In page http://loki3.com/poly/isohedra.html around six polyhedra with equilateral pentagons as faces are shown: a pyritohedron, icositetrahedrons… Is there a complete list of this kind of polyhedra? How to compute the angles of those pentagons?
[Math] polyhedra with equilateral pentagons faces
discrete geometrymg.metric-geometrypolyhedra
Related Solutions
Wonderful images, Edmund! :-)
A net $P$ can fold to a polyhedron iff there exists what I called in the book you cite an Alexandrov gluing, which is
an identification of its boundary points that satisfies the three conditions of Alexandrov's theorem [the subject of a recent MO question]: (1) The identifications (or "gluings'') close up the perimeter of $P$ without gaps or overlaps; (2) The resulting surface is homeomorphic to a sphere; and (3) Identifications result in $\le 2 \pi$ surface angle glued at every point. Under these three conditions, Alexandrov's theorem guarantees that the folding produces a convex polyhedron, unique once the gluing is specified.
Let me quote two results from the book, informally, the first quite disappointingly negative, the second compensatingly positive:
Theorem 25.1.2 (p.382): The probability that a random net of $n$ vertices can fold to a convex polyhedron goes to $0$ as $n \to \infty$.
Theorem 25.1.4 (p.383): Every convex polygon folds to an uncountably infinite variety of incongruent convex polyhedra.
In particular, for example, a square folds to an infinite number of convex polyhedra, whose space consists of six interlocked continuua, as detailed in our book (Fig.25.43,p.416).
The exact question you pose—Which nets can fold to convex polyhedra?—remains open. Although if you give me a specific net, we have an algorithm that will produce all the convex polyhedra to which it may fold. But note my emphasis on "convex," an adjective you left out in your question. That is Open Problem 25.1 in our book (p.384), on which topic I have written a separate note subsequent to the book's publication: "On Folding a Polygon to a Polyhedron." In a nutshell: every polygon folds to some (generally) nonconvex polyhedron, by a result of Burago and Zalgaller. But their proof is complex enough that I have no understanding what that polyhedron might look like.
Aside from the paper I cite above, you may be interested in this result:
Four of the five Platonic solids may be "unzipped" and "rezipped" to be doubly covered
parallelograms (which may conveniently be placed in your wallet!). See this MSE question for
the (difficult to find) icosahedron net [below]. The holdout here is the
dodecahedron, whose 43,380 edge unfoldings each may only fold back to the dodecahedron.
The answer to the question in the title is negative in dimension 3: it was shown by Grunbaum that every 3-polytope with congruent facets has an even number of facets. See p. 414 of his book "Convex Polytopes", 2nd edition. The original reference is:
Grunbaum, B. On polyhedra in $\mathbb{E}^3$ having all faces congruent. Bull. Res. Council Israel, 8F (1960), 215-218.
Best Answer
The pentagonal isohedra with sides of equal length listed on http://loki3.com/poly/isohedra.html are the regular dodecahedron, non-convex equilateral pyritohedron, equilateral pentagonal icositetrahedron, non-convex equilateral pentagonal icositetrahedron, non-convex equilateral pentagonal hexecontahedron and non-convex equilateral pentagonal hexecontahedron.
(I should note that the page jolumij references, is a page I built to summarize my findings. I had tried to learn about the isohedra from pages such as http://mathworld.wolfram.com/Isohedron.html, but all sources I could find were very incomplete when it came to describing the pentagonal isohedra. They offer names such as "octahedral pentagonal dodecahedron" without describing how to construct them or mentioning they may represent an infinite family of shapes.)
I assume by "this kind of polyhedra," you're referring to isohedra with pentagonal faces. Mathworld offers this definition of an isohedron:
For my list of isohedra, I relax the definition to include non-convex polyhedra. The isohedral transforms can also be used to create polyhedra with intersecting faces "with symmetries acting transitively on [their] faces with respect to the center of gravity."
As to whether those 6 are the only isohedra with equilateral pentagonal faces, I believe the list is complete, but I haven't rigorously proven it. What I have done is start from the tetrahedral, octahedral and icosahedral symmetry groups and applied the isohedral pentagonal transform to them. This transform has two degrees of freedom. Then I explored the space for equilateral pentagons as well as other interesting symmetries or patterns. I haven't seen references to many of the shapes I found (including 5 of the 6 shapes listed here), so I'd be interested if other people know of any other references.
As to how to compute the angles of those pentagons, http://loki3.com/poly/transforms.html#penta gives a description of the transform and what my notation means. You can use the parameters and transform to derive the angles. For example, the non-convex equilateral pyritohedron is 4p(0.09549150, 0.6605596), which means you apply the isohedral pentagonal transform (p) to a tetrahedron (4) using the parameters 0.09549150 and 0.6605596. In this case, you get two 36 degree angles and three 108 degree angles.