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New one (currently under development): cosmicog.github.io I was kinda okay with bad mobile support but not supporting .svg images is what made me switch. Even the previous post Algebraic Proof That Higher Dimensions Should Exist is incomplete. Check the new website for the comple version.

This is a Clifford torus. In geometric topology, it is the simplest and most symmetric flat embedding of the Cartesian product of two circles \(\color{#6ab825}{S^1_a}\) and \(\color{#6ab825}{S^1_b}\) (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. A normal torus is basically a a tube shape that looks like a doughnut or an inner tube, created by revolving a circle in the \(\color{#6ab825}{3rd}\) dimension around a circle, which is why a torus is a "surface of revolution. Here🔗  is a link to a Desmos page that you can play with it. When you change the 3D properties (\(\color{#6ab825}{X}\), \(\color{#6ab825}{Y}\) and \(\color{#6ab825}{Z}\)), you can just see the doughnut rotating. But when you play with the \(\color{#6ab825}{4th}\) dimension (\(\color{#6ab825}{W}\)), your doughnut seems to lose its 3D form, meaning, you can't 'imagine' what's going on physically. However, Before playing...

The Riddle In the 18th episode of the season 2, Capt. Holt presents this riddle: There's an island with 12 men, one of them is slightly lighter or heavier. You have to determine who has the different weight, using a seesaw. The exciting catch? You can only use it to measure three times. Anonymous Here's a 12 balls version and multiple answers I've found: Twelve balls and a scale question on puzzling.stackexchange.com . There are top 2 answers, the most upvoted one is creatively brute-forcing it. Numbering the balls and weighing them via following order: Weighing 1: 1 4 8 12 / 5 7 10 11 Weighing 2: 2 6 9 11 / 4 7 10 12 Weighing 3: 5 8 9 10 / 3 6 11 12 And then, there are 24 different outcomes, which forces you to map all possible solutions if you wanted to prove it with a computer code: ==L : 3L L== : 1H R== : 1L ==R : 3H L=L : 8H R=L : 5H =L= : 2H L=R : 5L R=R : 8L =LL : 9H LL= : 7L RL= : 4L =LR : 6H LLR : ...

TWAN   (The World At Night)   Guest Gallery  is open to photographers from around the world with outstanding nightscape images. Thingvellir Aurora ( Gísli Már Árnason ) From the photographer: "A historic site in Iceland, known for the Althing where the parliament was formed in the 10th century." Thingvellir, Iceland Sheet Lightning ( Simone Cmoon ) From the photographer: "It was a hard and long tour on and around the Gorner Glacier but the view was absolutely worth it." Gorner Glacier, Switzerland Moon Venus and Jupiter Alignment ( Dario Giannobile ) From the photographer: The Moon alignment with Venus and Jupiter in front of the Summer Milky Way that was rising on the horizon." Siracusa, Italy Lanzarote Winter Night ( Guido Lettieri ) From the photographer: "The sky above the volcano in a winter night of Lanzarote, Canary Islands." Canary Islands, Spain Enchanted Place  ( Francisco So...

Image Credit:  Event Horizon Telescope   Collaboration Explanation:  What does a black hole look like? To find out,  radio telescopes  from  around the Earth  coordinated observations of  black holes  with the largest known  event horizon s on the sky. Alone, black holes are just  black , but these monster attractors are known to be surrounded by glowing gas. The first image was released yesterday and  resolved  the area around the  black hole  at the center of  galaxy M87  on a scale below that expected for its  event horizon .  Pictured , the dark central region is not the event horizon, but rather the  black hole's shadow  -- the central region of emitting gas darkened by the central black hole's gravity. The size and shape of the shadow is determined by bright gas  near the event horizon , by strong  gravitational lensing deflections , and by the black hole's spin....

I ( Dr. Carol JVF Burns , actual owner of this page) prepared this page to thoroughly familiarize myself with the $\rm\TeX $ commands that are available in MathJax, and to provide a resource that may be useful to other MathJax users. Davide Cervone, the lead developer of MathJax, has most generously provided extensive edits, and this page is greatly improved due to his efforts; I owe him countless thanks. All mistakes on this page are my own (and I welcome suggestions and corrections). Please contact me via the contact form on my homepage . MathJax allows a syntax modeled on both $\rm\TeX $ and $\rm\LaTeX $. Therefore, web authors can use familiar and concise commands when creating mathematics with MathJax. Know the shape of a character that you want, but not its name? Draw it here! symbols # indicates numbered arguments in definitions ...