I asked [this question two months ago on MSE][1], where it earned the rare
[*Tumbleweed*][2] badge for garnering zero votes, zero answers, and 25 views over 61 days.
Perhaps justifiably so! Here I repeat it with slight improvements.
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Let $P$ be a polyhedron, all of whose vertices are at points of $\mathbb{Z}^3$,
all of whose edges are parallel to an axis, with every face simply connected, and the surface topologically a sphere.
Let $A(P)$ be the *area sequence*, the sorted list of areas of $P$'s
faces. For example:
<br />
<img src="https://i.sstatic.net/jDTJZ.jpg" width="150" />
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Using regular expression notation, this sequence can be written
as $1^4 2^2 3^2$.
In analogy with [golygons][3], I wondered if there is a $P$ with
$A(P)= 1^1 2^1 3^1 4^1 5^1 \cdots$. I don't think so, i.e.,
I conjecture there are no *golyhedra*. **Q1**. Can anyone prove or disprove
this?
Easier is to achieve $A(P)= 1^+ 2^+ 3^+ \cdots$, where $a^+$ means one or more
$a$'s.
For example, this polyhedron achieves $1^+ 2^+ 3^+ 4^+ 5^+ 6^+$:
<br />
![OrthoPolyhedronTwisted6][4]
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**Q2**. But can $A(P)= 1^n 2^n 3^n \cdots$ be achieved, for some $n$?
The above example is in some sense close, with $A(P) = \cdots 4^4 5^4 6^4 \cdots$,
but end effects destroy the regularity.
The broadest question is: **Q3**. Which sequences $A(P)$ are achievable?
Can they be characterized? Or at least constrained?
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***Update*** (*30Apr14*). **Q1** and **Q2** are answered by Adam Goucher's
brilliant example that achieves
$1^1 2^1 3^1 \cdots 32^1$. In light of this advance,
a more specific version of **Q3** may be in order:
**Q3a**: Identify some sequence that is *not* realized by any $A(P)$.
***Update*** (*9Jun14*): Alexey Nigin has constructed a 15-face golyhedron,
described on
[Adam Goucher's blog](http://cp4space.wordpress.com/2014/05/11/golyhedron-update/).
And later a [12-face golyhedron](https://yadi.sk/d/dmqoxL8kR6fSC). Finally, he found a <a href="https://disk.yandex.ru/d/Zzw_Q6gKTZjwk">11-face golyhedron</a>.
[1]: https://math.stackexchange.com/q/687147/237
[2]: https://meta.mathoverflow.net/q/484/6094
[3]: http://en.wikipedia.org/wiki/Golygon
[4]: https://i.sstatic.net/Va8uN.jpg