Introducing Patrick Honner’s Pi Day idea in 4 dimensions

This will be the 2nd of probably 4 blog post in a series about exploring Patrick Honner’s Pi Day activity in 4 dimensions.

The first project (which includes the background) is here:

Playing with 4 dimensional shapes using Zometool

and Honner’s original post came to my attention via this tweet:

and the main motivation for this 4th dimensional exploration was how my son reacted to working through Honner’s activity:

The point of today’s exercise was to remind my son about Honner’s interesting approach to calculating “\pi” for various shapes. The main idea is that the radius of a shape is difficult to determine, but for simple 2-dimensional figures we should always be able to determine the area and circumference. If we want to use this idea we’ll need to find a way to define \pi in terms of area and circumference only:

Having found a new way of defining \pi for circles, we now try to find a similar approach for spheres:

Now we are nearly to 4 dimensions – we just need to find the right way to define \pi for a 4-dimensional sphere. It seems like this task shouldn’t be so hard, but there is a little surprise:

We actually talked about 4-dimensional spheres a few years ago:

Showing the kids about the area of a circle

4-Dimensional Spheres

I really doubt that either of the kids remembers these talks, but it is kind of fun to look back on them now ๐Ÿ™‚ Tomorrow we’ll look at what our new formula for \pi tells is about the zome shapes we looked at yesterday – namely the 5-cell, the Hypercube (aka the 8-cell), and the 16-cell:

  

Patrick Honner’s Pi day exercise in 4d

[This is a quick post I wrote while my younger son was at a little math enrichment activity. Sorry it looks like it was written in a hurry and not proof read . . . ]

Earlier today my older son and I played around with Patrick Honner’s Pi Day exercise:

That project is here:

Patrick Honner’s Pi Day Exercise

After we finished my son wondered about extending the exercise to 4 dimensions!

But, extending to 4 dimensions isn’t as easy as it seems. For one thing, the “volume” and “surface area” of a 4 dimensional sphere involve $\latex \pi^2$ not \pi:

“Volume” = (1/2) \pi^2 R^4

“Surface Area” = 2 \pi^2 R^3

So, we’ll modify Honner’s 3d \pi formula to be \pi^2 = (1/128) (Surface Area^4) / (Volume^3). That’ll give us a value for \pi^2 and then we can compute \pi.

So, I found the “volume” and “surface area” of the 4 dimensional regular Polytopes here:

Polytopes

Calculating "\pi" for the regular 4 dimensional polytopes gave values of approximately:

5-Cell: 8.63

8-Cell: 5.66

16-Cell: 4.62

24-Cell: 4.00

120-Cell: 3.38

600-Cell: 3.24

We’ve actually made a 3D version of the 120-cell with our Zometool set:

Screen Shot 2016-03-14 at 6.48.08 PM

 

That project is here, and maybe helps see that the shape is getting sort of spherical.

A Stellated 120-Cell made from our Zometool set

Another way to see some of these 4-dimensional shapes is to check out the game Hypernom:

Using Hypernom to get kids talking about math

Anyway, thanks for Patrick Honner for a fun Pi day!

Patrick Honner’s Pi Day Exercise

Saw this tweet from Patrick Honner yesterday:

I liked the activity for a lot of reasons – it was a great way to review a little geometry and arithmetic, a nice opportunity to discuss more 3D geometry, and finally a lucky coincidence with our project from yesterday talking about the truncated cube and the truncated dodecahedron:

What Will It Look Like?

Screen Shot 2016-03-13 at 10.16.07 AM

The unlucky part is that my younger son slept in because of daylight savings time – boo ๐Ÿ˜ฆ Oh well, here’s what I did with my older son:

Part 1: We first talked about the surface area and volume formulas for a sphere and then discussed how you could use the ratio in Honner’s post to define \pi in an unusual way.

Once we had the idea that $\pi = (1/36) \frac{SA^3}{V^2}$ we extended that idea to other 3 dimensional shapes. The first shape was a cube.

Also, sorry for the bad camera shot initially on this one – didn’t realize I’d cut off the bottom of our whiteboard until mid way through the video – oops.

After the Cube discussion we moved on to a Tetrahedron. Since I wasn’t looking to study the tetrahedron too much today, we pulled the volume formula from Wikipedia.

Next we took a look at the two shapes from yesterday and tried to guess which one was more like a sphere:

Finally, we did the calculations for these to shapes to find out what “\pi was for the truncated cube and the truncated dodecahedron according to Patrick Honner’s formula:

I left some of the calculations at the end of the video as an exercise for my son. After he finished the calculations he wanted to try to extend the idea to 4 dimensions – yes!! Sadly this idea proved to be a little harder than he expected, but maybe we’ll look at it tonight.

So, a great exercise – thanks to Patrick Honner for posting it!

A really nice thing that happened this week

Earlier in the week my phone “pinged” because of this tweet:

Clicking through I learned that Joel David Hampkins hard turned the Fold and Punch and Fold and Cut projects into an amazing activity at his daughter’s school:

Math for nine year olds: fold, punch and cut for symmetry!

Hardly any of the projects we do involve any planning – in fact, they’d probably require two extra levels of planning to get to “fly by the seat of our pants” status. So, it was cool to see a long and incredibly well though out write up of this project, and especially cool to see where a professional mathematician took the project ๐Ÿ™‚

I hope that people are able to take advantage of the wonderful write up by Hamkins – this is a tremendously fun project for kids!

A nice series problem for kids from Five Triangles

Back in 2013 we did a neat problem on Numberphile’s “Pebbling the Chessboard” video:

That video also reminded me of a neat “proof without words” that Patrick Honner had written about:

Our project is here:

Numberphile’s Pebbling the Chessboard game and Mr. Honner’s Square

and Patrick Honner’s blog post is here:

Proof Without Words: Two Dimensional Geometric Series

Tonight I saw a neat tweet from Five Triangles that reminded me of the prior project:

I thought it would be a fun one to try out with my older son, though I didn’t quite know how to introduce the problem. I started with a slightly easier series as a trial: 1/2 + 2/4 + 3/8 + 4 / 16 + . . .

Since things seemed to go pretty well with the first problem I decided to go ahead and try out the series posted by Five Triangles:

So, a neat problem for kids building off of a the “simple” infinite series 1 + 1/2 + 1/4 + . . . . As our project from 2013 shows, the more complicated versions can have interesting geometric interpretations, but I’ll leave those for another time. Tonight it was just fun to see some neat arithmetic with infinite series.

Using Dan Anderson’s Moirรฉ Patterns Program to talk about rotations

A tweet from Patrick Honner inspired one of our projects and a slew of programs illustrating the pattern in different ways:

We used two from Dan Anderson in our project:

That project is here:

Using NumberPhile’s Freaky Dot Patterns video with kids

In the project the boys struggled a little with understanding how many degrees each shape needed to be rotated to end up in the same position. I was a little caught off guard by the difficulty they were having, but afterwards thought that these Moirรฉ Pattern computer projects (and the Numberphile video, too) were a great way to introduce rotations to kids.

So, tonight we revisited the idea of rotation to try to make things a little clearer. First up was an equilateral triangle. In the less abstract setting of the dining room table, the kids were able to talk through the rotational ideas a little more easily:

Next we looked at a square. After the discussion my older son gave about the equilateral triangle, my younger son was able to give a nice description of what was going on with the square:

The last shape we looked at was a regular pentagon. My older son thought that the rotation angle + the interior angle of each shape (or at least a regular polygon) would add up to 180 degrees. I asked the kids to figure out why that was true for the pentagon:

Finally, as a special little treat / challenge, I showed the boys a strange situation where you have to rotate something 720 degrees around the center to get back where you started. This surprising rotational trick is something that I learned back in my abstract algebra class in college from Mike Artin.

So, a fun project and a nice little surprise – the Moirรฉ Patterns idea is a great way to introduce kids to rotations!

Numberphile’s “Freaky Dot Patterns” video

Saw this neat tweet from Patrick Honner earlier in the week:

The video itself will blow you away:

I shared it with Dan Anderson yesterday who made a couple of computer versions of patterns from the video:

Sorry the video quality isn’t so great, but it was fun talking through these patterns with the kids. We started with the square pattern:

Then we moved on to the triangle pattern. It was surprisingly difficult for the kids to understand how to describe the rotations, but eventually they figured it out. I think I’ll revisit a bit more about rotations for a Family Math Project this weekend.

The patterns in the Numberphile video make a great project to talk through with kids. I can’t wait to try a few more ideas from their video.

Amazing math from mathematicians to share with kids

About two years ago I saw this Numberphile interview with Ed Frenkel:

One of the ideas that Frenkel mentions in the interview is that professional mathematicians haven’t done a good job sharing math with the general public. Although I’m not really the kind of professional mathematician Frenkel was talking about, I took his words to heart and have been on the lookout for math to share – especially with kids.

It turns out that there are some fantastic ideas that are out there for kids to see. Some surprising fun I had sharing Larry Guth’s “no rectangles” problem with kids earlier this week (see below) made me want to share some of the ideas I’ve found in the last couple of years, so here are a few examples:

(1) One of the most incredible lectures that you’ll ever see is Terry Tao’s “Cosmic Distance Ladder” lecture at the Museum of Mathematics in New York City:

I used Tao’s video for three projects with my kids – but there are probably 20 math projects for kids you could get out of it.

Part 1 of using Terry Tao’s MoMath lecture to talk about math with kids – the Moon and the Earth

Part 2 of using Terry Tao’s MoMath lecture to talk about math with kids – Clocks and Mars

Terry Tao’s MoMath lecture part 3 – the speed of light and paralax

(2) The Museum of Math’s public lectures are a great source beyond Tao’s lecture.

Here’s a project based on Bryna Kra’s lecture:

Angry Birds and Snap Cubes – Using Bryna Kra’s MoMath public lecture to talk math with kids

Eric Demaine’s lecture was part of our Fold and Cut theorem project:

Fold and Cut part 3

and I can’t say enough good thinks about Laura Taalman’s work – she’s inspired dozens of our projects.ย  Just search for her name on the blog:

(3) and Speaking of Fold and Cut . . .

Katie Steckles and Numberphile put together an incredible video about the Fold and Cut theorem. I used the video this week for project with 2nd and 3rd graders at my younger son’s school earlier this week.ย  Steckles’s presentation is so incredible – this is the kind of math that really inspires kids:

We used it for three projects (including the Eric Demaine one above):

Our One Cut Project

The Fold and Cut Theorem is Awesome!

In prepping for the grades 2 and 3 projects I also totally coincidentally ran across a “fold and punch” exercise that is a great activity to try with kids before trying out fold and cut:

(4) Another great success with the 2nd and 3rd graders was Larry Guth’s “no rectangles” problem. I had a great time playing around with this problem with my kids, but nothing prepared me for how enthusiastic the kids in the two programs were about this problem.

Larry Guth’s “No Rectangles” problem

After the 3rd grade night, Patrick Honner sent me this picture that I used to wrap things up with the 2nd graders.

(5) The Surreal Numbers

I’d seen John Conway’s surreal numbers previously via an amazing Jim Propp blog post:

The Life of Games.

and I wanted to revisit them after finally reading Donald Knuth’s book:

Revisiting the Surreal Numbers

Infinity + 1 and other Surreal Numbers

Playing with the surreal numbers via checker stacks is an incredibly engaging way for kids to learn about mathematical thinking.

(6) Speaking of John Conway –

In the 2014 edition of the Best Writing in Mathematics Conway had an article about variations on the Collatz conjecture. It was a fascinating article that even gave us the idea to translate some of the math into music.

The Collatz Conjecture and John Conway’s “Amusical” variation

I’ve also talked with the boys about the standard version of the Collatz conjecture:

It is a great way to introduce kids to an unsolved problem in math while also sneaking in a little bit of arithmetic practice!

(7) Occasional contest math problems

I happened to run across another MoMath lecture yesterday – this one by Po-Shen Loh. He was talking about “Massive Numbers.” I thought maybe he’d be talking about the book “Really Big Numbers” by Richard Evan Schwartz:

A few projects for kids from Richard Evan Schwartz’s “Really Big Numbers”

or maybe Graham’s Number:

An attempt to explain Graham’s number to kids

The last 4 digits of Graham’s number

but instead he talked about a neat problem from the 2010 International Mathematics Olympiad:

His presentation is fascinating and I even talked through the first version of the problem with my younger son:

Another math contest-like problem I really enjoyed talking about with the kids was this one:

Show that any positive integer n has a (positive) multiple which has only the digits 1 and 0 when represented in base 10.

A challenging arithmetic / number theory problem

(8) Building off of popular books by mathematicians as well as public lectures

I was surprised at how much great math writing and speaking there has been for the general public in the last couple of years.

Jordan Ellenberg’s “How not to be Wrong” inspired several projects – probably my favorite was using his idea of “algebraic intimidation” to talk about the famous 1 + 2 + 3 + . . . = -1/12 video by Numberphile. :

Jordan Ellenberg’s Algebraic Intimidation

Jacob Lurie’s Breakthrough Prize public lecture inspired two projects about a year apart from each other:

 

Using Jacob Lurie’s Breakthrough Prize Lecture to Inspire Kids

Using Jacob Lurie’s Breakthrough Prize talk with kids

And, Ed Frenkel, who got me thinking about sharing advanced math with kids in the first place has inspired a few projects, too:

Fine Ed Frenkel – you convinced me

Ed Frenkel, the square root of 2, and i

and one of my all time favorites:

A list Ed Frenkel will love

(9) Finally, it would be impossible to write a post like this one without mentioning the work that Evelyn Lamb is doing writing math articles for the general public. I’ve lost count of how many projects she’s inspired, but it is probably well over 20. I’m especially grateful for her talk about topology which have generated really fun conversations with the boys. For example:

Using Evelyn Lamb’s Infinite Earring with kids

Evelyn Lamb’s fun torus tweet

and

Henry Segerman’s Flat Torus

which arose after Lamb pointed out this video:

 

So, I’m really happy that mathematicians are sharing so many amazing ideas. I think this is the sort of math promotion that Frenkel had in mind. Hopefully it continues for many years to come ๐Ÿ™‚

Patrick Honner’s Geometry Problem Part 2

This morning we worked on a project based on a geometry problem that Patrick Honner tweeted yesterday.ย  Here’s the tweet:

And here’s the project:

Patrick Honner’s Geometry Problem

During the day today I saw this tweet about the problem from Henri Picciotto:

I was not familiar with “turtle geometry” so thank goodness for the second half of the tweet! It looked like an interesting idea and I tried it out with both of the kids when they got home from school.

Wow!! What a great way to introduce some new ideas about angles to the boys by getting them to think about a problem that they’d already solved in a slightly different way.

I’m sorry these are longer than usual (especially the video with my younger son), but I just let the camera run until we got to the end.

Here’s my older son’s thoughts:

And here are my younger son’s thoughts:

So, a really great idea from Henri Picciotto on Twitter. I’m really happy that I had the chance to explore “turtle geometry” with both kids tonight ๐Ÿ™‚

Patrick Honner’s angle problem

Last night Patrick Honner posted this introductory geometry problem on Twitter:

I was looking for a problem to talk through with each of the kids this morning and this one fit the bill perfectly. It is a nice review problem for my older son and a nice way to (perhaps) introduce some new ideas to my younger son.

Also, the watching the two approaches to solving the problems is a nice illustration of the difference in approach that kids gain with a little experience (my older son is 2 years older than my younger son).

Here’s my older son’s approach to the problem. As this is more of a review problem for him, my goal is to have him carefully explain all of his steps:

Here’s my younger son’s approach. This is definitely not a review problem for him since he’s seen only a little bit of geometry. He knows a few basic ideas about angles and triangles, though, and he puts those ideas to use in his solution. The problem provided a nice opening for us to talk informally about angles and parallel lines, which was nice:

So, a luckily-timed tweet from Patrick Honner led to two nice conversations this morning. People sometimes look at me like I have four heads when I talk about all of the great math stuff that people share on Twitter, but here’s yet another example that led to some good conversations this morning!