# The associahedra

Replacing the associativity property of multiplication with
a *process*

yields a richer notion of associativity. It also means that multiplying more things requires more care about how they are associated.

## K_{4}

There are five ways to multiply 4 things, and they fit into a
neat shape! This is known as *K _{4}*, and it
shows how the different ways of associating 4 things are related;
there are two different ways around the pentagon!

## K_{5}

For associating 5 things, you need a more complicated shape known
as the associa*hedron*. It has six pentagonal faces, plus
three quadrilateral faces. Wikipedia has
more information about the associahedra.

Here is a cut-and-fold
net for the associahedron K_{5}! It's based on an
earlier version made
by Cheng and Lauda, but with vertices labeled as in
the diagrams above. Also it has tabs for easier assembly, and you
can download the tikz
source for it too!

This uses the tikz folding library, with some custom extensions
to foldable Platonic and Archimedian solids
by *nvcleemp*. Those additional solids were written in 2009,
and added to tikz in 2012, but the instructions on
the blog post were really useful when I wanted to figure out how to
make my own!

## Caveat

If you make this carefully, you will note that this net actually doesn't quite fold into a polyhedron -- the faces aren't quite flat! There is a lot of interest in actual polyhedral constructions of this shape, and there are lots of different ways one can do it. Eventually I'd like to have some other cut-and-fold nets for those too. If you make one, let me know!

One thing I haven't seen *anywhere* (or at least, not
anywhere that I can understand) is a model for K_{5} which
has the vertices as in the diagram above (and in this net), but with
actual polyhedral faces. If I knew coordinates for the positions of
these vertices in space, I could find the angles between them and
make an actual polyhedral version of the net above. If you know how
to do this, please let me know!

#### Update

I tried
to do this homework exercise myself, using regular pentagons and
computing the positions of the resulting vertices. But I discovered
that the four vertices which would form a quadrilateral between four
pentagons *don't lie in a plane*. This means that to solve the
problem I would have to use a non-regular pentagon of just the right
size... Maybe I'll try that later.

#### Update 2

There's nothing like being wrong on the internet to motivate a
person. Now
I've completed the homework exercise!
Lengthening two sides of each pentagon by a factor
of *1.1140106* results in quadrilaterals which are "flat"
measured to 7 significant digits. I call these "isoscales"
pentagons, because they still have a reflection symmetry. The
three other sides are rhombuses, and the polyhedron still has
3-fold rotational symmetry. Here's a picture! (There are some
extra lines because I didn't bother to remove duplicate vertices
or encode which ones should be connected. Click through the
homework link above for an interactive version.)

To encode this in the folding library, you'll have to learn a little more about it than I already know. Here are the vertices of the pentagons and quadrilaterals for you though!

pentagon:

```
[[0.000000000000000, 0.000000000000000],
[1.11401060000000, 0.000000000000000],
[1.30901699437495, 0.951056516295154],
[0.500000000000000, 1.53884176858763],
[-0.344248207313832, 1.05948704035187]]
```

rhombus:

```
[[0.00, 0.000],
[-0.797, 0.5463],
[0.797, 0.5463],
[0.00, 1.093]]
```

tags: fun-topics | papercraft | mathart