Unless you know multivariable functions or are pretty slick, you probably think I'm crazy. However, I can assure you that these are simply two different perspectives of the exact same shape; the only thing that has changed from one to the next is the place from which you are looking at it. If you're a skeptic (and I hope you are), you still don't believe me. Fair enough, but look at the animation after the jump and you don't have to believe me--you will see it with your own eyes.
this is a large .gif, it takes a minute to load, and can probably be seen in full resolution if you click on it.
For the sufficiently curious, the graph is of the function
Pretty awesome, right!? This is a very interesting consequence of trying to put a 3 dimensional object onto a 2 dimensional space. In other words, this is a 2d projection of a 3d object. It makes an awful lot of intuitive sense that by trying to represent a 3d object in a 2d space that information will be lost--in other words, if anyone tries to draw a bog-standard sphere on a piece of paper (that is no shading, no contour lines, just an outline), all their mathematician friends will laugh at them for drawing a circle! In fact, you can't draw a sphere on a flat piece of paper because it lacks any indication that it ought to be jumping off the page. What is very interesting about this idea is that the addition of information can really help in tricking the human brain into thinking that the object does have some depth. In the above images we have the additional information of color gradients, a contour grid, and in the animation, changing perspective. Had I not included these things, it would have been very difficult to communicate what I'm talking about; the images would just look like a chaotic mess.
I think it is interesting to note that despite the added information, there are still a number of times that the shape doesn't make very much sense at all, and there are a number of other shapes I could use that would be even more confusing. Indeed, even the way in which you rotate the object is a form of information--had I been able to figure out how to rotate the object around only one axis, it would probably make less sense than the second image alone! In terms of being a confused human being, there does seem to be an easy explanation of what confuses us and why : were the image of a textured, roughly round object rotating around one axis (instead of something like the above), most people would be able to easily interpret its nature. The reason is simple--our lifetime experience is riddled with roughly round, textured objects, and certainly not many objects shaped like that.
I think this concept of additional information has some very interesting and underutilized ramifications, Especially in this modern digital era. For instance, perhaps you have heard of a phenomenon called synesthesia. Synesthetes have a peculiar sensory experience, the most common being the sensation of colors in combination with symbols. In other words, a person with color/grapheme synesthesia will see a letter as being colored, and always having the same color (when not adjacent to other letters, more on this in the "good resource" below). This sounds pretty novel, but is it useful? How about, hm... Absolutely! It only takes one example to prove it. This kind of perception has the added advantage of being able to recognize words and phrases just be seeing a sequence of colors, no reading necessary for a general conception of what the word might be. This type of cross-information processing is something most people already do unknowingly by guessing a word from its general shape or first and last letters. There is a further advantage of having letters associated with colors: memory. By having an additional layer of information, there is an additional path for that information to be recalled; it isn't too difficult to imagine that if angry words were on average more red than other words, we would quickly associate red with anger on a very visceral level, more than just the abstract idea we already have that red can conceptually indicate anger. If you'd like to read more about what it is like to be a synesthete, here is a good resource. I don't necessarily think that that person's experience is optimal for added information, but it makes me think what I feel is an obvious thought: we live in a time where the majority of letters and numbers can actually and easily be colored... when it confers such obvious advantages, why aren't we doing it!? This likewise leads me to another obvious conclusion: the extent to which we are under-utilizing the technology at hand is profound. When one considers (as I often do) that the extent to which we are utilizing technology is profound, well, there is only one conclusion left: we live in an inconceivably amazing time!