Substrate

a graph is planar if and only if its crossing number is zero.

The Four Color Map Theorem (or colour!?) was a long-standing problem until it was cracked in 1976 using a "new" method... computers! A little bit of extra footage from this: https://youtu.be/laMkuPrad3s This video features Dr James Grime - http://jamesgrime.com More Grime videos: http://bit.ly/grimevideos This video supported by Squarespace: https://www.squarespace.com/numberphile Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Subscribe: http://bit.ly/Numberphile_Sub Videos by Brady Haran Patreon: http://www.patreon.com/numberphile Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/ Brady's latest videos across all channels: http://www.bradyharanblog.com/ Sign up for (occasional) emails: http://eepurl.com/YdjL9

A counterexample to Hedetniemi's conjecture - featuring Erica Klarreich. Get 3 months of Audible for just $6.95 a month. Visit https://www.audible.com/numberphile or text "numberphile" to 500 500 More links & stuff in full description below ↓↓↓ Read Erica Klarreich's Quanta article on this subject: https://www.quantamagazine.org/mathematician-disproves-hedetniemis-graph-theory-conjecture-20190617/ And visit her website: http://www.ericaklarreich.com/ Yaroslav Shitov's breakthrough paper: https://arxiv.org/abs/1905.02167 Thanks to Stephen Hedetniemi for providing us with photos and pages from his original dissertation. Some more graph theory on Numberphile... Four Color Maps: https://youtu.be/NgbK43jB4rQ An Unsolved Problem: https://youtu.be/niaeV_NHh-o Planar Graphs: https://youtu.be/xBkTIp6ajAg Perfect Graphs: https://youtu.be/C4Zr4cOVm9g Friends and Strangers: https://youtu.be/xdiL-ADRTxQ River Crossings: https://youtu.be/ZCVAGb1ee8A Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. https://www.simonsfoundation.org/outreach/science-sandbox/ And support from Math For America - https://www.mathforamerica.org/ NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Subscribe: http://bit.ly/Numberphile_Sub Videos by Brady Haran Patreon: http://www.patreon.com/numberphile Numberphile T-Shirts: https://teespring.com/stores/numberphile Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/ Brady's latest videos across all channels: http://www.bradyharanblog.com/ Sign up for (occasional) emails: http://eepurl.com/YdjL9

A celebration of minor, automatic graciousness within a community, which can shine a penetrating light.

Mochizuki had created so many new mathematical tools and brought together so many disparate strands of mathematics that his paper was populated with vocabulary that nobody could understand. It was totally novel, and totally mystifying. “You don’t get to say you’ve proved something if you haven’t explained it,” she says. “A proof is a social construct. If the community doesn’t understand it, you haven’t done your job.”

Smarter Everyday (Destin's channel): https://www.youtube.com/user/destinws2 Makers: I Like To Make Stuff: https://www.youtube.com/user/iliketomakestuffcom Evan & Katelyn: https://www.youtube.com/user/EvanAndKatelyn Crafted Workshop: https://www.youtube.com/user/craftedmagazine HomeMadeModern: https://www.youtube.com/user/HomeMadeModern Four Eyes: https://www.youtube.com/channel/UC1V-DYqsaj764uBis9-UDug Peter Brown: https://www.youtube.com/user/kludge1977 John Heisz: https://www.youtube.com/user/jpheisz Jeremy Schmidt: https://www.youtube.com/channel/UC0T7tvy44mlQCjaTtparOZw Modern Builds: https://www.youtube.com/channel/UCIxAaCJ84uefATKmazDyIjw Science: Vsauce: https://www.youtube.com/user/Vsauce Ben Eater: https://www.youtube.com/channel/UCS0N5baNlQWJCUrhCEo8WlA 3Blue1Brown: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw Minute Physics: https://www.youtube.com/user/minutephysics Steve Mould: https://www.youtube.com/user/steventhebrave SciShow: https://www.youtube.com/user/scishow SciShow Space: https://www.youtube.com/user/scishowspace Everyday Astronaut: https://www.youtube.com/channel/UC6uKrU_WqJ1R2HMTY3LIx5Q Veritasium: https://www.youtube.com/user/1veritasium Other Great Stuff: Vlog Brothers: https://www.youtube.com/user/vlogbrothers CGP Grey: https://www.youtube.com/user/CGPGrey Homemade Wanderlust: https://www.youtube.com/channel/UCQhqmV26773qZhzqJz4VFcw Darwin onthetrail: https://www.youtube.com/channel/UC18exdGWh7piVWisrnDXiZg Andrew Huang: https://www.youtube.com/user/songstowearpantsto Rob Scallon: https://www.youtube.com/channel/UCyDZai57BfE_N0SaBkKQyXg Kitchenomicon: https://www.youtube.com/channel/UCLjhr0DXE69h6RRb98r-VAw Captain Dissolution: https://www.youtube.com/channel/UCEOXxzW2vU0P-0THehuIIeg MDS: https://www.youtube.com/channel/UCYMKd37bdxI-qj3f3z6U6_g

[Continuations generalize] the asynchronous value concept. If we plug in `R = Void` then we just get an [asynchronous] value, but if we use a non-`Void` `R` value[,] we will get a type that kind of mixes together aspects of synchronous computation and asynchronous computation

The perfect mix of function and form in a modern dive watch, and a rare opportunity to own a Japanese Domestic Model.

Look as long as you can at the friend you love, no matter whether that friend is moving away from you or coming back toward you.

and the moon is a spider tracking its white mud across the sky. We take Rome with us to Rome.

All can be measured by the standard of the capybara.

I think my limit for open-loop writing is about 14k words. For the book on temporality I’m working on now, I’ll probably serialize it online in some form before trying to put it together as a book. ​​ All in all, it was a wonderful outpouring of deep reserves of creativity and knowledge, the likes of which I haven’t seen online in a long time. ​ Twitter is where all the history-making, universe-denting social media action really is. It is as close to a pure ideas-commons/digital public as we’ll ever get. ​ Email today is now less a communications medium than a communications compile target.

If you had told me four years ago today that someone would eventually get a Blackbox tattoo I never would have believed you…

(Did you know you could play video games in the morning before work? I certainly didn’t. But you can.) I still struggle with instinct, the motivation to do the most, the feeling that I’m not a good reader if I’m not cycling through new books every few days. I want to keep slowing down, and keep giving words the time they deserve.

Math Subject GRE GRE Math Subject Test videos that provide test takers with tips and tricks to solve problems much more quickly.

Dropbox mirror: https://www.dropbox.com/s/iwpaq4k8xxue3rp/Knots_Notes.pdf?dl=0

ISSUE SIX features fiction by Christopher Higgs, Jennifer Kronovet, Kyle Minor, Mark Jude Poirier, and Maura Stanton; creative nonfiction by Priscilla Becker, Jehanne Dubrow, and Emily O’Neill; film writing by J.M. Tyree; poetry by Diannely Antigua, Sandra Beasley, Molly Bendall, Jericho Brown, Heat

US healthcare is a joke. Let's make it funny. Out of Pocket is a weekly comedic deep-dive and analysis into the rabbithole that is our healthcare system. Existing healthcare research is dry and boring, and raises questions like: Why are 75% of all healthcare conversations defining what a “digital therapeutic” is?

In January, when McMillan announced his intention to open Suckerpunch, Portland’s first entirely zero-proof bar, the response was similarly overwhelming. Within days, hundreds of people signed up for the bar’s mailing list, and a preview event at Roseline Coffee in February sold out in advance.

it’s interesting to note how powerful it can be to have an actual writer crafting narrative strategy. Articulation is the first product

The general term for any diagonal going top-left to bottom-right direction is 𝑘-diagonal where 𝑘 is an offset form the main diagonal. 𝑘 = 1 is the superdiagonal, 𝑘 = 0 is the main diagonal, and ��� = −1 is the subdiagonal. According to Mathworld, the general term for the antidiagonals seems to be skew-diagonals.

Mathematicians have proved that copies of smaller graphs can always be used to perfectly cover larger ones.

for a love that would hurt in a way she could translate into forever.