A Beginner's Guide to the Fourth Dimension

we hear it all the time that our world is three-dimensional it makes sense after all we can only define up to three directions each at a right angle to each other the Dune interact with each other no matter how far left or right you go you won't move up at all but the moment we try to add another dimension into the mix it all breaks down no matter how hard you try in our three-dimensional universe you can't lay out four or more directions that will interact now I know what some people are thinking can we consider time the dimension in simple terms it does make sense we can measure time and until one point in time from another in a linear sense four hours later is always four hours later just like how four feet to the left is always four feet to the left hey you could even make time act like a spatial dimension take a flipbook each frame of the animation is a two-dimensional image but by traveling through the 3ds stack of these two-dimensional frames at a consistent rate we can make time act like a spatial dimension where heighten the stack of 2d frames is the same as time between two events in that sense this 2d animation just became 3d right not necessarily just because time can match up with a spatial dimension doesn't mean that it is one notice how in the animation time only moves some way from the bottom of the stack to the top but in regular 3d space one can move either way up or down this doesn't mean it's wrong to imagine time as a spatial dimension since it clearly can be done but rather it's wrong to assume a spatial dimension in time are exactly the same so now that that's cleared up what would four dimensional space be like the easiest way to answer that question is actually to do the opposite rather than how to dimension in let's take a dimension out try to picture what it would be like to live on the surface of a sheet of paper you can no longer jump up or down off the paper because up and down don't exist to you anymore you can't look up off the paper either all you can do is look forward across the paper when you do look in front of you you'll see a horizontal line and in that line you'll see the edges of other 3d figures imagine a circle in a box clear away to us as three-dimensional beings we can see the circle in the box but to a two-dimensional beam the size of the box would prevent anyone from seeing inside also consider that whenever something casts a shadow on to say a wall the shadow forms a line on the wall of course to you all the walls will be lines anyways so you wouldn't really notice let's bring that same situation into three dimensions as you look around all you can see are two-dimensional objects just like how even the Box in two dimensions look like nothing more than a line since you can only view it from one side at a time the front of the box on three dimensions only looks like a square also suppose the ball was still in the box now its sphere inside a cubic box we still can't see it now because all the sides are still covered the shadow the box casts on the wall looks different instead of just being a lot in like before now it's a flat 2d shape if you wanted you could even trace it on the paper and get a 2d drawing out of it here's where things get tricky let's try to add a new dimension that we haven't ever seen before let's just dip our toe into the four dimensional water and use the differences between 2d and 3d space to determine what would happen in 4d space one of the first things you would notice have you looked in a 4d space is that you would see the world as a 3d shape in a similar way to how you view a 3d world and can put it onto a 2d picture but there's one catch you can see all the sides of a shape remember how if you were to put a circle on a 2d box you could still see it looking down from above in 3d well now in 4d you can see every side of the 3d box when looking down on it including seeing the ball inside are you freaked out yet after you get over the bit of Wonder you experience being able to see inside of the box you might also notice that another forty box is casting a shadow on the wall and the shadow it's a 3d shape yes just like how to these spaces linear shadows 3d space has to be shadows here in 4d space we have three-dimensional shadows but one of these shadows look like here's an example this is called the tesseract which is the shadow you would get from casting light through a 4d cube notice how it's made of two cubes connected at the vertices just like how you can draw a cube on paper from two squares and connecting the corners now the tesseract looks weird enough sitting still but watch what happens the moment you start rotating I'll give you a moment to your mind which is most likely at this point exploded yes that's what the shadow of a 4d cube looks like when they keep rotates why does it do this where the middle cube stretches out to become the outer cube let's take a look at the shadow of a 3d cube first note the two squares one is smaller than the other because of perspective making further objects appear smaller but if you follow the small square you'll see that it moves to the other side before eventually moving to the front of the key and becoming the big square now watch the tesseract again the little cube because of 4d perspective when rotated to be closer becomes bigger well the big cube rotates to be smaller so just like in a 3d cube where the big and little squares switched places in a 4d cube the big and little cubes will switch places all in all four dimensional space isn't all that bad in fact with the proper understanding of what will happen one can devise some clever shapes that do things we can't even imagine doing the 3d space ever hear of a mobius strip it's a plane that thanks to being bent around in 3d space has only one side because what would normally be the top and bottom sides are not connected making them the same things to a little 4t trickery a figure called a Klein bottle can exist where the inside and outside are the same ever notice how you can mere writing on the page by flipping it over in 4d space you can mirror a whole 3d object just by flipping it over four dimensionally 4d space shows us what is arguably the best part about mathematics it works regardless of your bias in a previous video I showed how we could use two digits to represent any value that 10 digits came in a similar way just like how we can do any mathematical function possible in base 10 in base 36 we can have geometry that functions consistently whether it has three dimensions four dimensions or possibly even more

Channel: The Science Elf

Views: 3,216,721

Rating: 4.8262467 out of 5

Keywords: 4d, math, animation, dimensions, geometry, education

Id: j-ixGKZlLVc

Length: 6min 21sec (381 seconds)

Published: Thu Jun 30 2016

Reddit Comments

[u/lostvanquisher]

additionally here's a video from 3blue1brown that goes into more detail.

https://youtu.be/zwAD6dRSVyI

[u/coupdeturnt]

Murph!!!!

[u/[deleted]]

Or if we're gonna do this, lets at least do it with the best.

https://www.youtube.com/watch?v=N7K5KjOdLD8