Patrick Honner’s Pi Day exercise in 4d day 4: The Hyperdiamond

This is the 4th (of 5) in series of 4 dimensional explorations inspired by Patrick Honner’s Pi Day exercise:

The first three parts in the series are here:

Playing with 4 dimensional shapes using Zometool

Introducing Patrick Honner’s Pi day idea in 4 dimensions

Patrick Honner’s Pi day exercise in 4d: part 3

Also, since I didn’t want to really dive into the “surface volume” and “hyper volume” calculations, this website was critical for today’s project:

Regular Convex Four-Dimensional Polytopes

Today we looked at the 24 cell – aka the Hyperdiamond. We’ve looked at the shape previously after seeing this great video from Matt Parker:

Using Matt Parker’s Platonic Solid video with kids

Unfortunately my older son didn’t remember the previous exercise – ha! Oh well, luckily we started with a quick review of the hyperdiamond and the rhombic dodecahedron:

After that quick talk about the 24-cell we returned to our whiteboard to talk about the value of “$\latex \pi$” for this shape. The “surface volume” and the “hyper volume” for the 24-cell turn out to be fairly simple numbers, and that but of luck gives us an easy value of “\pi for this shape!

So, one last project tomorrow. We’ll look at the 120-cell and the 600-cell. Can’t wait!

Patrick Honner’s Pi day exercise in 4d – Part 3

We are spending the week working through Patrick Honner’s Pi day exercise in 4 dimensions. The first two parts of our project are

Playing with 4 dimensional shapes using Zometool

Introducing Patrick HOnner’s Pi day idea in 4 dimensions

Also, since I didn’t want to really dive into the “surface volume” and “hyper volume” calculations, this website was critical for today’s project:

Regular Convex Four-Dimensional Polytopes

The main idea today is to calculate “\pi” for the first three 4-dimensional platonic solids – the 5-cell, the 8-cell (aka the hypercube), and the 16-cell. A fun twist is that the 5-cell and the 16-cell have some 3d projections that are quite similar, but give quite different values for “\pi

So, we started with a super quick review of the 4d formula for \pi^2 and then took a look at the 5-cell. Although we didn’t go through the calculation, I liked my son’s guess that the hyper-volume of a 4-dimensional pyramid would be given by (1/4) * volume of base * height.

Next we looked at the 8-cell, or hypercube. Luckily this shape has really easy “surface volumes” and “hyper-volumes.” That allowed us to calculate “\pi^2 exactly without too much difficulty – plus we got a little bit of exponent review πŸ™‚

The last shape we looked at today was the 16-cell. This is the most difficult shape to understand, and understanding it is made even more confusing because we have a couple different 3-dimensional projects and they don’t look anything like each other! Also, as noted above, one of them looks a lot like the 5-cell.

It was fun to think about the “spherical-ness” of this shape prior to doing the calculation.

We are really having a lot of fun with this project. Tomorrow we’ll probably focus on the hyperdiamond because it is such a cool shape. Then we’ll talk about the 120-cell and the 600-cell for the grand finale πŸ™‚

A challenge for professional mathematicians

[March 24th, 2016 update – I’m going to link some articles at the end of the blog as I see them. There are two from today. I’m really happy that people are writing about this!]

I saw this article on gravity waves via a Steven Strogatz tweet this morning:

Seeing the article reminded me of the interview that Numberphile did with Ed Frenkel a while back – in particular, the part from roughly 5:00 to 7:00 when Frenkel discuses the need for mathematicians to do better at sharing their ideas with the public:

Frenkel’s point is that even though the ideas in fields such as biology and physics are just as complicated as the ideas in math, these other areas of science are much better at communicating with the public than mathematics is.

I was reminded of Frenkel’s point again this morning when I learned that earlier this month Maryna S. Viazovska solved the 8-dimensional sphere packing problem. Viazovska’s paper on arxiv.org is here:

The sphere packing problem in dimension 8

Maybe I’m a little biased – especially right now because I’ve been spending this week playing around with 4-dimensional shapes with my kids . . .

img_0947

but I think that the sphere packing problem (i) is something that can be explained to the public (it certainly seems less complicated than gravity waves) and (ii) is something that the public would find to be interesting. There’s not been much of any coverage of Viazovska’s result, though. Here’s what I found doing a simple Google news search:

Screen Shot 2016-03-22 at 2.24.44 PM

So, it sure seems this new result is something that would be great to share with the general public. There are, of course, many different directions an article could go – just off the top of my head:

(A) Jordan Ellenberg does a great job explaining the sphere packing problem and the connection to things like the Leech lattice and Hamming codes in How not to be Wrong,

(B) John Cook and Keith Devlin both have recent blog post with connections to higher dimensional spheres / cubes:

The empty middle: why no one is average by John Cook

Theorem: You are Exceptional by Keith Devlin

(C) Two years ago, Steven Strogatz shared this wonderful paper on N-dimensional spheres:

(D) The 2-dimensional problem of circle packing is something anyone can understand and is pretty fun to play with – here’s an old project I did with the boys using disc golf discs, for example:

 

Screen Shot 2016-03-22 at 2.51.32 PM

Sphere packing (well . . . circle packing)

Also, a version of the circle packing problem was in Jim Propp’s most recent blog post about mathematical thinking:

Believe it, then don’t: Toward a Pedagogy of Discomfort

So – come on professional mathematicians!! – here’s a great opportunity to promote a neat result and bring some really cool math to the public’s attention. Don’t let the physics crowd have all the fun!

A few articles that I’ve seen:

On Gil Kalai’s blog:

A Breakthrough by Maryna Viazovska lead to the long awaited solutions for the densest packing problem in dimensions 8 and 24

Kalai’s blog post also led to a question on Quora:

Why is the solution in dimension 8 such a breakthrough?

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 πŸ™‚ 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:

  

Playing with 4 dimensional shapes using Zometool

We had a 2 hour school delay due to snow this morning – perfect time to . . . .

My goal for this week with the boys is to work thorough Patrick Honner’s Pi day exercise in 4 dimensions. I already did the 3d version with my older son:

Patrick Honner’s Pi Day Exercise

and worked through the 4d version on my own last Monday:

Patrick Honner’s Pi Day Exercise in 4d

The path to getting the boys to understand the 4d project goes something like this:

(1) The 4d versions of volume and surface area for a sphere involve \pi^2 rather than \pi, so we have to figure out how to modify the formulas in Patrick Honner’s 3d exercise to take this 4d feature into account.

(2) We have to understand some of the simple 4d shapes – I’m looking at the 6 4d platonic solids. Our Zometool set helps us understand those shapes.

By luck, we’ve already done a project on the one Platonic solid that exists in 4d that doesn’t exist in 3d:

 

Screen Shot 2016-03-21 at 8.25.39 AM

Using Matt Parker’s Platonic Solid Video with Kids

(3) Finally, we have to come to understand the “volume” and “surface area” formulas in 4 dimensions. That is the part that I worked through last Monday after finding this awesome website:

Polytopes

Today’s project was mainly about point (2) above.

We started by looking at the 5-cell:

The the 8-cell, or Hypercube:

And finally the 16-cell which is a little more difficult to build and visualize. Luckily we found this helpful website:

The 16-Cell and some relatives

It was nice to have the surprise snow day today to play with these shapes!

The last digits of triples of consecutive primes

I found last week’s news about patterns in last digits of consecutive primes to be really interesting. Here’s Evelyn Lamb’s piece about the new paper:

Peculiar Pattern Found in “Random” Prime Numbers

Although the main results in the paper are way over my head, I thought it would be fun to try to understand the results a bit more. I decided to look at the last digits (in base 10) of triples of consecutive primes. Mathematica makes this task pretty simple since there is a function Prime[i] which tells you the i^{th} prime number.

I was surprised by some of the patterns.

Looking at the primes from 11 until the prime number 100,000,000 (that’s 2,038,074,743 if you are interested!) I found these results in the last digits of triples of consecutive primes:

(1,1,1) occurs 752,992 times, and
(9,9,9) occurs 752,902 times.

(3,3,3) occurs 737,172 times, and
(7,7,7) occurs 735,435 times.

and for the most frequently occurring patterns:

(9,1,7) occurs 2,429,154 times, and
(3,9,1) occurs 2,429,802 times.

(the full list is at the bottom of the page)

I was amazed at how close the counts were, so I ran the same code again, but this time went up to prime number 1,000,000,000 (which is 22,801,763,489) and found similar results:

(1,1,1) occurs 8,143,311 times, and
(9,9,9) occurs 8,139,168 times.

(3,3,3) occurs 8,013,553 times, and
(7,7,7) occurs 8,006,387 times.

and, again, for the most frequently occurring patterns:

(9,1,7) occurs 23,285,442 times, and
(3,9,1) occurs 23,285,599 times.

I’m surprised both by (i) how different counts for various triples are from either other, and by (ii) how similar certain pairs of counts are to each other.

Can’t wait to see what results end up following from last week’s paper. It certainly has been a fun couple of years for prime numbers!

Here’s my count for the triples occurring between 11 and prime number 100,000,000:

(1 , 1 , 1) = 752,991

(1 , 1 , 3) = 1,473,963

(1 , 1 , 7) = 1,338,238

(1 , 1 , 9) = 1,057,849

(1 , 3 , 1) = 1,605,399

(1 , 3 , 3) = 1,424,130

(1 , 3 , 7) = 2,139,275

(1 , 3 , 9) = 2,260,634

(1 , 7 , 1) = 1,829,358

(1 , 7 , 3) = 2,085,911

(1 , 7 , 7) = 1,330,194

(1 , 7 , 9) = 2,259,149

(1 , 9 , 1) = 1,606,073

(1 , 9 , 3) = 1,474,440

(1 , 9 , 7) = 1,304,421

(1 , 9 , 9) = 1,057,410

(3 , 1 , 1) = 1,152,478

(3 , 1 , 3) = 1,637,164

(3 , 1 , 7) = 1,916,059

(3 , 1 , 9) = 1,305,280

(3 , 3 , 1) = 1,121,750

(3 , 3 , 3) = 737,172

(3 , 3 , 7) = 1,253,949

(3 , 3 , 9) = 1,329,690

(3 , 7 , 1) = 1,830,430

(3 , 7 , 3) = 1,817,402

(3 , 7 , 7) = 1,253,361

(3 , 7 , 9) = 2,142,502

(3 , 9 , 1) = 2,429,892

(3 , 9 , 3) = 1,819,555

(3 , 9 , 7) = 1,916,051

(3 , 9 , 9) = 1,337,398

(7 , 1 , 1) = 1,123,346

(7 , 1 , 3) = 1,954,359

(7 , 1 , 7) = 1,821,161

(7 , 1 , 9) = 1,475,115

(7 , 3 , 1) = 1,660,559

(7 , 3 , 3) = 1,190,879

(7 , 3 , 7) = 1,818,700

(7 , 3 , 9) = 2,085,057

(7 , 7 , 1) = 1,089,299

(7 , 7 , 3) = 1,191,047

(7 , 7 , 7) = 735,435

(7 , 7 , 9) = 1,423,574

(7 , 9 , 1) = 2,362,556

(7 , 9 , 3) = 1,955,794

(7 , 9 , 7) = 1,638,318

(7 , 9 , 9) = 1,475,202

(9 , 1 , 1) = 1,594,226

(9 , 1 , 3) = 2,363,951

(9 , 1 , 7) = 2,429,154

(9 , 1 , 9) = 1,604,100

(9 , 3 , 1) = 1,623,274

(9 , 3 , 3) = 1,090,380

(9 , 3 , 7) = 1,831,771

(9 , 3 , 9) = 1,827,515

(9 , 7 , 1) = 1,624,894

(9 , 7 , 3) = 1,660,835

(9 , 7 , 7) = 1,120,365

(9 , 7 , 9) = 1,606,645

(9 , 9 , 1) = 1,592,910

(9 , 9 , 3) = 1,123,151

(9 , 9 , 7) = 1,153,949

(9 , 9 , 9) = 752,906

Another nice counting problem from Jim Propp

Saw another great post from Jim Propp yesterday via this tweet from Steven Strogatz:

Here’s a direct link to the blog post:

Believe It, Then Don’t: Toward a Pedagogy of Discomfort

Part of the post discusses a problem about inviting people to a party:

Talking through some of the ideas in this problem seemed like a great exercise to go through with the boys today – and it didn’t disappoint!

After watching Propp’s video, I started today’s project by simply having the boys discuss what they saw. Right away you can see that the problem is a great way to get kids to talk about math. My older son immediately wonders if we can find an optimal strategy, though both kids have a hard time explaining / defining what an optimal strategy is (which is fine, not exactly a super concept for kids . . . .).

Also, my younger son has an awesome answer to why the method in Propp’s video is called the “greedy algorithm” πŸ™‚

[note: Sorry for the blurry start to this video (it only lasts about 10 seconds) – I didn’t notice until it got published 😦 ]

The first new exploration that the kids wanted to try was a slightly modification to the process in the original video – start by inviting person 2 rather than person 1. The discussion here is mostly a repeat of the discussion in the first video because the new process is so similar to the original one. The main difference is that it my older son (6th grade) speaking rather than my younger son (who is in 4th grade).

At the end of the video both boys are starting to think that 9 is the largest number of people you can invite to the party. They offer a few ideas to justify this answer, though my younger son is less certain than my older son.

For the third part the kids decided to modify their counting algorithm by changing who they invited in the middle of the invitation process. This part shows, I think, what you can learn from watching kids do math – not in a million years would I have thought this particular change to the invitation process is something that anyone would have considered.

At the end of the video the kids are beginning to think about how modifying the greedy algorithm changes the number of people who are invited. Then my younger son has a really interesting idea – let’s “invert” the algorithm!

To study the “inverted” algorithm we began by looking at the original set of people who were invited / not invited in Propp’s video. We then tried to invite the non-invited people. I really like how this idea played out – and I love how even simply describing this strategy was a challenge for my younger son.

Then . . . after all this talk about the greedy algorithm-related strategies, all of a sudden my older son suggests a new strategy at the end of the video

The project couldn’t have ended on a higher note. At the end of the last video my older son suggested a new strategy that was not related to the previous greedy algorithm strategies. We worked through that strategy and discussed the various ways it was different from what we’d tried before.

The discussion here led to my younger son making a fantastic observation about how the various numbers got crossed out in the new algorithm! Yes!!

I”m really happy with how this project went today. The problem is accessible to kids (only simple arithmetic is required!) and leads to several nice discussions on pretty advanced topics like algorithms and optimization.

It reminded me a lot of our project on Larry Guth’s “No Rectangles” problem:

Larry Guth’s “No Rectangles” problem

I used the “no rectangles” problem for both the 2nd and 3rd grade Family Math nights at my younger son’s elementary school and the kids absolutely loved it. I think I’ll have to add the problem from today to next year’s program. It is so fun to find math that is interesting to mathematicians that you can share with kids!

Looking at the new discovery about primes with kids

Earlier in the week Robert J. Lemke Oliver and Kannan Soundararajan of Stanford announced a totally surprising discovery about prime numbers. Evelyn Lamb has a fantastic article about the paper here:

Peculiar Pattern Found in “Random” Prime Numbers

The new paper suggests that the prime numbers are not distributed quite as randomly as mathematicians had previously expected. In particular, the last digit of consecutive prime numbers has a distribution that is different from what you’d expect if the distribution of primes was random.

I thought this would be a fun result to discuss with kids. One surprising thing about this result is that it is pretty easy to understand. In fact, you can double check the result on a computer really quickly.

Before jumping in to the result about primes, though, I wanted to spend a few minutes talking about probability and probability distributions. This part of the project turned out to have some extra fun when my older son asked a neat question about dice.

Here’s the introduction to today’s project and my son’s question – what is the probability of seeing at least one 5 when you roll two dice?

With the complimentary counting problem behind us, we moved on to talk a little bit about the difference between probability and probability distributions. Once again we had a little detour following a statement that my older son made. I was happy to have these extra little conversations about probability this morning:

The next part of the project involved talking about prime numbers. We talked a little bit about how mathemticians viewed the primes. Number theory and prime numbers are not my field – hopefully the details in this part are right. A great (and accessible) read about prime numbers and randomness can be found in Jordan Ellenberg’s How Not to be Wrong.

It was really fun to hear what the kids had to say about the last digit of consecutive prime numbers here. This problem is a great way to get kids thinking about math.

Now we moved to the computer and used a little Mathematica program to study what the distribution of the last digit of consecutive prime numbers looked like. We chose to look at prime numbers that ended in 1. It was neat to see the results. Loved hearing what my younger son observed: “it seems like there’s a lot less primes ending in 1”:

We wrapped up by looping through the first 25,000,000 primes and compared our results to the results that were given in Evelyn Lamb’s write up from above. Our results were really close – yay!

So, I think that talking about this new discovery makes for a really fun project for kids. I’m sure that our project could be improved quite a bit by any mathematician who had a good understanding of number theory (since my understanding of that subject is essentially zero!), but even the high level walk through that we did today was fun. It is pretty amazing to find a new discovery about prime numbers that can be understood by kids!

Trying out 4 dimensional bubbles

At the end of the project with my younger son this morning he remembered that we’d see some of the 4 dimensional shapes we were looking at in our Zome Bubble project. He went on to wonder if we dipped our 4 dimensional shapes in the bubble solution would we get a 5 dimensional shape. Well – We had to try that!

First, though, we looked at what happened when you dipped a cube and tetrahedron in bubble solution:

Next we tried the 4 dimensional shapes – what happens when you dip the zome versions of the 5-cell and the Hypercube into the bubble solution?

Ahead of the dipping, my younger son had this thought:

“I think we are going to see a 5 dimensional shape”

Here’s what happened:

I’m really loving just playing around with the 4 dimensional shapes with the boys. Soon we’ll move on to looking at the 4d version of Patrick Honner’s Pi Day project – can’t wait for that!

Finally, here’s the project from this morning that led to my younger son wondering about bubbles:

Sharing 4d shapes with kids

Sharing 4d shapes with kids

Earlier in the week – on March 14th, in fact! – we were playing around with Patrick Honner’s Pi Day Project:

Here’s a link to that project:

Patrick Honner’s Pi Day Exercise

When we finished, my older son wondered what a similar exercise would look like in the 4th dimension:

I played with it a little, and the idea was a bit more difficult than I expected. Here’s what I was able to write down while my younger son was at a little enrichment math program he does on Monday evenings:

Patrick Honner’s Pi Day Exercise in 4d

We have also – thanks to a great video from Matt Parker – looked a bit a some fairly complicated 4 dimensional shapes before:

Using Matt Parker’s Platonic Solid Video with Kids

So, with that all as background, last night and this morning I spent some time with each of the kids looking a the “smallest” 4d platonic solid – the 5-cell. It is fascinating to hear kids describe the shape.

Here’s my older son:

and here’s my younger son:

I love that my younger son remembered that we’d seen these shapes in our soap bubble project:

Zometool and Minimal Surfaces

It even turned out that our friend Paula drew the 5-cell after seeing our project !!