New sections on nspheres, sum of a polygon's internal angles

[josephlindenberg]

Mike, I recently added two new sections to my tau website on topics that were addressed in the Pi Manifesto. When you have a moment, would you take a look and tell me what you think? (Anyone else still here, feel free to chime in too.) Thanks.

http://sites.google.com/site/taubeforeitwascool
"The sum of the internal angles of a polygon - A rebuttal"
"A different pair of formulas for every dimension. Their common link?"

[SpikedMath]

Very nice discussion on "uninternal angles". I'm not sure if that's the best name for them but can't think of anything better -- too bad external angles is already taken! However, since we're changing the circle constant we might as well change the definition of 'external angle' as well!

The next section is nicely written too. The diagram (Nspheres.png) was confusing at first until I realized the red things were arrows (which it does say after the diagram). Let me think about it for a bit and see if I can come up with a clever rebuttal, hmmm....

[xander]

How about "outside angle"? Or, if we want to be really clever, "outernal angle"?

xander

[josephlindenberg]

Since it has done without one this long, it would appear mathematics doesn't really need a name for the "uninternal angles". I just needed something to call them for purposes of this discussion, and I intentionally chose an awkward name. I considered using "outside angles", but I didn't want people getting too comfortable with the name and then absent-mindedly swapping in synonyms like "external". You know how our brains can work -- outside, external, same thing. That's also partly the reason I color-coded the names.

[bmonk]

It's nice, though. And fits my thoughts on the subject. The formula for the angles of a polygon is not real obvious or simple to my mind, but going by the uninternal angles makes it very simple: the sum is 720° or τ, no matter how many angles are involved.

[josephlindenberg]

Thanks for pointing out the problem with me not explaining the diagram, Mike. I've rewritten that n-spheres section, and I added a better introduction of the diagram.