Information Science: Visualizing x^4 and Higher Powers

Someone online asked "We can imagine x^2 as a square and x^3 as a cube. How can we imagine x^4 and other high powers?" and I had considered this thought experiment in college working with multidimensional arrays.  Many people get hung up thinking about physical dimensions of height, width, length and fail to realize that many attributes of an object or system can be considered a dimension.  So here's my answer:

Hopefully it's clear the upper left (of the 9) boxes is your first cube.  Then you can see you have arrays and eventually cubes of cubes (x^6). 

This works well for data representation, at least to x^5.  For example let's take x^4, and let our dimensions be length, width, height, and time:

We can see the length, width, and height at each point in time.  The object we're representing is getting smaller as time progresses.  Maybe it's an ice cube.

Maybe in the x^5 scenario, our 5th dimension could be temperature, so the top row could be some "medium temperature," the middle row could be "very hot" and the bottom could be "very cold" so the ice cube doesn't melt at all:

If you are working in higher order math, being able to visualize (and hopefully the need to visualize) goes out the window pretty quickly.