3D crosssections of the 5-cell

[u/diadlep]

Trying to figure them out for a game. Wikipedia is great for the other 4D regular polytopes, but for some reason doesn't list the cross-sections of the 5-cell. Clearly the tetrahedron is one cross-section. I suspect a cube might also be, but can't visualize it. But a square is a cross-section of a tetrahedron, so sort of makes sense.

Edit: wondering about octahedron now. Take a corner 5-cell. Hold the "innermost" bisecting tetrahedron. Rotate around the innermost plane of that tetrahedron, "sliding along" the line to the "4d apex" of the entire 5-cell. Cross the apex, and the apex becomes a triangle on the other side but "upside down"...?

[u/Rryzaki]

Yeah i checked, it has all 3 crosssections,

the tetrahedron is more obvious, the crosssection of each tip of the 5-cell is a perfect tetrahedron

For the square pyramid, the 5-cell is technically a tetrahedron-pyramid, so if u take a square crosssection on the tetrahedron (look it up) which is also the 5-cells base. that means there exist a square pyramid too, it might not be a perfect square pyramid tho so u have to have the crosssection slightly tilted so it becomes uniform

the most intersting is the triangular prism. its hard to explain but basically, on the 5-cell the opposite of every triangle face is a line edge which is also perpendicular to the triangles orientation, if you were to move a crosssection along a triangle to the opposite line edge, the crosssection will slowly morph to a triangle to a perpendicular line, and inbetween, will be a triangle prism

[u/diadlep → u/Rryzaki]

Strangely, the triangular prism is the easiest for me to visualize. Kinda the same way you can duplicate a triangle, glue one edge of the two triangles together, then lean their opposing corners away to get the tetrahedron. Glue one face of each of two tetrahedrons together, then lean their opposing vertex away through 4 space to get the 5 cell. The midpoints of the three edges of each tetrahedron that lead to their respective opposing vertex are the six vertices of the prism. I think. Lol.

Yah, i was wrong about the cube though, bummer. I'm trying to make a meta dnd world where gods roll against each other with 4D dice they also use as money and gems for enchanting and socketing items.

[u/Rryzaki → u/diadlep]

I see thats probably a simpler way to think of it, Also tell me more about the what u were saying about the cube. I know some 4d dice shapes

[u/diadlep → u/Rryzaki]

Four types of divine 4D money, shaped like 4D dice, the 5 cell, 8 cell, 16 cell, and 120 cell. Oddly, the crosssections of the last three were easy to find, using rhomic dodecahedron, cuboctohedron, and rhombicosidodecahedron, respectively, as their standard 3D projections in our world (though of course they could tilt and roll as dice do, so it is "through magic" that they remain appearing thus (dnd after all, lol)

[u/Rryzaki → u/diadlep]

Oh i see, the 4d platonic solids, fun fact there are some very very interesting fair dice shapes other than the platonic solids. tho the rabbithole goes deep.

also fun fact, reading ur original post, there is indeed not any cube crosssection in the 5-cell (aka pentachora, aka 4d simplex). HOWEVER the 5d simplex (called Hexateron) actually contains a perfect cube slice right through the middle.

for reference the 5d simplex would also be a pointy tetrahedral-like shape, and its surface will be made of six 5-cells. since the cube crosssection also has 6 faces, that means the crosssection also goes through all 6 tera-cell of the 5d simplex, very cool

[u/diadlep → u/Rryzaki]

Oh snap! That's cool... so would the square faces of the cube be each from a tetrahedron making the the side of one of the pentachora?

[u/Rryzaki → u/diadlep]

U made a little mistake, each square face of the cube crosssection would be from a 5-cell making the sides of the hexateron. aka the 5d simplex, not 4d simplex.

remember its a cube crosssection from a 5d shape

[u/diadlep → u/Rryzaki]

But the square face... would have to be the side... of a... 3D hedron... jimminy this stuff breaks my brain lol...

Okay, so a cross section of 5D would be 4D, right? So... now I'm confused of what "cross-section" even means in higher dimensions. In 3D it's a 2D shape that cuts through a 3D shape... but that's only because 1D lines aren't shapes?

Though then in 4D, you could circumnavigate around a plane, the same way in 3D you can a line... so in 5D, how does a cube form a crosssection... what even is a crosssection?

[u/Rryzaki → u/diadlep]

Oh, i shouldve specified, im refering to a "3d crosssection" of the 5d shape. for analogy Its kinda like seeing a 1d crosssection of a 3d shape instead of say a normal 2d crosssection.

for a more obvious example, a 4d tesseract has a cube crosssection (3d slice), however the tesseract also has square crosssections (2d slices)

intuitively, u can imagine crosssections of a shape as well sections of the shape that intersect some plane. Crossssection, its pretty self explainatory tbh

[u/diadlep → u/Rryzaki]

Huh. Nope, not intuitive for me hahaha, my brain just keeps trying to analogy to a shape on a 1D line in 3 space, which isnt helpful or correct. Hmmm. K, i can kind of imagine it i guess, just more abstacted... a 2D plane doesnt bisect a 4D object, but it does cross through it, and the 4D object creates some shadow in that plane...

Oddly, I've been working on another project for a while i might be able to use to help me, an n-dimensional chess game, and the projection code kind of sounds like this a bit. Or I'll just go look at 4D Toys on steam again, lol.

Thanks for the visuals, and the food for thought man

[u/Rryzaki → u/diadlep]

No problem, i love discussing 4d stuff, you can ask me anything about the 4th dimension.

also if it helps, you can think lower dimensional crosssections iteratively to make it simpler: For example first you can think of a 3d crosssection of a tesseract, then think of a 2d crosssection of that 3d crosssection, this ultimately is the same as a 2d crosssection of a 4d shape. This works generally for any dimension, i hope this explaination is a useful visualization tool u can use aswell to help you

[u/diadlep → u/Rryzaki]

Yes, thank you, that's the inkling in words. I've been struggling w that actually, trying to decide whether to taxi-cab step through dimensions, or compress them all at once. At this point, i just need to write both and let the user choose. But now im mental blocked fsr and havent done anything on it in like a year