TITLE: Open problems in Euclidean geometry?
QUESTION [54 upvotes]: What are some (research level) open problems in Euclidean geometry ?

(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a bit what I mean by "Euclidean geometry". By this term I mean, loosely, the study of the geometry of certain subsets of Euclidean space $\mathbb{E}^n$ from a point of view which is either the "classical" one (i.e. axiomatic), or one that involves more modern tools, but the problem in question has not to be "clearly" a problem within some other branch of maths such as differential or algebraic geometry, algebraic or general topology, analysis, or measure theory.
Some examples to clarify:

the study of configurations of lines or affine subspaces is EG; but the algebro-topological study of hyperplane arrangements is not.
plane conics as defined via their metric property are objects of EG; but "algebraic curves" are not, unless they're defined by some "elementary enough property" (intentionally vague) involving the Euclidean metric.
root systems of Lie algebras are EG. 
polyhedral cones are EG.
polytopes are EG.
tessellations of space with polytopes or analogous objects are in EG.
minimal surfaces in $\mathbb{E}^3$ are not EG.
fractal geometry (Julia sets, self-affine fractals...) is not EG.
not sure about convex bodies. If they're polyhedral I'd say their study fits in EG.
packings of spheres are EG.

REPLY [9 votes]: The classification of all pentagons tiling the plane is not completely finished,
see for example MR3037862. (Bagina, O. G. Convex pentagons that tile the plane (types: 11112, 11122). (Russian) Sib. Èlektron. Mat. Izv. 9 (2012), 478–530.)
Update: This problem is now probably solved: see e.g. https://en.wikipedia.org/wiki/Pentagonal_tiling and references therein.