In 4d

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