Trinagon, being completely new, has not yet been explored !!  (by a proper mathematician at least)

The maths involved are mainly group theory (like all permutation puzzles) apart from the spatial niceties of the polygons. The symmetries involved are pleasant to look at, and do impact on the symmetry & 'puzzle' groups of each puzzle.
First some more basic concepts and ideas, then below you'll find some answers, which mostly may not have been too obvious at the start.

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We would definitely be interested in finding two (or three) algorithms :

• One that can check whether a proposed solution pattern can actually be achieved in play, without the actual solution needed.
  (A basic theory (more an idea) for this exists already. Still an occasional work in progress. 
• A better algorithm, to find the Gods Number for bigger puzzles. 
  It gets impossible very quickly to find it, when a puzzle has more than 100 million permutations
  The smallest number of moves needed to solve puzzles, when found is usually astounding elegant. 
  These solutions are great to lkearn from & are included in the game. 

Some mathematically interesting features within trinagon may hopefully be a source of inspiration for many students (of all grades) :

For example :

• platonic bodies (and other polyhedra) and many of their properties.
  Even just visual representations, getting a feel for it and learning to 'see' 3D, will greatly improve the ability to work with polyhedric shapes, be it in art or in architecture (and maybe even both) or others.
• general symetries in physics, chemistry and mathematics,
• vector calculus
• group theory. (Trinagon may even be a nice diversion for  some particle physicists, using their understanding of dihedral groups ?)
• probability theory

A few simple 'ponderances' for students :

• How many different solution patterns (only the end positions) are there to a given colors-only puzzle ? 
  How does this change when you add facemarkers, and directions ?  
  
  
• ... and can all of them be achieved / reached in normal play ? (mostly yes, if no. Why not.).
  How many can be ? (That's where it gets difficult). 
  
  
• Why are there exactly 6 rotationtypes ?
  Or could there be more ?  (Yes there could be more ! but only if ... ) 
  
  
• How many turns does it take to move a triangle from A to B, and how many shortest paths are there, depending on the playgound ?
  
  
• In the Icosahedron. Do 2 triangles that are as far away as they can be from each other have the same spatial orientation ? Why (not) ? 
  Is there a connection to how many moves it takes to move from A to B ?
  
  
• Is there a useful relationship between the distance of two triangles and the number of moves to bring one to the others place, or exchange them (in the hexagon) ?
  
  
• And one out of the game : 
  In the tetrahedron puzzle. Use one flip-rotator (4,5 or 6) and 3 flat rotators.
  Is it possible have only one triangle face flipped ?
  Yes / No  &   Why ? :)   
  
  .. Are you sure :) ? I thought i was !
  
  
• ... make up your own. There are enough questions for all and for everyone.

After playing a while, a few good guesses might already answer some of these questions or give you an idea of how the get there.

If you have come up with questions and maybe even followed through with some answers, feel free to tweet (or X) Lumpazy @ the Trinagon Puzzle Collections.

Some things that have been considered and mostly solved :

• What makes a puzzle Unique ?
  In a recent update another algorithm was integrated that can actually reduce ('collapse') the puuzzles to their actually defining base state.
  The basic idea is obvious and simple. For example if a puzzle has only colors, then it doesn't matter what types of rotators it has. And the other way round : if a puzzle has no facechangers, then the faces of triangles don't matter. And so on.
  Also spatial symmetry needs to be checked. E.g. The hexagons have a spatial symmetry of 6. So turning a hexagon puzzle 60 degrees, doesn't make it any diifferent.
  There are some other small pitholes though : Some triangles absolutely may look the same, but on some puzzles they do NOT belong to the same group ! 
  
  
• Same Group. What group ?
  Triangles are grouped together when they look the same, and can be moved to the same place, and there have the same orientation. Then they are interchangeable. I decided to call this a 'group'. 
  This also leads to the maths involved in calculating the permutations for every puzzle.
  and also helps greatly with the ideas involved in solving the really hard puzzles. 
  
  
• Permutations of each Puzzle : 
  Now five years after the first launch of the game a few rather simple algorithmns have been found / made to count the total number of triangle configurations on a puzzle.
  The numbers are from as small as one thinks on the smaller polyhedra to extremely huge for the larger ones. Usually we all tend to underestimate the huge amount of possible permutations.
  The algorithmn only calculates exactly up to floating point value. Everything above is estimated in powers of 10.
  Comparing to the Rubis cube. Trinagon has way more permutations. This is even true when triangles are interchangeable (same group, see above).
  
  
• Gods Number for each puzzle : 
  Within all this also a solution-finding algorithmn has been checking for the best possible solutions to many puzzles. The algorithmn uses brute, but intelligent, force :).
  The results of the best solutions are in my opinion : Totally astounding.
  This information and the possibility to also load 'gods solution' are currently being worked into the game.
  Gods solutions will only become available once a puzzle was solved. There is a learning curve and a self-rewarding curve there : It's much easier to find a really short gods solution if you know how many moves it takes. Especially on the smallest polyhedra. Therefore, let the surprise come slowly.
  Comment : For the rubiks cube many solving systems have been created. Simulations were run and a lot of thinking & processing power got spent on figuring out the rubiks cubes 'Gods Number', and many mathematical papers have been written on this one single puzzle (one resource here ..), and other permutation puzzles.
  There are many apps that will autosolve it for you, and there a numerous tutorials where you can learn to apply solving patterns for the rubiks cube.
  
  
• Ad Brute Force : 
  With n rotators active there is a maximum of (2*n -1) new configurations ('states') with every move. So the number of states increases rather fast by the power of the number of moves.
  Depending on the basic setup, size and rotationtypes, many states will be repeating themselves, which leads to at best a 1/3rd less that need to be stored. But this comparison is exponentially more effort.
  So it is pretty clear why finding gods number is not a matter of minutes on the harder & bigger puzzles, but would take days or years to cumpute (on a regular PC). Obviously the search for a perfect solution goes from the start up, as well as from the solution downwards, and will meet in the middle.
  There will often more than one gods solutions, but only one needs to be found. 
  
  
• Open Question : Without actually brute force Gods solution searches :  Maybe any proper mathematician would like to find a proof of the existence (or absense) of a 'fitting' solution, which hasn't been played to, but simple set up with the above knowledge of interchangeables. The assumption is, that yes.
  Most puzzles can be solved then, but there already is a rather simple example where this is NOT true, and quite easy to see : The impossible tetrahedron puzzle above.