I realized after posting the 4D Pong game yesterday, and watching peoples’ reactions, that the tricky concept is hypervolume — the equivalent of volume, but in one higher dimension.

The problem is that we have no direct experience of navigating in that very rich space. It’s deceptively simple — go up/down, left/right, in/out, and also one other dimension, but that extra dimension is one for which we simply have no intuition, since we don’t live there.

Something as simple as playing Pong — moving a box around in a hypercube and bouncing off its hyper-walls (ie: cubes, which serve the same function as the square walls on a 3D cube) — is already stretching beyond what makes sense to most people on any intuitive level, even if they get what’s going on intellectually.

I’m starting now to think about what might be good scaffolding experiences, even simpler than something like Pong, by which one could gradually “level up” to having the intuition required to navigate with ease through a hypervolume.

6 thoughts on “Hypervolume”

  1. This is in line with the comment I just posted on yesterday’s entry before seeing this one:

    I wonder if there is a “games for learning” opportunity here. Could you create the game in such a way that it progresses from 2d pong to 4d pong and, in the process, teaches the player about 4d hypercubes?

  2. Is the following correct? In any pong game the paddles are always 1 dimension lower than the space (so, in 2d pong the paddle is a line, in 3d it is a plane, and in 4d it is a cube). The puck has the same dimensionality as the space. The paddles are always fixed at the extreme values in one dimension (e.g., w=red/blue in your 4d pong, or x=0/1 in a horizontal 2d pong where the axes range from 0 to 1). You successfully “hit” the puck if you move your paddle to a position that intersects the puck by the time the puck hits the extreme value in the paddle’s fixed dimension. (So actually, time is an additional dimension in the game that we are not counting.)

    I think there are a few things that are hard about playing this version of 4d pong. The biggie is understanding the hyperspace enough to predict where the puck will go if it bounces off the other “walls” before it gets to the paddle’s fixed “wall”. The others might be easier to fix. For me, simply moving the 3d paddle in 3d space effectively using the keyboard is hard. I wonder, if one could play this with a Wii controller or a Kinect, if that would make things noticeably easier? Or at least, use the mouse for two of the dimensions with the keyboard only doing the third. Another difficulty is being able to tell the paddle’s actual 3d position from its 3d projection on the screen—your moving figure helps somewhat with that. Also, a better visual cue for a successful block of the puck would help. I have trouble determining when I’ve hit it and when I’ve missed.

    I think the real learning to be had is in understanding the hyperspace, and getting the other things out of the way could help with that.

    One more random thought: the red/blue dimension takes some getting used to. In terms of scaffolding experiences, how about a 3d pong that uses (x,y,w) instead of (x,y,z), if you see what I mean.

  3. A couple of other ideas for scaffolding steps:

    – Allow the game to be vastly slowed down, or even single-stepped so that the player can take time to think about what is going on.

    – Have only 1 additional surface, other than the paddles, that the puck can hit. More like tennis than like squash.

  4. I found my biggest difficulty was determining when a collision determined a score, and when it did not. to some degree this was due to both the oscillation of the 3d projected hypercube, and the aforementioned difficulty in visualizing hypervolume, even though I understand at least 2 3d representations of the hypercube. Sometimes it seemed that I was guarding a volume, and sometimes a plane depending on the ball’s motion. (‘Course thinking about it more this makes perfect sense)

    I played the computer to 100 points.. by the end of it I was keeping within 3, but the machine still won.

    So thinking more about it: The defended goal is actually the 3d volume located at the W end (w=0) of the hypercube, while the target goal is the 3d volume located at the far end of the w axis. The paddles are also 3d cubes which can move within.about the surface of their respective goal volumes, and points count whenever the ball enters that volume…

    Yeah it really is just a visualization problem, as the 3d projection loses a bit of the w axis along the way.. its like playing tennis from the top down.. cant tell when it hits the net except by feedback from the net. 🙂

    Fun game though. Thanks!

  5. This discussion reminds me of Miegakure (http://marctenbosch.com/miegakure/), “a puzzle-platforming game in four dimensions”, which I had the opportunity to play at PAX East. I’m very much looking forward to it, as the author seems to be approaching it as a teaching opportunity as well as entertainment.

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