How to explain 4 Dimensions
A Beginner's Guide to the Fourth Dimension
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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
