Changing perspective is difficult for people to do. For instance, I remember a playful little visual test that was given to a group of us PhysEd students back in my high school days. Ms. Macho drew a three-dimensional cube on the blackboard, comme ça:
She then asked if we could see the cube from two different perspectives. The trick was to notice the two squares, which represented the front and back sides of the cube, and their relation to the uniquely shaped rectangles, the trapezoids, comprising the other four sides of the cube.
The first perspectival point of view would be to see the front of the cube as the solid line square on the bottom left. The dotted lines, then, represented the inside edges of the hypothetical solid. The second way to view the cube would be to see the front of the cube as the square on the very top right, which has two sides that are solid lines, the top and right sides, and two sides that are dotted lines, the bottom and left sides. Seeing this second square as the front of the cube results in the first, solid-line square becoming the far side or back of the cube. The teacher asked us if we could, in our minds, see the cube flip back and forth between these two different perspectives. Most of us, about twenty students, agreed that we could manipulate the cube in our minds and see it from the two suggested points of view.
Now, you might want to argue that it’s not really fair to ask us to flip the cube when it’s drawn this way, because the dotted lines make it confusing to see the cube from the second point of view. You might further argue that it would be better, and far more fair for those manipulating the image in their minds, if all the lines were drawn in solid. I might agree with you, but this is the way the instructor drew this particular test. I suppose some research could be done, that would, of course, probably require a government grant and some high-tech lab equipment, computer screens and timing devices, a control group and some statistical analysis. These, in turn, would render correlations between the solidity of the lines and the speed with which they can be mentally manipulated; however, at this point I will have to leave the solution of the problem up to you. Perhaps an interdisciplinary post doc at Princeton University would be the ideal place to undertake this experiment, perhaps by someone whose primary field is in the study of Geometry, with an interest in Sociology.
You may additionally venture to comment that there doesn’t seem to be any good reason to view the cube from different perspectives. As far as you’re concerned, one perspective is just fine. In fact, inasmuch as anyone can see, and I think you would agree, the two perspectives are practically identical; I mean, yes, they’re slightly different but they are definitely symmetrical. It could be argued, therefore, that it is only necessary to see the cube from one perspective.
However, others might object that the two perspectives are, indeed, quite different. They will describe how, even though the two views are technically symmetrical, they are, in fact, the complete opposites of one another; therefore, this group would surmise, both views are important and necessary. The three-dimensional cube simply cannot be fully understood from just the one perspective. Moreover, there may be other perspectives from which to view, understand and discuss the cube. This group might add the pejorative opinion that perhaps it’s just fear or laziness that leads to the argument that only one perspective is necessary or right, or whatever.
Anyway, Ms. Macho then asked us to practise for a few minutes flipping the perspective of the cube in our minds. I was quite familiar with this cube. I drew it a lot, sometimes even in class. I also liked to draw another puzzle which was of a square with an ex through it and a little roof on top:
You were supposed to be able to draw it in one go, with one single, continuous line from start to finish, without lifting the pen from the paper. I found all the different ways to draw it, again, sometimes even in class. So when the teacher asked us to flip the cube, I found it relatively easy. I mean, it was a bit sticky at first – try it – but then it became easier and easier, until I was flipping it back and forth in the blink of an eye.
I found this fun but I don’t think my compatriots were on the same page. They probably hadn’t drawn the cube nearly as many times and they probably couldn’t care less that you could see it from two different perspectives. I’m sure they weren’t into exercising their minds because almost all of them were jocks, not geeks. I don’t know if they were trying very hard or even paying attention. Typical teenagers, they were too cool to learn anything and way too cool to have fun doing it, so I think the next part of Ms. Macho’s game took them by surprise.
We were asked to see how many times we could flip the cube in thirty seconds. Out came the stopwatch and suddenly it was a competition. No one told us there was going to be a quiz! Thirty seconds of strange silence was followed by a quick tally. Some could flip the cube once or twice, some five or six times. I must have flipped that cube thirty-five or forty times in thirty seconds.
Since the group had obviously been unprepared for the Electric Acid 3D-Cube Test, the instructor ran the experiment a few more times. Each time, the students improved a little. It was evident from the improvement seen over just a few tests that changing perspective became easier for the students with more exposure and experience. It seemed obvious to me that with practice, anyone should be able to strengthen their perspective changing muscles.
This story was first published in Voices 2018, A Toronto Writers’ Co-Operative Anthology, John Miller, et. al. eds., Toronto Writers’s Co-operative, 2017.