When you are first exposed to the First Isomorphism Theorem for Rings, a thousand words can't do what that simple triangle diagram can.
Mathematics is the study of patterns. Humans have amazing powerful visual and spatial reasoning centers in the brain which can process huge amounts of semi-structured information.
Writing words down instead forces everything through low-bandwidth serialized language processing apparatus instead.
Mathematical notation (and computer code, and music notation, and ...) benefit tremendously from even slightly exploiting spatial reasoning skills. But better is some kind of diagram, interactive simulation, or similar.
The fastest mathematical explanation is direct tutorial, person-to-person, where the amount of symbolic formalism can be reduced to a minimum, and hand gestures, chalk drawings, colorful verbal analogies, etc. can convey most of the message. When limited to pure symbolic formalism in a scholarly paper, mathematics teaching and learning is a much harder slog.
The reason formal mathematics uses symbolic formal language as a canonical way of recording results is that it’s easier to figure out how to verify it step-by-step and make sure that the logic is correct.
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I’ll give you an example, from a paragraph a friend of mine wrote several years ago in a blog diary:
(I posit that you could decipher and understand this paragraph at least an order of magnitude faster if I drew a 30 second napkin sketch.)
> The tea room I went to is an “eight tatami design”, a large square space. Each side is two tatami lengths (four widths, two widths and a length) long. The eight tatami are arranged in the following way: from the entrance—which is in a corner, second forward on the left—two tatami extend into the room to meet the opposite wall on their short sides; the tatami touching the opposite wall is met on its right long side by a third tatami’s short side. This third tatami’s short side in turn meets a fourth tatami’s long side, whose two short sides meet the opposite wall and a fifth tatami’s short side. The fifth tatami’s second short side meets the room’s the fourth corner and its long side meets the sixth tatami’s short side. The sixth tatami’s second short side meets the first tatami’s long side. These six tatami form a square with an square empty middle. The empty middle’s sides’ dimensions are all one tatami length long. The middle, then, is filled by two tatami whose orientation is like that of tatami one, two, four and five—short sides to the opposite wall and the wall with the entrance. I tried out alternative arrangements and I might be wrong but I think that this solution (or a rotation of it) is the only possible tatami arrangement for the room’s dimensions that fulfills the rule “four corners must never converge”. Going in a circle around the room, starting at the door and following the direction in the description there, we can number the outer tatami one through six and the two inner tatami seven (left from the entrance) and eight (right from the entrance).
It helps to know that tatami are rectangles with length 2x width (something a casual and/or non-Japanese reader might reasonably not know).
What's described is eight tatamis (length 2x their width), arranged in a square. Much of the confusion comes from describing each tatami's relation with its neighbors, rather than the space as a whole, and referencing other tatami rather than the room as a whole.
A simpler description:
A square room, four units per side, filled with eight mats, each one unit wide and two units long. Looking away from you, down either wall are two mats placed lenthways. On the near and far walls, a single mat between those, long edge to the wall. In the center square, two mats placed long edges parallel to the side walls.
Or the 3600x3600 diagram in this image: http://www.tatami.com.my/layout.jpg
The problem is your friend's blog description. Not inherent limits of language.
I don’t think it’s possible to construct a textual description or mathematical formula of a collection of tatami patterns which takes less than 40–60 seconds per pattern to unpack into a clear mental image.
If I spend about 5–10 seconds looking at each of the 14 tatami mat diagrams in your picture, I’m confident I could reconstruct the patterns from memory a few minutes later. I can’t imagine doing the same without spending a significant amount of time on a textual explanation. Additionally, after examining an image with a bunch of tatami patterns for a couple minutes, I could probably tell you whether additional patterns satisfy the criteria to be acceptable or not, without doing any explicit analysis of those criteria. I’d have to think a lot harder if presented only with prose or formula descriptions.
Yes, often a picture is the clearest way to express something -- I'd be better off showing you a blurry potato of the Mona Lisa than trying to describe it to you. At the same time, there are circumstances in which images aren't available.
When two people are conversing without having a visual medium -- in the dark, one is blind, over a radio or phone. When the text itself is all you have: in an anthropological or forensic context. And in such cases, with effort and refinement, that text can often be quite clear.
Which gets to the point: a diagram or symbolic notation is the result of effort and refinement. It's a reduction of a situation to its essential parts. It's also possible for such diagrams to be far other than clear or accurate. Look to ancient maps or images of animals or digrams of equipment or mechanisms, or even of the old-school maths notations that current notations have replaced. Albrecht Durer's illustration of a rhinoceros looks very little like a rhinoceros (though he was working from second-hand reports). It is a poor representation. An overly ornate diagram, chart, or illustration communicates poorly often because it seems to have little idea of what it wants to communciate.
If you've ever read or heard Edward Tufte's exposition of visual presentation, he offers many examples, some old, some new, of both good and bad graphics. His own designs are so excellent because they reduce the topic to its very essence. To draw on a reasonably contemporary example, his comparison of what NASA had used to portray O-ring erosion, prior to the Challenger shuttle disaster, and what a simple plot of erosion depth vs. temperature offered.
Which still leaves us with the question of whether or not any textual description of your tatami problem would be clearer than an image. I'm suspecting that one could get quite close.
Note, this is what I was responding to: “[...] people naturally communicate using language. In your head you may think in symbols and spatial relationships, but if you want to get a concept out of your head and into someone else's head you have to use language.”
It is often the case that words + diagrams is the clearest way to explain something, and in most cases where you have a choice, you can choose both. Symbolic expressions fall somewhere between the two, and when used properly, further increase your chances of getting the point across. There's no need to turn every choice into an exclusive dichotomy, regardless of how common that is on HN.