot.org/Curriculum/Geometry/ThreePyramids.shtml )

The problem is easy enough to state, and I’ll borrow the wording from Cut the Knot – “A solid square-base pyramid, with all edges of unit length, and a solid tetrehedron also with all edges of unit length, are glued together by matching two triangular faces. How many faces does the resulting solid have?”

At first glance it seems that joining two solids along one congruent side will result in a new solid having a number of sides equal to two less than the sum of the side of the two original solids (since you’ve covered up two sides).  In this specific case that reasoning leads you to believe that the number of sides in the new solid will be 7.  However, there is a surprise . . . .

The second fun project was constructing a Excavated Dodecahedron, which I happened to see while flipping through an old geometry book last night.  That book – “The Penguin Dictionary of Curious and Interesting Geometry” by David Wells is really fun to play around with.  I’d recommend it to anyone interested in geometry.

The book has a nice picture of an Excavated Dodecahedron (which it calls a “deltahedron”), but the animated gif on the Wikipedia page is easier to understand:


My youngest son and I spent the morning yesterday building a small version of the shape from the blue pieces in his Zometool kit:


Definitely a fun morning, but the shape was so interesting that we thought it would be fun to try to make a larger version.  The larger version revealed a hidden (to us anyway!) icosahedron inside of the shape.   I’m not sure if that icosahedron will show up well in the pictures or not, but we tried to highlight it using small red pegs going from the center to each of the 12 verticies.  The edges of the icosahedron are the shorter blue pegs.  Here’s a top view and a side view:



Finally, here’s our short talk about the project and a few other fun 3d shapes:

All in all, the Zometool set has been a hit.

Numberphile’s “Pebbling the Chessboard” game and Mr. Honner’s square

Yesterday I saw an amazing math post on Twitter by Dan Anderson:

I love seeing math games with surprising outcomes that are simple to explain.  NumberPhile’s video on the problem is a masterpiece:

Solving the problem in the game involves summing a fairly complicated infinite series:

3/4 + 4/8 + 5/16 + 6/32 + 7/64 + . . . . .

The Numberphile video shows one way to sum that series, and eariler this year Patrick Honner published a nice visual proof showing how to sum (nearly) the same series:


Here is his beautiful picture that 1/4 + 2/8 + 3/16 + 4/32 + 5/64 +  . . . . = 1

I thought the game in the Numberphile video would be a super fun project to work through with kids.  I spent a little time last night trying to figure out how to talk about it with my boys and then spent the morning today going through it.

The first thing we talked about was Mr. Honner’s visual proof.  I wanted to do that at the beginning so that we wouldn’t get too distracted by the series when it came up during the game:

Finally, introducing the concept of invariants and connecting the game with Mr. Honner’s series:

The “Pebbling the Chessboard” game is such an amazingly fun and instructive exercise for kids.  Wish I would have known about this game back when I was teaching!

The Museum of Math, Zometool, and dimensions

About a year ago we visited Einstein’s Workshop (known as H3XL at the time) in Lexington, MA to do a “Zome” project.  We didn’t know what a Zome project was exactly, but it sounded fun so we drove up.  Over the course of about 3 hours several adults and kids build this really neat structure:


Today we visited the Museum of Math in NYC and were reintroduced to the Zometool kits.  We bought one of the small sets that were sold in the gift shop and my youngest opened it up the second we got home  (I’d actually already bought the boys a larger Zometool kit as a Christmas present, but since I’m a sucker for mathy stuff, an extra kit a few days early was fine with me!).

Before moving on, though, I have to say that the Museum of Math is an absolutely fantastic place to visit.  If you happen to find yourself in Manhattan, block out a few hours to spend there.  You’ll be glad you did.

In any case, when we got home I hopped on the computer to catch up with work and my youngest son went in the back room to play with the Zometool kit.  After about 30 minutes he came into the kitchen and announced that he’d built a hypercube:


How fun.  Seems like we’ll be using our Zometool kits for lots of exciting geometry.  The obvious lesson that the hypercube presented was a short talk with the boys about dimension:

and then a quick follow with this video we found on youtube that shows a computer generated image of a hypercube moving in space.

A super fun day thanks to Zometool and the Museum of Math!

Evelyn Lamb’s fun Torus tweet

A neat post on twitter today by Evelyn Lamb got me thinking about simple topology:

One of the people replying to her post pointed out the amazingly neat video here:


These posts  got me thinking about the fun square diagrams you can make for a torus, the Klein bottle, and the projective plane.  I decided to play around with some of those ideas with my kids tonight and it turned out to be super fun.  It is so great to have the  chance to introduce them to math that isn’t usually part of the pre-college curriculum.  Here’s what we did:

Since we were reminded of Vi Hart’s amazing “Wind and Mr. Ug” video in the middle of our video, here a quick link to it:

This video was actually the first Vi Hart video that I ever saw.  It is one of the most amazing pieces of math for the masses that I’ve ever seen.  Actually, I thought she deserved a MacArthur “Genius” grant after seeing it.  Even today it is mesmerizing.

There are also a couple of other fun videos on youtube that give a little insight into the Klein bottle and projective plane.  Riding a bike around a Klein bottle –

and neat visualization of the projective plane:

Though this wasn’t the field of math that I studied, I’ve always found it to be amazing.  It has a particularly fond memories for me because of one of my former students, Ali Gilmore.  She came to the University of Minnesota when she was in high school for a special math program.  She wasn’t all that interested in the multi-variable calculus we were studying so I gave her a copy of Massey’s Algebraic Topology to read.  She was hooked, and many years later is a post Doc at UCLA focusing on low dimensional topology and knot theory.

The Golden Ratio jumping the shark

Yesterday evening I saw a post on twitter about a new “Golden Ratio” video:

My first reaction while watching the video was to check if the upload date was April 1st.  Sadly (and surprisingly), it wasn’t.  Patrick Honner’s comment on twitter summed it  up best:  “I am so tired of the Cult of the Golden Ratio.”    Indeed.

For reasons I don’t completely understand, I found it hard to stop thinking about this video out last night.  It bothered me more than most contrived math stuff usually does.  Here’s my best attempt to explain why.

The so-called “Golden Ratio” is the number (1 + \sqrt{5} ) / 2, which is about 1.618.  My first introduction to this number actually had nothing to do with a classroom setting at all.  I was a kid – either in junior high or in high school, I don’t remember – just playing around on a calculator.  I’d enter any positive number, hit “1/x,” then add 1, then hit “1/x,” and repeat.  No matter what number I started with, I always eventually ended up at 1.618.  It was both strange and compelling.  What was special about this number?

Later, this time definitely in high school, my math teacher, Mr. Waterman, introduced several different fun computer models involving iteration.  We played with the Mandelbrot set, Julia sets, and also studied these interesting bifurcations:



From those bifurcation diagrams I learned that the convergence that I’d first seen on my calculator didn’t always happen.  Sometimes an iterative process converged to one point, other times there was some sort of “convergence” to more than one point, and other times there was absolutely no structure at all.   It was amazing to learn about this stuff, and quite possibly the first time I’d seem math that wasn’t either just plain vanilla high school textbook math or math contest math.
Around the same time, one of the legendary former students from our high school, Anita Barnes, came back from Wash. U in St. Louis and gave a lecture about recurrence relations.  She explained how you could take a simple recurrence relation such as the one for the Fibonacci numbers:

x_1 = 1, x_2 = 1, and x_{n+1} = x_n + x_{n - 1}

and write down an exact solution for each x_n.  It really blew me away.  After her lecture half of the board had a list of Fibonacci numbers and the other half showed a way to calculate each one of them using an amazingly complicated formula involving the square root of 5.   Right in the middle of that messy little formula was my old friend 1.618 . . . Wait, what??

As an aside, even today I see neat stuff about these type of iterated processes.  This past summer, for example, Steven Strogatz of Cornell posted the following fun question on twitter:

Anyway, back to high school and another fun topic we studied:  continued fractions.  Looking at one of the simplest continued fractions, the Fibonacci numbers and the golden ratio pop up again.  It is such a fun example (and not that hard) that I used it with my older son when we talked about the quadratic formula a few weeks ago:

The main point, of course, is the really interesting math, not the specific numbers that come out of that math.   Pretty sure that the words “golden ratio” do not appear in the video.

Thinking about that fun talk with my son got me thinking about another one we did a few months ago.  This one was on pentagons and tied together lots of seemingly unrelated topics that we’d run across over the years.  This talk showed how you could use quadratic equations to calculate that cos(72) = (-1 + Sqrt(5)) / 4.   The same math shows that cos(144) = (-1 – sqrt(5) ) /4 which I guess was yet another missed opportunity to mention the golden ratio:

Luckily, “Donald Duck in Math Magic Land” connected the golden ratio to pentagons, so I don’t have to feel too bad about my missed opportunity.

At the end of the day, I guess what bothered me about the new Khan video is more than it being another example of the  “Cult of the Golden Ratio.”  It is the missed opportunity that Feynman gets to in this interview:

There are many interesting and surprising places where the “Golden Ratio” pops up in mathematics.  I’ve mentioned a few above, and I’m sure there are 10x as many that I’ve failed to mention.   There are so many twists and turns you can take from these specific examples, too.    However, the fact that the radius of the Earth to the radius of the Moon happens to have some contrived relationship to the golden ratio just doesn’t seem that interesting to me.  Putting the focus on these odd coincidences rather than on other interesting math where the golden ratio also appears puts knowing the name of something ahead of actual learning, and that’s really a shame.


Neat math online and some fun number patterns for kids

I’ve mentioned several times before that I’m amazed by all of the great math that gets shared on Twitter.  One specific thing that I’ve done more of with my kids as a result of seeing all of the work that teachers are sharing is spend time talking about patterns.

Three examples showing some of the fun math that people are sharing on line are:

(1) Fawn Nguyen’s site, visualpatterns.org,  which I’ve mentioned a few times before,

(2) Patrick Honner’s visual proof that the series 1/4 + 2/8 + 3/16 + 4/32 + 5 / 64 + . . . . = 1.  That post is here:  http://mrhonner.com/archives/10239

(3) Evelyn Lamb’s post about Cycloids:  http://blogs.scientificamerican.com/roots-of-unity/2013/12/04/hypocycloids-make-you-happy/

With that background, I was talking through a problem from an old MOEMs math contest with my kids last night.  The thing that you needed to notice in the problem was that 1 + 2 + 3 + . . . + 14 + 15 = 120.  Not a particularly interesting fact all by itself, but as we were talking through the problem my younger son started asking me about several different pattern that he saw in the numbers.  That short conversation last night led to a slightly  more in depth conversation this morning:

I’ve not really known what to make of all of the math education articles flying around in the NYT (and various other places) during the last few weeks.  Sort of feels like we’ll be having the same conversations about math and science education for years and years and years.  What I do know for sure, though, is that I’m glad that I get to see all the cool math and science stuff people are sharing on Twitter and also glad that I get to use that material to help me have these fun conversations about math with my kids.

Follow up to Prime Numbers and Infinity

I got a nice question from an old high school classmate asking about yesterday’s blog post on prime numbers and infinity.  He asked me how you prove that the sum of the inverse squares was pi^2 / 6 and that the sum of the inverse primes was infinity.

Unfortunately both proofs are a little too advanced for kids.  The first requires understanding of the Taylor series for sin(x)  (at least the proof that I know), and the second requires a bit of number theory.  I wasn’t trying to present the ideas as something for the kids to prove, but rather just trying to show some fun facts that will come down the road (way down the road in this case).

However, there is one series that is pretty easy to understand that also has an infinite sum -> 1 + 1/2 + 1/3 + 1/4 + 1/5 + . . . .

After the question yesterday I toyed around with the idea of doing a second video, but decided to do it another time.  Then, as with yesterday’s blog, some funny coincidences happened today.  The first being that I started a new section on fractions with my younger son and the second one being a short proof of the divergence of the above sum that Dave Radcliffe put on twitter:

I’d never seen this clever little trick before and couldn’t wait to get home to show it to the boys:

I’ve been following a bunch of math folks on Twitter for about a year now and just can’t believe how many fun  examples I’ve found to share with the boys.  This little community on twitter has been a great resource for me.

Prime numbers and infinity

Yesterday I wrote about a really neat site showing patterns arising out of factoring integers:


A friend of ours on twitter, Federico Chialvo, made me aware of this site Friday on twitter.  As we watched the patterns play out yesterday, my younger son asked a nice little question –  how many prime numbers are there?  Fun question, of course, and I thought I’d use that as the topic for one of our  Family Math videos in the future.

Next up yesterday was a neat post from Patrick Honner about infinity.  Patrick is a high school math teacher in NY who seems to have an endless supply of interesting ideas about both math and teaching math:


His post reminded me of a curious comment in my college analysis book, “Principles of Mathematical Analysis” by Walter Rudin, or  sometimes “Baby” Rudin.  Problem 10 in Chapter 8 asks this gem of a question:

Prove that the sum of (1/p) diverges, where the sum extends over all primes.  (note – sorry I don’t know how to format math equations in the blog, yet).   Following the problem is an interesting comment from Rudin – “This shows that the primes form a fairly substantial subset of the positive integers.”    The comment struck me when I was in college, and the discussion in Patrick Honner’s post reminded me of it at a time when I already had the subject of why there are infinitely many primes in the back of my mind.

In any case, with that background, here’s what the boys and I talked through this morning:

Definitely a fun start to the day.

Sierpinski’s triangle and snap cubes.

Our math friend (and professional Ultimate player !!) Federico Chialvo sent us the following link on twitter last night:


Such a neat visual pattern.  The kids really liked it.

As we watched the different patterns fly by, my youngest son noticed that the pattern for the number 243 looked a lot like Sierpinski’s triangle.  That observation gave rise to our Family Math exercise this morning:

So glad I bought these snap cubes!

Interestingly, after we finished my youngest asked how our pattern today related to the one from the Chaos Game:


In the Chaos Game we had a little computer program that ran 10,000 iterations and produced a picture that looked like Sierpinski’s triangle.  He knew that 10,000 wasn’t a power of 3 and that didn’t reconcile with today’s exercise in his mind.  Good question.

On a side note, check out Federico’s fun blog post about the Collatz Conjecture:


The Collatz conjecture gives us such a fun opportunity to introduce kids to an usolved math problem.  We used it for one of our first Family Math videos.  So fun!

PISA and V.I. Arnold questions from twitter

Yesterday Fawn Ngyuen posted a link to some sample questions used by PISA.  Here’s the website:


I was pretty surprised by the Level 6 question for 15 year olds and the fact that only 2% of US students performed at or above Level 6.  When I got home from work yesterday, I gave the Level 6 sample question to my older son:

He had a little difficult working through it, but did manage to get to the right answer.   The biggest struggle was finding a way to check his answer.  That particular struggle made me happy since one of the things that we’ve been working on is slowing down and double checking his work.

Overall, I like the question, but I’m still struggling to understand how only about 1 in 50 US 15 year olds are able to work through this problem correctly.

While I was publishing the video least night, Steven Strogatz put a link to Tanya Khovanova’s blog on twitter:


This blog post discusses the following problem from V. I. Arnold’s book “Problems for Kids from 5 to 15″ :

(From an American standardized test) A hypotenuse of a right triangle is 10 inches, and the altitude having the hypotenuse as its base is 6 inches. Find the area of the triangle. American students solved this problem successfully for 10 years, by providing the “correct” answer: 30 inches squared. However, when Russian students from Moscow tried to solve it, none of them “succeeded”. Why?

According to Khovanova’s blog post, Arnold expected kids under age 10 to be able to solve this problem.  Wow.  I’d be pretty surprised if 1 in 50 10 year olds could solve this one, but it is a very interesting problem.  Since we were in the middle of studying quadratics anyway, I took a break from our Algebra book today and used this problem as a jumping off point to talk about the arithmetic mean / geometric mean inequality.  Definitely fun:


Definitely a fun and educational day for me on twitter yesterday.