The Holiday Box has evolved! Now you can support Vsauce, your brain, Alzheimer's research, and other YouTube educators by joining THE CURIOSITY BOX: a seasonal delivery of viral science toys made by Vsauce! A portion of all proceeds goes to Alzheimer's research and our Inquisitive Fellowship, a program that gives money and resources directly to growing science channels here on YouTube! null SOURCES AND EXTRAS: my twitter: http://www.twitter.com/tweetsauce my instagram: http://www.instagram.com/electricpants Gabriel’s horn: https://en.wikipedia.org/wiki/Gabriel%27s_Horn http://blog.plover.com/math/gabriels-horn.html Gabriel’s wedding cake: https://oumathclub.wordpress.com/2015/01/25/gabriels-wedding-cake/ PDF: http://people.emich.edu/aross15/math121/misc/gabriels-horn-ma044.pdf Supersolid PDF: http://people.emich.edu/aross15/math121/misc/love-1989-supersolids.pdf Zeno’s supertask paradoxes: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes https://math.dartmouth.edu/~matc/Readers/HowManyAngels/SpaceTimeMotion/STM.html https://www.youtube.com/watch?v=u7Z9UnWOJNY Staccato Zeno runner with no velocity/acceleration discontinuities: https://goo.gl/2mYdcT General supertask reads: http://personal.lse.ac.uk/robert49/ebooks/PhilSciAdventures/lecture25.html http://plato.stanford.edu/entries/spacetime-supertasks/ http://mathpages.com/rr/s3-07/3-07.htm https://math.dartmouth.edu/~matc/Readers/HowManyAngels/SpaceTimeMotion/STM.html https://en.wikipedia.org/wiki/Supertask http://www.vordenker.de/gunther_web/achill1.htm Big book of collected supertask writing: https://books.google.co.uk/books?id=0AzP9WLLJLcC&dq=order+type+of+the+integers&source=gbs_navlinks_s PDF’s about supertasks: http://pitt.edu/~jearman/EarmanNorton1996a.pdf http://www.interciencia.es/infinity/thomsonlamp.pdf http://arxiv.org/pdf/1309.0144.pdf http://www.pitt.edu/~jdnorton/papers/onjpl.pdf http://personal.lse.ac.uk/ROBERT49/teaching/ph103/2013-2014/pdf/EarmanNorton_CommentsOnLauraudogoita.pdf http://arxiv.org/pdf/1503.05847v1.pdf http://www.phys.nthu.edu.tw/~pep/download/Drawer/187.pdf http://arxiv.org/pdf/math/0212047.pdf Plank scale: https://en.wikipedia.org/wiki/Planck_scale http://astronomy.swin.edu.au/cosmos/P/Planck+Units https://www.reddit.com/r/explainlikeimfive/comments/1oetkk/eli5_why_is_a_planck_length_the_smallest_possible/ Hyper computation and supertasks: http://georgschauer.com/2013/04/03/overcoming-constraints-on-reliability-in-nomologically-accessible-m-h-relativistic-hypercomputers/ http://www.hypercomputation.net/download/1996a_hogarth.pdf Thomson’s Lamp: https://en.wikipedia.org/wiki/Thomson%27s_lamp http://alexanderpruss.blogspot.ro/2009/05/thomson-lamp.html Benacerraf on supertasks: [PDF] http://joelvelasco.net/teaching/hum9/benacerraf62-supertasks.pdf Ordinal numbers (omega and omega+1): https://en.wikipedia.org/wiki/Ordinal_number http://www.sjsu.edu/faculty/watkins/ordinals.htm http://math.wikia.com/wiki/Ordinal_number Ross-Littlewood paradox: https://en.wikipedia.org/wiki/Ross%E2%80%93Littlewood_paradox https://angryfaic.wordpress.com/2011/02/25/four-paradoxes-involving-infinity/ New York Times Neanderthal genome article: http://www.newyorker.com/magazine/2011/08/15/sleeping-with-the-enemy Neanderthals: https://en.wikipedia.org/wiki/Neanderthal http://ngm.nationalgeographic.com/2013/01/125-restless-genes/dobbs-text https://rogerdhansen.wordpress.com/2011/08/21/the-wandering-gene/ and boats: https://www.newscientist.com/article/mg21328544.800-neanderthals-were-ancient-mariners/ Another fun infinity paradox: http://lesswrong.com/lw/isp/on_the_importance_of_taking_limits_infinite/ SMBC Zeno comic: http://www.smbc-comics.com/?id=2759 MATH! null null null null music from: null and http://www.audionetwork.com/