I asked this question two months ago on MSE, where it earned the rare
*Tumbleweed* badge for garnering zero votes, zero answers, and 25 views over 61 days.
Perhaps justifiably so! Here I repeat it with slight improvements.

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:

Using regular expression notation, this sequence can be written as $1^4 2^2 3^2$.

In analogy with golygons, 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 $1^+ 2^+ 3^+ 4^+ 5^+ 6^+$:

**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?

**(**

*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. And later a 12-face golyhedron.

golygons: lattice polygons with edge lengths 1,2,3,... $\endgroup$ – Joseph O'Rourke Apr 28 '14 at 1:17