A one semester-long visual mathematical project

Question

For the incoming semester, my school gave me an amazing new classroom, where all walls are covered by white boards. Even windows are covered by movable boards. It is great to have that much space and I will use lateral boards to keep information such as classroom life (homeworks, examinations,...), and keep important theorems, definitions or figures all lecture long.

I am not sure what to do with the board in the students'back. Since their chairs have wheels, I could use it normally but I don't feel it will be convenient.

My idea is to leave this board to the students and let them use it to present a math related project. The two main constraints are:

1. Only this board can be used to present the project. In particular, it should be self-contained, and the students should assume they would not be present to answer to the audience's question. The aim is that any passer-by could enter into the classroom and learn something from the project (maybe work on it - see below).
2. The project will evolve during the semester, the students improving or modifying it during small classbreak, or during their lunch break,... The idea is that the project does not take too much of the students time. Also, I planned to take photos of their work regularly.

My question: Do you have examples of this kind of project? Or resources on how to organize it?

Specific constraints related to this precise class (but I am interested in answers in another setting, addressing the two main constraints above):

1. They are high school students. They will start the study precalculus, but already know quite a lot about functions and trigonometry. A project involving multivariable calculus would also be interesting (for another class).
2. It is a special high school for science students.

My idea so far would be a timeline about Asian history of mathematics (Asian because I am a European teacher in Asia, and I don't know anything about this part of the history of mathematics).

Answer 1

A study of the Platonic solids can be a nice visual project. A challenge to the students could be to find cartesian coordinates that correspond to the vertices of the these 5 solids. The octahedron and cube would be quite easy. Finding the vertices of the dodecahedron and icosohedron with trigonometry is challenging and fun. Another Platonic solid project would be finding radius from center to face center, edge center and vertice. Also finding dihedral angles. Coxeter's Regular Polytopes is an inexpensive Dover book and would be helpful for such a project.

Answer 2

How about a gallery of triangle centers? From the Wikipedia page:

In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles. For example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of them has the property that it is invariant under similarity. In other words, it will always occupy the same position (relative to the vertices) under the operations of rotation, reflection, and dilation. Consequently, this invariance is a necessary property for any point being considered as a triangle center.

Also,

As of 3 July 2014, Clark Kimberling's Encyclopedia of Triangle Centers contains an annotated list of 5,875 triangle centers.

Of particular interest are the 1st Liu point and the 2nd Liu point (X(3598) and X(3599), respectively) which "were discovered by Kang-Ying Liu of St. Andrew's Priory School, Honolulu, Hawaii, during 2010" while she was still in high school.