A concrete reflection in combinatorics

Selected screenshots, some accidental, some intentional

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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 ðŸ˜¦

“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)

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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…

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”).

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

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

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

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

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