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1 point

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8th May 2019

Further to this, perhaps the most well-known analogy was neatly illustrated in the book Flatland.

Check it out if you haven’t. The book considers the perspective a two-dimensional object on a two-dimensional plane surrounded by three dimensional space and objects. From that perspective, objects in z-space that do not intersect the xy-plane cannot be “seen” on the plane. Moreover, their intersection with the plane, while perceptible, is not perceived as a fully three-dimensional object. So, for example, a sphere that intersects the plane is perceived on the plane as a circle with a diameter equal to that of the sphere’s great circle at the intersection.

1 point

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8th Oct 2017

No, Monster's inc. isn't a good portrayal of 4 dimensions. It's where one three dimensional manifold meets another along a plane (the door).

Using his two D example, it's like two planes intersecting along a line. A Square, the main character from Edwin Abbot's Flatland might go from one plane to another by going through the common line. But he wouldn't really be moving around freely in 3 dimensions.

As usual, Tyson is B.S.ing on topics he knows only a little about.

1 point

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2nd Jan 2017

The Carl Sagan video below is also pretty good. In fact, he and I used the same reference Edwin Abbott's *Flatland*, a book written in 1884. When Sagan cuts the apple, that's the same as the cheese slicing conveyor belt.

You can pick up a copy of Abbott's *Flatland* for pretty cheap. It's easy to read and is pretty short.
https://www.amazon.com/Flatland-Romance-Dimensions-Thrift-Editions/dp/048627263X

1 point

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26th Sep 2016

and extrapolate to 3 dimensions. You'll have a great understanding, I promise, and it's fun to read. I'm assuming here you're wanting an expression of a 4th SPACIAL dimension, and not an exposition on "time as a 4th dimension of spacetime."

Think of a safe in 2 dimensions...a 3 dimensional person can hover OVER the safe and see everything that's in it. That same person could pluck an item out of the safe with ease. The 2 dimensional person would crap themselves when they opened the safe only to find that object mysteriously missing.

I doubt there are 4 dimensional people who can look into our safes and steal stuff, because, well, they haven't so far. Unless you count my socks that are constantly being stolen out of my dryer.

1 point

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18th Sep 2015

As I understand it, it's not really accurate to say that time is the 4th dimension.

Time is a temporal dimension but in terms of spacial dimensions there is a 4th that is separate from time.

The 4th dimension is impossible for us to completely comprehend because we live in a universe (as far as we can see) with only 3 dimensions, just like a circle living in a 2-dimensional 'flat-land' universe, would not be able to comprehend a sphere.

As mentioned already on this thread, the book flatland describes this concept in a clever and fascinating way. This book blew my mind and I reread it maybe once a year (its a small book). Written in sort of old English so maybe not for everyone but I recommend it.

4 points

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6th Oct 2013

I've been exploring this recently. I'm not an expert, but I'll do my best to explain it.

The shape or object represented in the gif you posted is called a **tesseract** or a **hypercube**. You can search for these terms for more information.

To explain this, some basics about 2D and 3D must first be established to understand how to continue the explanation to 4D.

A super-brief explanation of the gif above as the four dimension object (spatially) is that it is a *representation* or *projection* of viewing a 4D object/shape in a 2D view. (That gif as displayed on our computer screens is 2D because our screens are 2D and it's not encoded as 3D to be viewed with 3D glasses) and a 4-D object or shape actually appears to us to be 3D objects inside of 3D objects, just as if we look at a 2D object - say a square drawn on a piece of paper - we are able to see inside of the 2D object and see additional objects drawn inside of it and just as we are only able to draw a 3D object on a piece of paper if it is drawn as a transparent outline, this gif shows the 4D object drawn as a transparent outline in which we only see the many sides folding in and outside of itself. A being that is capable of seeing four spatial dimensions would be able to look at you and see inside of you. The following demonstrates this concept pretty well:

**Fourth Spatial Dimension 101 (video, 6:27)**

To better understand the concept of the fourth dimension, read on. I also included more videos below, including an excellent one by Carl Sagan.

First, some facts / definitions:

0D (zero spatial dimension) is simply a point. It either exists or does not exist. There is no concept of a point moving in 0 dimensions because there is no space for it to move.

1D (one spatial dimension) is simply a line. It has length. A point can move along the line from side to side, left or right.

2D (two spatial dimensions) is a plane. It has length and width. A point can exist and/or move from side to side lengthwise and side to side width-wise, left or right, and (if we imagine the plane as a flat surface that's level to the ground,) then we can call the width direction either forward and back, if we imagine looking at the plane on a wall, we might call it up or down. Either is fine. Two dimensions.

3D (three spatial dimensions) is technically called "3-dimensional Euclidean space" but since it's what we commonly perceive, we often just refer to it as "space." It has length and width and height. Other words can be used for these directions, as long as it's three separate directions not in the same plane, such as left-right, up-down, and forward-back.

4D (four spatial dimensions) is known simply as four-dimensional space, probably because we don't use it in conversation enough to have a nifty, shorter term for it. There is also a non-spatial version of four dimensions commonly referred to as "spacetime" which is a combination of 3D space and time.

A special note about the fourth dimension... Space vs time as a fourth dimension are differentiated as such:

*time as the fourth dimension*is referred to as the Minkowski continuum, known as spacetime, and the*spatial-only dimensions*are referred to as Euclidean space or dimensions. Spacetime is not Euclidean space; it is not only spatial. (*The gif you linked above is a representation of the spatial fourth dimension.*..yes, it includes time to show it rotating. If you were to consider it as a spacetime dimension then it would be 5 dimensions: 4 spatial plus time, but it is commonly referred to simply as spatial in my understanding.)

**Conceptualizing the limitations and advantages of dimensional perception:**

Beings that can perceive in 2D can see inside of objects that are 1D.

Beings that can perceive in 3D can see inside of objects that are 2D.

Beings that can perceive in 4D can see inside of objects that are 3D.

Beings that can perceive in 1D can only see representations or projections of 2D objects.

Beings that can perceive in 2D can only see representations or projections of 3D objects.

Beings that can perceive in 3D can only see representations or projections of 4D objects.

We are able to perceive objects spatially in 3 dimensions (3D). By spatially, we mean that we're interpreting the environment or world's space, and not considering the fourth dimension as something other than space, such as time. (The gif linked above is of a four-dimensional object of which the fourth dimension is also space.) When we look at a drawing of a square on a piece of paper, we are able to see not only its length and width, but also inside of it because we are viewing it from above - from height. If we look down at it and draw a triangle inside of it, we can see both at the same time. We are able to see inside of 2D objects. A 3D object is comprised of several layers of 2D objects stacked upon one another. So imagine the 2D drawing, and stacking many papers on top of each other until it's several inches or centimeters tall. That's a 3D object now. Then, shape it into a square at each sheet of paper (so cut through all sheets) and you will end up with a cube of paper. Shape it into a triangle and it will be a triangular, pie-like shape. Angle it more narrow on the way up and it will be a pyramid-like shape. With any of these shapes, we cannot see inside of it. But now imagine this: just as we in the 3rd dimension looking at a shape in the 2nd dimension can see inside of it, a being in the 4th dimension looking at a shape in the 3rd dimension can see inside of the 3D object. That is because just like there is only length and width in the 2nd dimension, but no height; in the third dimension we have length width and height, but no ______. I'm unaware of whether there is a name for the additional direction that would exist in the fourth dimension.

I also don't know whether a 4th spatial dimension actually exists or is just an abstract concept, nor do I know whether it is possible or known to be possible to detect. *As far as I am aware*, the fourth spacial dimension is only known of abstractly, meaning that there is no evidence for it actually existing.

**These videos explain how to understand what the 4th dimension would look like**:

**Dr. Quantum explains the 4th dimension (video, 5:09)**

*An oversimplified explanation from the movie "What the bleep do we know: down the rabbit hole" in which the character, Dr.Quantum, first explains what an (imagined) 2D world (flatland) would look like to us - who are able to see the 3D world, as a way of understanding (or extrapolating) how a being that could see in the 4D world would be able to see through and inside of 3D objects.* (**note:** *I've been warned that this is part of a video that goes on to some cult-like recruiting, so please be forewarned about the video's conclusion and entirety.*)

**Cosmos - Carl Sagan - 4th Dimension (video, 7:24)**

*Carl Sagan explains how to imagine what the 4th dimension looks if we were able to see it and how it would allow us to see inside 3D objects. An important part of this video is explaining and showing exactly how and why we can only see a distorted version of 4D objects since we only see in 3D*

**4th Dimension Explained By A High-School Student (video, 9:05)**

*An excellent description of the first through fourth dimension and how we can perceive them.*

**Unwrapping a tesseract (4d cube aka hypercube) (video, 1:39)**

*Watch the above two videos to see how we can conceptualize a 4D object in 3D space.*

Videos mentioned elsewhere in this comment:

**Fourth Spatial Dimension 101 (video, 6:27)**

**Videos, Books and Links mentioned by other redditors**:

**Flatland: a romance of many dimensions (Illustrated) by Edwin Abbott Abbott (book, free, ~230kb)**

*Amazon description & reviews*

*hat-tip to /u/X3TIT*

*Looks interesting.*

*hat-tip to /u/karoyamaro*

*(Edited: 1- to add video lengths; 2- added book links, 3 - readability more videos, 4 - a warning about the Dr. Quantum video.)*

2 points

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3rd May 2021

“Flatland” is a very thin book and worth the read. Don’t even need to check it out of a library, you can finish it in a few hours.

https://www.amazon.com/Flatland-Romance-Dimensions-Thrift-Editions/dp/048627263X

1 point

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14th Sep 2022

There's a cool little book called "Flatland - A Romance of Many Dimensions" which describes a group of creatures living in a 2D just as you describe.

https://www.amazon.com/Flatland-Romance-Dimensions-Thrift-Editions/dp/048627263X

1 point

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16th Jul 2022

1 point

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11th Nov 2021

1 point

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10th Jul 2021

"Flatland" is a book by Edwind Abbott that preceded Carl's Sagan video about the 4th dimension in the series Cosmos by almost 100 years.

1.1) https://en.m.wikipedia.org/wiki/Flatland

1.2) https://www.amazon.com/dp/048627263X/ref=cm_sw_r_wa_awdb_imm_5A6AVE874MVWTF1Q2DNG

1.3) http://www.gutenberg.org/ebooks/201

Subsequent books have been published on this topic:

Planiverse: 2.0) https://www.amazon.com/dp/0387989161/ref=cm_sw_r_wa_awdb_imm_8ZRE5E50H4KDMSN9TGMH

Flatterland: 3.0) https://www.amazon.com/dp/073820675X/ref=cm_sw_r_wa_awdb_imm_GJDKA6CMXQ30ACD39NG6

1 point

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31st Aug 2020

1 point

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12th Aug 2017

How do you get a 3D display? As others have mentioned, computer screens are 2D. Perhaps a separate screen for each eye? There are also auditory cues to suggest 3D.

It is an interesting question. Some suggest we have a hard time imagining 4 spatial dimensions since we have no experience outside of our 3 spatial dimensions. See Edwin Abbott's Flatland as well as Dewdney's Planiverse.

A computer game world could be set in 4 spatial dimensions. Presently a lot of 4 dimensional polychora exist in digital form. Perhaps a player of this game might become accustomed to 4 spatial dimensions and have a better understanding.

1 point

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1st Sep 2016

I heartily recommend the book Flatland on the subject. Make sure you get the edition with illustrations as the others are rubbish. You can also read it for free on Project Guttenberg.

1 point

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9th Feb 2016

1 point

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25th Nov 2015

0 points

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20th Dec 2017

Distance between two objects in three dimensions is

sqrt(x^2 + y^2 + z^2 ).

Which follows in a straightforward fashion from the Pythagorean Theorem.

Distance between two points in 4 spatial dimension would be: sqrt(w^2 + x^2 + y^2 + z^2 ).

If w were zero and the rest of the terms were non zero, the two points would still be a distance from one another.

A way to look at it in three dimensions, imagine two points in the xy plane:

(0, 0, 0) and (0, 0, 1,000,000).

Regardless of the fact the two points are coplanar, they're still a million units apart.

If you're interested in worlds with other than three spatial dimensions, two wonderful stories are Flatland by Edwin Abbott and Planiverse by A. K. Dewdney.

If you don't mind a little math, Coxeter's Regular Polytopes is a great book.

0 points

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31st Mar 2016

For the love of... we had to take tests. I'm jealous. You wouldn't be able to really appreciate most of them without some mathematical training, though. I'd go for the Monty Hall problem or "Set theory and different 'size' infinities". <em>Flatland</em> is certainly still worth reading, though, and there are fun videos about the type of person who sells Klein bottles.

If you're honestly interested in the Set Theory stuff, I'll try to find you some better resources. The Wikipedia articles are probably far too technical.

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