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Projex - More Info

Welcome to Projex!

Overview:

Projex, or Projective Hex, is a variation on the classic game of Hex played on the real projective plane, RP2, instead of on a rhombus in the euclidean plane. It was invented by the late Bill Taylor. The key mathematical observation, analogous to the Hex theorem in the plane, is that if the vertices of a triangulation of RP2 are 2-colored, exactly one color will contain a cycle which is non-contractible as a loop in RP2. Thus, in the game, players take turns coloring these nodes, with the winner being the one to make such a cycle. As in usual Hex, it looks nicer to draw the dual of the triangulation, and have players color faces.

Here are two ways to construct RP2: (1) Quotient a disk by identifying opposite boundary points. (2) Quotient a sphere by identifying opposite (antipodal) points. Implementations of Projex using (1) exist, but I find them difficult to use due to the confusing nature of the boundary. The goal of this project is to create an implementation based on (2). Thus, the "board" is some tiling of the sphere with antipodal symmetry, representing a tiling of RP2, and making a move on some tile also colors the antipodal tile. With this representation, the following are equivalent characterizations of a cycle being winning. The last two are valid only for minimal (equivalently, induced) cycles.

  1. The cycle is not contractible in RP2.
  2. On the sphere, the cycle is its own mirror image.
  3. On the sphere, the cycle contains a path between some tile and its antipode.
  4. In RP2, no neighborhood of the cycle is orientable.
  5. In RP2, the cycle has a mod 2 self intersection number of 1.
  6. In RP2, the complement of the cycle is connected.
  7. On the sphere, the complement of the cycle has exactly two connected components.

There are many interesting symmetric tilings of the sphere, and so there are many boards one can use to play Projex. I have included a decent selection of them for you to try. The code for this website runs the game automatically given an ".obj" file to use as the board. Thus, it is very easy for me to add additional boards, provided I have access to a 3D model of them. If there are other polyhedra you are interested in trying out, please let me know and I can add them!

Euler characteristic:

People are sometimes confused about the presence of pentagons on the Projex boards. Because the sphere/RP2 have positive Euler characteristic, this is unavoidable; it is impossible to make a Projex board using only hexagons. One must include 6 pentagons (12 for the sphere), 3 squares, or some other combination.

Pie rules:

Projex is subect to the same strategy stealing argument as Hex showing that, with no modification, the first player always wins with optimal play.

For usual Hex, it is common to balance this by introducing a so-called "pie-rule": Player I makes a move, then Player II can choose to either continue as normal, or switch places with Player I. Thus Player I is encouraged to make a not-too-strong first move. Though the resulting game is now provably a win for Player II with optimal play, the situation has improved in that Player I has the power to make Player II's starting advantage as narrow as possible.

The extra symmetry of Projex boards may require a more complicated pie rule for a fair game. Though I am not sure, my intuition is that, because each tile is similar in strength, in Projex with the usual pie rule Player II will always choose swap and end up with a decent advantage, and thus Player I will always vainly start with the weakest tile possible (probably a pentagon) to make this advantage as small as possible.

Thus, if you play Projex, you may want to experiment with other pie rules. For example, in the three move pie rule, Player I makes two moves for themself and one for Player II, then Player II decides whether ot not to swap. This can presumably create situations in which Player II will not want to swap, e.g, if the two Player I stones are right next to eachother. Pressing "R" or "B" will allow you to manually set the color of the next tile placed, which could be useful in implementing these.

Scenarios:

In the "Scenarios" tab you will find several game states with certain tiles already colored. Starting a game of Projex from these states allows one to essentially play all-new connection games. Give them a try!

Credits:

Projex was invented by Bill Taylor. Most though not all of the models used by this website were generated by polyHédronisme. Some of the scenario and Board ideas are due to or inspired by Mark Steere.

This website was made by me, Felix Weilacher. Thanks to Forte Shinko for extensive playtesting and feedback. Thanks to Eric Demer for feedback on this page.