# Change et Mada Mada Dane

A concrete reflection in combinatorics
Selected screenshots, some accidental, some intentional

Change et Mada Mada Dane
6/23/17, while showering in Grenoble, France

“As you can clearly *see*, the visual cortex is the largest system in the human brain.” – reality 😦

“But, but, but… Pontryagin was blind! The best perspective is the missing one, not necessarily the visual one”, felt the blind pattering droplets on a sunny day. “Shall we optimize for eyeballs and clicks, or for a deeper cause?” The naive fish swam one, five, four more times through the (current) https://www.expii.com/map/752 (current), Finning in Blue yoga, hastefully tagging: East Side, skepticism, neuroscience, ideaflow, expander graphs, traveling salesman, planarity testing, mapping research ideas (Sarnak: there’s a small number of fundamentally distinct good ideas), ergobrain, following convictions, video-game-like flow/creativity while editing maps (plush & nuggets, serious games), data structures/representation (spreadsheets and graphs are both 2-d visualizations)…

Eventually, the fish fell asleep, charging for a cloudier day.

(loop ensues)

Here’s a compilation of top-level views of the Geometry map (the fruit of an on-and-off collaboration over the last 3-4 years with several other team members). It’s not complete (I didn’t think to document progress through screenshots until around 2015-09-03), but anyways…

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Throughout, the degree of the synthetic-analytic divide varies a bit.
The key organizational players in that respect are the “Area” (metric) group and the “Similarity, altitudes, and/or slopes” group, whose contents and relations change over time.

1/28/17 might be the best for an efficient, pseudo-rigorous transformations approach to Congruence & Similarity.
(For example: negatively-oriented congruence rules “derived” from positively-oriented rules using “reflection-is-isometry” as axiom.
Or as a trickier example: SSS and HL, the least obvious congruence rules, explained/”proven” using Pythagoras.)

6/21/17–6/23/17 might be the best views for speculating topic placement/flows beyond the scope of HS geometry: say, regular polytopes, algtop, linalg (several possible directions/flavors), more transformations, non-euclidean geo, diffgeo, alggeo (real, complex, or otherwise).

The two 6/30/17 views have the same underlying map data.

The overall most “precise & honest” dependency graph is probably around 7/1/17 or 7/9/17, however it’s definitely too scary for public navigation in 2017. Tradeoffs…

Ultimately we prioritized friendliness over absolute rigor (which is rather annoying to achieve synthetically, for sure), but one fun note is that once you have the Pythagorean theorem (either proven in Triangle Types, Similarity, or Area fundamentals), the order of everything else from then on starts to matter much less (because you can efficiently understand the hardest congruence rules, SSS and HL, using Pythagoras—thus completing the congruence/similarity rules, and also giving the most fundamental computational tool).
In early versions (even over a year before the first map shown, 2015-09-03), I think I tried to put Pythagoras as early as possible. One difficulty with Pythagoras is that there doesn’t seem to be a “positive” proof (i.e. proof without using negatively-oriented similarity and/or subtracting areas), so it’s tempting to talk about properties of reflections, perpendicular bisectors, and/or isosceles triangles in a nonstandard way (which can conflict with desirable top-level groups like “Triangles” or “Special Lines & Centers”).

Anyways, here are some deeper views (top two levels + epsilon).

2017-07-19 Geometry (moderately precise version):

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2017-07-19 Geometry (closer to current map, except with advanced circles stuff included in epsilon):

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2017-07-24 Geometry (closest to current live map at https://www.expii.com/map/752, except with advanced trig stuff included in epsilon):

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In the last slide, there’s a list of non-HS geometry topics siphoned off into https://www.expii.com/map/5525 (Advanced Topics) for now… together with how one might use or reunite those topics in the future:

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# Personal website, and sharing (esp. learning) resources

I’m more or less cross-posting from my Art of Problem Solving (AoPS) blog.

In particular, my personal website has a Dropbox link to some of my handouts, notes, etc. as well as other googled ones.* (The link will break if I accidentally move files around, so let me know if that’s the case. But I’ll try to at least keep it updated on my website.)

I’ll probably add some math notes once I figure out how to do $\LaTeX$ to WordPress conversion.

*On this note, I wonder how difficult it would be to set up a centralized global repository of (say) learning resources, perhaps organized via labels/tags (similar to Gmail inbox labels/“folders”—actually I think “multiple label” capability in general would be quite helpful; EDIT: apparently Google Drive allows thisjust bad UI). Actually, this is perhaps sort of the idea behind the selected papers network (see homepage and Baez’s and Gowers’ blog posts). (I suppose Google already does this to some extent, but…)

This is only one way that better communication and collaboration could help everyone out a lot. For an example about how to improve learning via contests, see my comment under Evan Chen’s recent blog post. (The point is that contests are (ostensibly) heavily “problem-solving/creativity”-based, but I think it would be easy for authors to offer extremely helpful insights behind their problems (story, background/context, interconnections, etc.), which might then spur an overall healthier contest environment, etc.)