a set of visuospatial illustrations demonstrating some interesting properties of numbers, such as that (*n*+1)×(*n*-1)=*n*^{2}-1, that the perfect squares (0,1,4,9,...) go up by successive odd numbers (1,3,5,...), & that the area of a triangular number (1+2+...+*n*) has a closed form. these properties are easy to show using algebra, but make more sense when visualized.

see also number spirals & regular expression visualizer.

[osteele.com]

should be (n+1)x(n-1)=n^2-1

Oh Stefan, you beat me to it.

stefan & aaron, I'll believe you as you seem so certain. I have changed the post.

ok, there's something wrong here. you're posting an example of good communication (that explains some basic maths). right? and yet it has completely failed.

the aim of the diagrams you posted was to explain exactly what you got wrong. you missed the point to the extent that you have to rely on "trust" rather than understanding.

looking at the page in question, part of the problem is that in the first text block (4th para) it gives the (incorrect) expression you quoted.

but that's not the only mistake. in the section you took your diagram from ("Product of Alternates", mid page) the expression is correct (first para of that section), but the labelling under the diagram is confusing. a confusion made worse by your dropping of the first part of the diagram so that it sits more prettily within the design of your site.

the combination of the original poor design and your insensitive cropping is to make the image caption and (to a lesser extent) the diagram itself meaningless.

i'm not sure what the moral is here. maybe you were simply too rushed.

but i'm worried that it's worse than that. that "this is good communication" has comes to mean "pretty thing, me like" and that whether or not any useful communication has occured is being ignored - communication certainly failed here.

perhaps there's another lesson in the "fragility" - even small errors ("-1" missing in some early text; bad cropping) can confuse the message. i suspect this is connected with the subject matter - that something with as crystalline a structure as logic/mathematics amplifies errors of this kind.

sorry if this seems preachy, but the disconnect between "this is good communication" and "i didn't understand a thing" is disturbing.

one more related point:

from my point of view (educated as a scientist / mathematician) it seems clear that the page author is trying to show that mathematics is not complex and opaque, but intuitively obvious. that basic truths like these can be grasped by anyone simply by looking at diagrams.

taking that further, i'm sure the author hopes that this understanding will give people the strength to question "obscure" notation.

i'm not sure i'm making myself clear, but what i'm trying to convey is the level of disappointment in how poorly this has worked, given the apparent quality of that page.

Andrew, thnkx for your comments. I understand your disappointment.

the real reasons behind this mishap are more banal than you describe: after ‘understanding’ half of the visualizations (& just skipping the other ones), I decided to post. as you mention, I chose the most aesthetic diagram & cropped it without respecting the mathematical expression. subsequently, the post itself was based on the original author’s text, rather than my understanding.

it should be obvious that this series of mistakes should not reflect on the perceived effectiveness of the diagrams, which have great quality.

so my excuses to you & the original author.

this event shows that the tension between beauty & effectiveness is intriguing, & one of the driving forces behind this blog.