4th Dimension Explained ► Tesseract Hypercube Visualized More Than 4 Ways

856,674 views Jan 28, 2021 #tesseract #1D #2D

Introduction to the 4th dimension, where we visualize a tesseract (in the family of hypercubes) by building from 0D to 1D, 2D, 3D, & 4D. Many fourth dimensional illustrations are utilized for visual learners. We also share examples from Flatland, how a 2D object views 3D shapes. 🚩 / carlosfarias1D, we have 2 0D points 2D, we have 4 1D lines 3D, we have 6 2D squares 4D, we have 8 3D cube 0D to 3 4th Dimensio The Hypercube: Projections and Slicin 4 Pieces of Art The Channel Trailer • Channel Trailer for People Who Have n...Emergence Theory Defined (A Layperson's Guide) ► Bird Murmurations & Your Body • Emergence Theory Defined (A Layperson...Joseph Campbell POWER OF MYTH ► Mythology, Hero's Journey, Monomyth, & Follow Your Bliss • Joseph Campbell POWER OF MYTH ► Mytho...Dream Theories Explained ► Facts, Psychology (Freud/Jung), & Lucid Dreams • The Mystery & Science Behind DreamsDavid Foster Wallace WORSHIP ► Commencement Speech Clip (This is Water audio) [ • David Foster Wallace WORSHIP ► Commen...](https://www.youtube.com/watch?v=-AK87zdMzRI&t=0s)1984 vs Brave New World ► Neil Postman's Amusing Ourselves to Death [Orwell & Huxley] • 1984 vs Brave New World ► Neil Postma...flow state by MIHALY CSIKSZENTMIHALYI ► The Secret to Happiness & Psychology of Optimal Experience • flow state by MIHALY CSIKSZENTMIHALYI...🎥 Watch Further Your Mind Is Eight-Dimensional - Your Brain as Math Part 3 | Infinite Series • Your Mind Is Eight-Dimensional - Your...The Simpsons, TreeHouse of Horror VI (1995) Part 1: • The Simpsons - Homer 3DPart 2: • The Simpsons - Homer in Real LifeInterstellar (2014) Lady Gaga's 911 (2020) • Lady Gaga - 911 (Short Film)🚾 Works Cited Intro song from "The Fourth Dimension" movie, actually a collection of 3 short films by Harmony Korine, Jan Kwieciński, and Aleksei Fedorchenko. It's weird but good. Val Kilmer really swings for the fences. • The Fourth Dimension Official Trailer..."Drawing the 4th, 5th, 6th, and 7th dimension", Physics Videos by Eugene Khutoryansky. (2012). Creative Commons Attribution license (reuse allowed). • Drawing the 4th, 5th, 6th, and 7th di..."Exploring other dimensions - Alex Rosenthal and George Zaidan", TED-Ed. (2013). Animation by Cale Oglesby. • Exploring other dimensions - Alex Ros...By Vitaly Ostrosablin - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index...By A2569875 - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index...By CrypticGamma - Orthogonal Tesseract Gif, CC BY-SA 4.0,https://commons.wikimedia.org/wiki/Fi...By Spiritia - Hypercube-construction-4d, Public Domain,https://commons.wikimedia.org/wiki/Fi...By JasonHise 2007 - 8-cell-simple, Public Domain, https://commons.wikimedia.org/wiki/Fi...By JasonHise 2007 - 8-cell, Public Domain, https://en.wikipedia.org/wiki/File:8-...Tom Moody (2007) http://www.digitalmediatree.com/tommo...https://celine.thejaida.com/2019/12/1...Banchoff & Strauss. "The Hypercube: Projections and Slicing" (1978) • "The Hypercube: Projections and Slici...http://matematita.science.unitn.it/4d... (2007)https://commons.wikimedia.org/wiki/Fi...Dali, Salvador. "Crucifixion (Corpus Hypercubus)" 1954.https://en.wikipedia.org/wiki/File:Da...©1999 Artists Rights Society (ARS), New York / VEGAP, Madrid 🚨 Legal Copyright Disclaimer under section 107 of the Copyright Act 1976, allowance is made for “fair use” for purposes such as criticism, comment, news reporting, teaching, scholarship, education and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. 🚀 What is this channel? Exploring Truth in philosophy, science, & art. We'll uncover concepts from psychology, mythology, spirituality, literature, media, and more. If you like Lex Fridman or Curt Jaimungal, you'll love this educational channel. Inspired by psychologist Jerome Bruner, the syllabus will circumambulate core topics; the spiral curriculum will regularly revisit ideas, building incrementally on earlier foundations. The curriculum may look scattered now, but eventually themes will converge and the giant map will make sense. The goal is a better understanding of life, the universe, and everything. Maybe together we can get somewhere closer to humanity's grand search for meaning and chip away at life's greatest mysteries. -CF p.s. Please subscribe, young channel! =)#4thdimension #4D #tesseract #hypercube #3D #2D, #1D #0D #flatland #cubes #fourthdimension #4Dexplained #carlsagan #4Dvisualization

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[Music]

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next we cover

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with dimension did i mention

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the fourth dimension did i mention

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the fourth dimension this

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is probably the simplest

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four-dimensional object

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also known as a tesseract part of the

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family of

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hypercubes it might look bananas to you

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right now

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but trust me it's legit we're gonna

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build your understanding of it with

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multiple examples and we're starting

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with the lowest dimension

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the zeroth dimension so at zero d

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we have just a single point no length

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height or width now if we double these

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zero d

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points we have one dimension and we now

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have

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two points connected by a line

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and bounded by length for the second

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dimension we're getting an assist from

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professor frink

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here is an ordinary square whoa slow

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down egghead so we double the zero d

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points again and we have the second

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dimension

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now we have a square bounded by length

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and height

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notice we have four points as well as

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four lines

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onto the third dimension we double the

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zero d points again

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now we have a cube bounded by length

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height

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and width notice we have eight points

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and six squares

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observe that in each dimension we are

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adding two

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sides of the lower dimension to form a

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new shape

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one d we have two zero d points

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two d we have four 1d lines

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3d we have six 2d squares

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before we go up a dimension i want to

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share a couple visuals of how a

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three-dimensional object would appear to

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someone

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in a two-dimensional world imagine the

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red cube is a real

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3d object the red dot flying around

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is a light source now you can visualize

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how the shadow of the 3d object

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would appear on a 2d plane

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the next example is even better and

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inspired by the 1884 book

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flatland by edwin abbott abbott

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i guess his parents decided to square

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his name

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imagine a 2d shape in a two-dimensional

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world

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as visited by a three-dimensional cube

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on the left is a visual from the cube's

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perspective penetrating the 2d plane

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on the right is a bird's-eye view of the

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cube's entry

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and exit from the 2d world and the best

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illustration is the view of a sphere

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passing through a 2d plane

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from a flat square's perspective

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the bar at the top of the screen shows

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you what the flat square

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sees cool right

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[Music]

4th Dimension

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back to the three dimensional object

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double the zero d

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points again and now we have a hypercube

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bounded by length height width and a

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fourth dimensional axis

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which we'll call the w axis

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it's very difficult to wrap our

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three-dimensional brains around an

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object a full dimension above us this

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tesseract has

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eight sides which are all cubes so

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to review 1d we have two 0d

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points 2d we have four 1d

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lines 3d we have 6 2d

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squares 4d we have 8 3d

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cubes well not just the cubes but we had

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chris daughtry jeff probe super chef

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bobby flay i mean it was insane

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every dimension we're going up by two

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sides

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and each side is made up of the

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dimension below us

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this needs extra emphasis every

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dimension

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simply adds two sides of the lower

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dimension

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a few more examples imagine we have two

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cubes

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one red and one blue they are connected

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to each other via the points we doubled

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from the third dimension

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it almost looks like they envelop each

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other all the while maintaining

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eight 3d cubes

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now here's a suspended tesseract we can

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count each of the eight color-coded

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cubes

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uno dos

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remember the flatlanders from before

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well we have a visualization of how a 4d

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object

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passing through a 3d plane would appear

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to us in 3d it's probably not as

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intuitive as the flatlanders

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because our brains are three-dimensional

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though as an aside there's a real theory

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our minds are actually

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eight-dimensional it is muy complicado

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i'll link to it in the description for

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those eggheads out there

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i find it helpful to see the progression

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from lower dimensions up to 4d

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so here's one good example and the

The Hypercube: Projections and Slicing

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second

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much longer example comes from a 1978

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award-winning computer animation

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the hypercube projections and slicing

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this was made by thomas banchov and

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charles strauss

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at my alma mater brown university

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we start with a square which is really a

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head-on view of a cube

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which begins to rotate our first

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impression of a rotating 3q

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is that of squares sliding this is the

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revolving door illusion

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if we rotate the three-dimensional cube

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about a different axis we get

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parallelograms and rectangles

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images that we readily interpret as the

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shadows of a rotating three-dimensional

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cube

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but now we rotate in four-dimensional

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space

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our first view of a rotating

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four-dimensional cube gives us the

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impression of three-dimensional cubes

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sliding

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the revolving door illusion in one

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higher dimension

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if we stop the four-dimensional rotation

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and revolve around an axis in

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three-space

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we get something which resembles a

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wireframe model like the frame of a box

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kite

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but now we rotate in a different plane

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in four dimensional space

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this is our first really good view of a

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four-dimensional cube it has

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16 corners eight on the red cube

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eight on the green cube with

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corresponding corners connected

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but as we rotate in three space we see

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that one three-dimensional face is still

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quite flat

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we rotate again in four dimensional

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space this time leading to truly general

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position

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at almost any instant the four edges

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that come out of one of the corners

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have no three lines lying in a

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stopping our four-dimensional rotation

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we show a three-dimensional spin around

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an axis

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where we see the red cube and the green

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cube interlaced

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at the end of our four-dimensional

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rotation we fold all the way back

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come back to three space continue back

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to the plane

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and end at a square

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viewing objects in three dimensional

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space we are aided by the devices

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of perspective as we rotate a three cube

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in perspective

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we see each of the six bounding squares

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as the smallest furthest away square

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and then again as the largest closest

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square

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this is not the top view of some

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two-dimensional creatures swimming in on

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itself

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it's a rotating three-dimensional cube

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and we are quite used to these

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perspective distortions

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but now we go to four space

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we change our perspective

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and we proceed to walk around that

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four-dimensional cube

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as we walk around we see each of the

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eight cubicle faces

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as the smallest cube when it's furthest

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away or the closest cube

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when it's largest this is not a

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three-dimensional creature swimming

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through itself

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it is the shadows of a rotating

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four-dimensional cube

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this is the best rotation with the red

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cube now largest

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now flattened out now turned inside out

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as the smallest furthest away cube

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and if we become comfortable with

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rotations every once in a while we see a

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rotation which confuses us once more

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rotating directly toward us and away

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from us

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we find another very strange view of the

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rotating

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four-dimensional cube in perspective

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at the end we go back and then at our

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three-dimensional cube

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i also want you to see this

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visualization of what's called the

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net of a hypercube if you spread out the

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tesseract

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you get its net as an arrangement of

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eight cubes

4 Pieces of Art

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and before we depart four pieces of art

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the first is the 1954 painting

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crucifixion

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corpus hyper cubists my surrealist

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salvador dali

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i could spend a whole video on this

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striking image alone

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fun fact it was iron rand's favorite

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painting she'd spend hours at a time

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admiring it at the met the second is the

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2014 film

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interstellar in which christopher nolan

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goes full m night shyamalan

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and dazzles us with one of the best

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representations

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of the fourth dimension in motion

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picture history

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season 7 episode 6 of the simpsons aka

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treehouse of horror 6. the third act

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finds 2d homer in a 3d realm

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it's only 8 minutes well worth the watch

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and it's linked in the description below

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cool man fourth and also linked below

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one of my favorite music videos of all

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time lady gaga's 9-1-1

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i'll explain why another time

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lastly a huge thanks to all the creators

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of the technical videos and gifs used in

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this video

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thank you for making them reusable via

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creative commons and fair use

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sources linked below thank you so much

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for watching

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[Music]

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like and subscribe