# How to build a Hexastix in 72 easy steps

I'm in a completely nondescript hotel room
that has been decorated in the theme of generic,
and I'm here because my wife has a physics conference in Vermont, USA
so I've come along for a bit of a trip
and as tradition dictates, I'm going to make a mathematical object here in the hotel room;
and this time, I'm going to make ... that
Now, I have no idea what that is yet
In fact, the person over there you can see showing it off
That ... yep there you are ... that is future Matt.
At the moment, I haven't got a plan
My only "proto-plan", is that this time I'm not going to use the things lying around the hotel room
to make my mathematical sculpture,
because across the road, is a Staples;
so it is a massive warehouse full of office stationary
So I'm going to go over to staples, find whatever I can that I think I can make something mathematical out of,
bring it back here and then build ... that.
So hopefully it's nice and impressive, whatever it is. I'm about to who you how to make it.
*clap* Let's do it!
[Stand-up Maths theme]
This is how convenient I hope it's going to be
That's the hotel where the conference is, and we're staying,
and then, over there, is a sign that says "Staples"
[Stand-up Maths theme]
Alright!
[Stand-up Maths theme]
OK, so I'm here in Staples, and it is the stationary wonderland I had hoped for
so I'm going to cruise around, and see what i can make a maths sculpture out of
[Stand up Maths theme]
I could get ... I mean, imagine ... what I could make with that
Oh, that is officially the world's ... that is insane
That's a Post-It note
OK, this ... let's ... hang on ... if I ... if I put this down
Ah! So this ... they actually come in a box of 72
So, I remember George Hart did a sculpture called "72 pencils", I think
They actually come in boxes of 72. That's brilliant.
These ones are \$8.49, these are \$13.49.
I'm going for the \$8.49
and some spares
And, elastic bands
Can I just get those, thanks
[Stand-up Maths theme]
Alright, how hard can this be,
putting 72 pencils into the one sculpture
So a single pencil is a hexagon shape, but if you get 7 of them together with one in the middle
and then 6 around the outside, you can make this hexagonal stick
We're going to basically build four of these, but all crossing in the centres.
Yeah we're going to look at the way this breaks down
You've got two on the bottom, and three across the centre, and then two at the top
And we're going to use that to our advantage
I'm going to split them 2-3-2
So I'm going to wedge two in the bottom ... here
two on the next one up ... there
OK, next step up, I'll split the original 7 to go 2-3-2
By two pairs of two that I've now elastic band as well to hold them in place
I'm going to put another two pairs the other way: straight down
and, ok ... so that's two there, three there, and then those two up top
OK, I've never done this before,
but so far, so good
Now for the difficult bit, I've got three sets all together, I'm going to put the fourth set through,
and if you stick with our original set,
you can see I'll split it this way; it goes 2-3-2 with these sticks
and I'll split it this way; it goes 2-3-2 with these sticks
These have to split at the remaining third direction to go 2-3-2;
one way to work out where that is is if you put it so the big original one, all 7, lines up with you when you've got these on either side.
If you flatten that out, you'll see a whole bunch of holes kind of appear,
and so what I can do now, is start sticking these in here
and so now, when I lift, (is that four; yep), when I lift it up,
There we go, there is my new direction
I'm actually getting a little over-excited, and I'm going to put 5 in there.
Hopefully that's going to ... that's going to come back to haunt me, but I'm doing it anyway
I'm a maverick when it comes to pencil structures
So they all cross roughly in the middle,
and their still our original seven,
and this is the last one I put in; I did five
and then these are still four
I've now got to build on the basic structure we've got going.
So I've got that set of 7 going in every direction
Ok, I'm sure everyone else who's ever made one of these sculptures, who isn't me
Was a lot more deliberate and precise with their arrangement of the pencils.
I'm just kind of looking at and going: "Well, it could make that a bit more 'hexagony'".
So here you look at that, and you're like "well obviously two more on this side is going to round up that hexagon"
So I'm going to chuck one roughly ... there?
That looks like it lines up
And then I'm going to put one roughly ... over here
So now you're like "yeah yeah, that looks a lot more like a hexagon; that's moving in the right direction"
Ok, 7 ... 7 ... 7 ... 7.
Every single one of these has 7 pencils in a hexagon shape, and they all cross in the centre there.
All we need to do now is to gradually expand them out in unison,
and every individual step, the elastic bands will keep it stable, until we end up with this, just much more spaced out
How difficult can that be?
[Stand-up Maths theme]
I started with something with that, where you can see the complete hexagon. Look, I've tried to move as many out as I could
And I ended up with THAT'S NOT BAD!
It's kind of ... the centre one is still where it was, and the rest I've moved out except ...
you ... I mean, what is going on over? ...
I mean, that is not ... that's not ... *breathes in heavily*
Let's just do another side, and see what happens
This is kind of fun, for a maths sculpture...
That these things don't need to be incredibly precise.
The fact that you can just kind of keep grabbing pencils and moving them into whatever looks like the most obvious spot
I quite enjoy that...
So far I've not needed any instructions, and ...
uh, that one's really stuck
Come on...
[Stand-up maths theme]
Alright! That ... THAT, there you are!
So, now everything's expanded out by ... one step!
So you can see now ... for some, but not all, of the pencils round the outside
there are Staples spots one further out
But as you do some on one, and then some on another one, then you'll reinforce the other ones;
Every pencil goes out one, each side gets 6 more pencils, and we're pretty much done
Sometimes when you take it out, before you have a chance to put it back in again, everything just shifts slightly, and you can lose track of the path
So what I've started doing now is eyeballing: "right where I think it's one out, it's going to go THERE. I can see the hole. I can see the hole"
"Taking out the pencil ... I can see the hole ... I'm shoving it in."
I have no idea if there are people in the hotel room next to me listening in to this"
[Stand-up Maths theme]
Ok, here's a problem. I'm currently putting in my last pencil.
So, if I can ... come on, fit through there ... ok, right.
So once that's in ... erm, there's still gaps.
So ... the problem is that I haven't got enough pencils.
You see, the original pack came with exactly 72.
Which is exactly what you need for four lots of eighteen.
So these centre ones, which have been our guiding reference points the whole way, I know, we have all become very emotionally attached to them,
now they have to all come out; and they're used to complete the very last bit
Now the last one, alright!
So that one comes out of there ... goes in there
and that's it! I'm done! Check it out!
And there it is! The finished mathematical ... pencil ... sculpture
So there are a few ways you can think about what shape this actually is
There's the way that we built it; so it is four intersecting hexagonal tubes
Or ... you could try and work out what the inside void is
So that central space ... what shape is that?
And I'm going to give it away. It is a rhombic dodecahedron.
The greatest of all the dodecahedra.
And so, I've actually seen pictures online where people have taken these, and cut off all the bits of the pencils that stick out,
and you end up with a rhombic dodecahedron right in the centre.
And if you want to look up these online it's called a "hexastix"
I actually first came across it as a mathematical sculpture George Hart did called "72 pencils"
You can see what he did there.
And I mean it as an absolute compliment that this is possibly one of the less impressive mathematical sculptures George Hart has done,
because he has done some absolutely incredible stuff. Do check out George Heart's work
If you want to make one of your own one of these,
I recommend the instructions done by a guy called Alejandro Erickson
So Alejandro did the early steps that I used to build it myself.
He came up with the "putting 7 pencils in one direction, and the splitting them 2-3-2, and then building up like that".
Only he then puts in far more detail into his instructions than I did in this video.
So if I was a bit vague, go check out his instructions, there will be a link below
Actually if I wasn't clear enough about my method, it was to start the same way Alejandro did, and then just kind of wing it.
So see if you can work it out as you go along.
I found that surprisingly satisfying
If I wasn't sufficiently clear, what I realised by the end worked the best, was for each of the directions, when you want to go up a step,
you move all the corners out one; and that's reasonably easy to do if you've still got the centre marked;
provided there is a rigid spot to put it. If there is not a rigid spot, do another side for a while, come back later, it'll be fine.
Once you've moved the corners out, put in the extra pencil into all of the edges, it should be clear enough now
and then move out the old edges to the new position, and that will give you this reasonably quickly.
There are actually two reasons why you should try making your own one of these
1. It is surprisingly satisfying.
It was really good fun putting ... slightly frustrating, but good fun
And the fact that all the wholes kind of opened up just when you needed them,
and the shape of the the pencil's a hexagon
helped put this whole thing together.
It's really ... I highly recommend building it, just because the procedure is so deeply enjoyable.
And on top of that, it's a really nice finished product
What I find particularly interesting is that it's solid
It's a proper ... I mean ... it's a rigid thing, and it's not fragile
but yet it kind of is fragile, because there is no way you could build it in this finished form
Because it takes too many pencils, all working together to hold it together, to be this strong
The only way you can build it is by starting small, and then growing up.
Now, I'm actually not planning on taking this all the way back to England with me.
I will keep it with me on my travels around the US.
If you would like to meet this hexastix,
I will have it at the talk I'm doing at the National Museum of Mathematics (MoMath) in New York on Sunday, 23rd of October 2016
So if you're watching this video shortly after I've uploaded it, it's this coming Sunday
And so if you want to come along to my talk at MoMath I'll put a link in the description, you can see this,
And after that I have no intention of taking it home with me
So I'm going to give it away to someone at the MoMath show.
So if you email me or contact me somehow before the show, and say
"Hey, I'm coming to your show, I would give the hexastix a wonderful home"
If you can provide a nice, safe life for it, and if I get a particularly good description of a place this could go and live out it's retirement
I would give it to that person.
If I get loads of valid requests, I will pick one at random on the night
through whatever randomised process I deem appropriate.
And, if you can't make any of my live shows, I have another gift for everyone
so the music you're currently hearing, [Stand-up Maths theme fades in]
That is my Stand-up Maths theme song.
People keep saying, "Oh, where did it come from?" and "Where can we get it from?"
So it came from the brain of Howard Carter
and now you can get it from somewhere!
Howard and I have turned it into a proper song.