Stellation

Introduction

Technically all that stellation is, is extending the faces of a polyhedra untill they meet. It might be easier to visualize in two dimensions; if you look at a pentagon and extend it's edges till they meet and form a pentagram. This doesn't work for all polyhtopes as you can see if you look at things like a triangle or a cube. Now for an example in three dimensions it's easiest to look at the octahedron. To make things "simple"(perhaps just more generalizable to more complex polyhedra) think about expanding one one face out infinitely so you just have one flat plane, then think about where the other faces intersect as a line on that plane that you can extend as much as we need(this is the same as expanding the face but it's much easier to think about and visualize.). The faces along the edges of our chosen triangle expand out and never intersect with each other like if you atempted to stellate a triangle, the faces that you could think of as on the next "lower level(or ring(neither very relevant here but again you'll see this same concept in later polyhedra))"(or as the ones that share a vertex but not an edge with the chosen triangle) will extend lines perpendicular to the vertices of the original face which transcribes another triangle to make something akin to the triforce or a simple sierpinski triangle. These are all of the intersections for the octahedron as the final face is paralel to our starting face and all of the lines we've drawn are actively extending away from wach other or are parallel. So this shows where all of the faces intersect each other and there's only one "sections of pockets"(I guess is how you would put it) and so if you made the large triangle a new solid face on every face of the octahedron(because doing this process on every face yields the same result) you get a new polyhedron called(you're never gonna guess this) the stellated octahedron(or the stella octangula which you might not have guessed.). The resulting diagram is known as a polyhedrons stellation diagram and it's realistically the only way that you can discover all of the possible stellations for a polyhedron without your brain exploding after staring at dice for hours(totally not speaking from experience/s). For polyhedra with more than one kind of face there needs to be a different diagram for each face which makes things more complicated so as far as examples go for now I'll be using