Wikipedia:Reference desk/Archives/Mathematics/2014 August 7

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August 7

Notation for points in 4D space

When giving coordinates in four-dimensional space, does the dimension perpendicular to meatspace, or to the only triad of dimensions renderable spatially, conventionally come before or after the dimensions x, y and z? For example, should the center of the default spawn region in Minecraft (which, in JSON, is {'dimension':0; 'x':0; 'y':128; 'z':0;}) by expressed as (0,0,128,0) or (0,128,0,0) [when a w-axis is reserved to distinguish the overworld from the Nether and the End]? NeonMerlin 00:55, 7 August 2014 (UTC)

Alphabetically the w-coordinate comes before x, y, and z. Bo Jacoby (talk) 02:22, 7 August 2014 (UTC).

And even so, I believe it's customary to put w fourth. -- Meni Rosenfeld (talk) 05:47, 7 August 2014 (UTC)

Incidentally, for the hyperboloid model of hyperbolic space, which uses one "redundant" timelike dimension, I've seen (and used) both conventions: t,x,y,z and x,y,z,t. —Tamfang (talk) 07:30, 7 August 2014 (UTC)

(x,y,z,w) is the canonical form I was taught for euclidean 4-space, but Minecraft is a different construct from that, isn't it? I mean, if we want to be precise, I'd think the overworld is a (subset) of R^3, and the netherworld is another copy of R^3. So, mathematically, I think this world would be isomorphic to the Disjoint_union_(topology) of X,Y, where each X,Y is a subset of R^3. --Except not quite, as we'd have to identify the points where the player can travel from the overworld to the netherworld. Anyway, representing this as 3+1 space where the +1 is just a flag for netherworld seems reasonable enough, but I thought the actual topology of the world might also be interesting to consider. (It doesn't work well anymore, but there was a time when discussing Asteroids_(video game) was a good way to illustrate the topology of the torus ;) SemanticMantis (talk) 17:09, 7 August 2014 (UTC)

Further- if there were only one point to transport between the overworld and netherworld, the game space would be the Wedge_sum of two copies of R^3. I may be missing something, but I think that construction could be generalized to take a quotient by n-many pairwise point identifications. The point is, the game world is not really a 4-space, it's a quotient space of a disjoint union of 3-spaces. SemanticMantis (talk) 20:01, 7 August 2014 (UTC)