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OK, welcome to the Talk.
If you're coming in, please close the
door behind you,
please give a warm applause to Mark knows
he's going to talk about higher
dimensional 40 geometry and fractals,
which is he has an electric sheep and
other exciting pictures.
All right. Hello, everyone.
Um, so this,
as you might may have seen in
the abstract, the stock will be split
into three distinct sections.
The thing about the topic is that it's
extremely wide.
I could probably feel a lecture for about
a week, but we're not going to do that.
So I
am going to assume you roughly know how
to do prospective predictors and predict
if you do. And you might as well believe
me, that that would be good, too.
So for forty part, I'm going to I'm going
to throw some formulas in there and
how to do that.
Then I'm going to talk about some
information systems and how they
generally work.
And in particular, after
that, I'm going to talk about fractal
flames, which are a special kind of
I try a different system.
So the twist we're going to do here is
that for a system and
for effective flames, we're going to talk
about how to do those in poverty
now.
And when I say 40 interdimensional,
I mean, ah, to the power of
four and higher.
So it's not about higher
dimensional stuff as you would have it in
other fields of math.
It's more it's more condensed
into something specific.
All right. And there's going to be some
intermission slides where, uh, circa
some demo program I got like the cheesy
cereal possible.
So you can follow that along in your
browser if you feel like it.
And let's get started.
So when I said one
at three parts, I am going to tell
you what I'm skipping.
So this is
a cube.
We've all seen a cube.
Um, so
this stuff I'm skipping because we don't
have enough time is had
to do the actual matrix American
made products, which is like the
basis for most of the linear algebra
stuff you've heard of.
I'm going to skip across product and 3D
and I'm going to assume you're
off. You know what in coordinates are
if you don't, it's basically I don't no
coordinate any hyperscale.
I'm going to assume you trust me on
rotations. You basically
in three days you need a free cross-breed
matrix to have a rotation.
Um, there's going to
be a reason for being organized.
Cornets are that we want to have some
fine transformations, which is basically
the same. But we can also like translates
of, uh,
there's look at, uh, look at
transformations in jail.
So also skipping those.
But we're going to extend them later.
And there's something called perspective
transform, which
is.
Yeah. Which is basically what we saw on
the slide before, which it
reduces the number of dimensions.
So in Winfried case, you get a two day
picture, which you can sort of your
screen and.
Well, yeah. So the perspective transform
is something you do so you can learn to
do predictions.
There's going to be a lot of slides like
this.
Uh, so for example,
I'm going to skip over it, but I'm not
one of indentation like vectors in our
stuff is going to are going to be bold,
uh, vertical bars around the Matrix.
I mean, that that we're going to do
determinant.
Uh, and
when we have when we have rotation
matrices, we can just extend
them to be so you can multiply
them with
a fine transformation matrices.
So you just add another row
and you add another column,
all zeros except for Lessel.
So it's going to be interchangeable.
So just just a heads up
and this is a matrix we're also skipping.
I'm just throwing that here because it
looks kind of intimidating.
It's documented in your own
jail manual. So,
yeah, just throwing
it out there.
OK, and
Mexican-American anyway.
So now part one, um,
you guys are going to believe you can do
prospecting for Texas and Freddy,
probably because you've been playing golf
duty or something at some point.
So, so
we can do that in 40.
And this, for example, would be
a tesseract or a hypercube.
So in order to render this, we need
to do some math to take our 40
object, in this case, Tesseract, and
reduce it to a 3D object, which we
can then turn into a two
object.
And if we want to do it in five days,
it's going to be the same thing.
So again, what do we need for that?
It's basically the same list
shit.
But and it gets worse because
we don't actually have a cross product
for the damnit.
Well, we don't need a cross product.
We just need normal's.
And so I'm just going to do it different,
which is a lot shorter.
OK, so we want to do Normal's
I was I was just showing him a slide from
across product and it's pretty much the
same, the same thing.
But like in Fredi,
we had a cross product and we take two
vectors and we get an offer going
back to that.
So this is the property we we need
to do a FedExes.
Now, in 40, we can
have two vectors and multiply
them in some way to get something
original. Because when we get a plane,
because we didn't fix free
of a vector, so we have one left over.
Well, we can take three vectors and
do the determinant with this matrix here,
and that'll do it.
So this gives us one of two vectors, just
like we normally cross party.
So we just solve the normal
Grosberg problem.
So, yeah,
and of course we need no base vector
because usually we just needed three of
them.
So I'm called I call this term because
we all like quantum physics anyway.
We can do this in higher dimensions.
It basically looks the same, except we.
So what we do is if we have
anything in and we
we need one less than that and then we
can fix all the coronate and
then we get a vector back,
that's going to be your problem with
that. So which is part of the
formulas we need, we need later.
I'm not gonna go into delve into
the whole formulas and most of the time
you can just read a read of and later I'm
just going to figure out from version out
that I use for this stuff.
So, uh, another thing we need is a look
at transformations in Fredi.
I mean, you have like you have a point
you're looking at and you have your
camera, so you need to turn
around and face that point.
Fortunately, we don't actually need
trigonometric functions because
we can just, uh, we can
just build a rotation matrix by
filling in the last, uh, the last
column of a matrix with, uh,
what we're looking at minus where we're
from.
And then we need some orientation vectors
and 40 we need two
orientation vectors.
If you're if you're looking at your
Auburndale manual, you you have a
function that does that for you.
You get the three vectors, you give it
where you're looking at, where you're
looking from and which side is up.
So because you could rotate the camera
like this.
So in 40, you need,
uh, you need to fix another vector for
that.
So we have two from
open back and we feel
from right to left, we calculated
the last column and
we do our normal
or cross product over, uh,
of those vectors.
And then we go to the left and we slide
out one of the vectors
and then we keep doing that.
And the last one we have another normal
for for for everything.
Now, you can actually find this in
some publications online, but you
can't find this,
which is what we do.
And we want to do it in like days and
stuff.
It still looks the same,
but it's ugly.
But, uh, basically we still have
we have to lessen the number of
dimensions. And then we, uh, we
slide.
We slide.
We slightly calculated columns
in from the right and calculate towards
the left.
Right.
So we also want to do a prospective
transformation, which is what
basically, um,
basically you have a camera like this
instead of like this.
So it goes like two sides.
So that's part of the respective thing.
Fortunately, since
Auburndale is actually taking care of all
the cut offs and everything, and we we
just want to free the object, we can skip
a lot of stuff. And The Matrix doesn't
look nearly as our base of predicates.
So we can use a simple
matrix like this.
What we want to do is we only
have a triangle
which is like how wide our
I can look and everything.
But the last but the last two
dimensions, I mean the last one
dimension and this is V homogenous part,
they're fixed with the eye angle.
So the last coordinate
in our forty five fifty whatever
is always kind of like a death.
So yeah.
So basically that works.
The five decades is pretty much trivial.
We just have the last two fixed and
whatever we have is uh corrected
with the angle formula we had in
affricate.
So that takes care of that.
And then we need some kind of
matrix which is basically like we take
all of this stuff, put it all together,
and then we're happy because we only have
one matrix that we have to multiply away
for a vector.
And it's not that that is just a couple
of matrix multiplications.
This is actually a focus for Matrix.
And this is the five and this is a five
plus five.
But that's not so bad because we can just
so we implicitly expand this one to five.
S5 with, as
I previously explained, so
not to that four and the five decades
is pretty much pretty much the same.
So we don't really have to worry about
this. So I mean, five D and higher.
So what
do we do once we did our
matrix vector matrix multiplication?
So we still need to actually convert.
So we gave it a homogenous 40
coronate and our matrix
and we multiply those and
then we get a still a five coordinate
back.
And at this point
we need to take me home locking this part
out.
And we also need to do a prospective
division.
So this is the same function in both
cases. So we just apply that twice and
then we get free director back
and that's basically it.
So now we have a 3D vector.
We do the 3D thing again.
We have cameras for every dimension and
we're going.
So, yeah,
and I just show how that is done
or at least what it looks like.
So this is we have a clip from here
and we can rotate
this and pretty nothing too fancy.
A little program I wrote a while back
and we can do it in 40.
So it turns inside out
yay terms and set up.
If we did this with a 3D
thing and 40,
we would basically turn the whole thing
inside out because we do a 40 rotation
Afridi's object and
we can also do this in 50.
And I think it
gets kind of confusing after five, it's.
The rest of it for.
Seiner.
All right.
OK, now I promise
I'll read it from some systems, we're
going to have to skip over a couple of
these things so it's not too bad.
So, Nightrider, folks, this is the basic
idea is that you have you have like
two free four functions and you
have a set of starting vectors.
So you multiply
all of these with all of them and
they do that recursively and they
function like this and sum it all up.
And this is actually from a fractal flame
paper.
I don't quite agree with that because we
don't we don't really do that ad
infinitum when we destroy it,
because then we wouldn't see anything
because it would never stop.
So that. Yeah, but that's what the
effect of flame paper had.
So we can basically
choose any kind of function we want for
this.
A simple one is we
are fine transformation's,
which are basically we can we can
like take your point, we can
translate it and we can rotate it.
So normally you Dukas
game to render these things, which is
which is explain later.
But I'm not going to do that because we
can't it doesn't
work in 40 or Fredi because
you need too many points.
So what we can do is we take a
sufficiently divided, primitive
and then apply all of these functions
recursively and then we get pretty
pictures.
So, for example, if we tried to do a
Stravinsky gasket, we have this
square and then we can
we have free functions.
And they were like this where we go
on it and then make it smaller, which
we can all do with fine transformations.
We do that a couple of times recursively
and we can never, ever nicer, never
sharper picture.
So does this work and pretty enough?
Sure.
Uh.
We can't do that.
There's no problem here.
I'm going to stick with a pretty one,
these are randomly generated just
with the seeds. So it's just, you know,
so nothing too fancy here.
We converted this.
It looks pretty.
You can do this in 40 and turn
it inside out.
And so
you can turn it inside out, as you see
here, sort of,
well, some systems in too deep,
but they use a special kind of function
called a variation, which is
defined in that fructify
paper by Scott Scottrade, which
is reference NBA. And so you can read it
because it's cool.
And the variations kind of work like
this, you have your afine transformation
and you also apply
a set of their
variation. Functions which are
defined in appendix was like 40
noncanonical ones.
They kind of look like this.
So, for example, we
can have a function that does not think
we can have some smet apply signs to
to all of the coordinates
and also like an almost
Feargal transform all sorts
of kind of funky things.
There's a couple of weird concerns and
they thought about some cool stuff there.
So what you do is you you
take a number of these variations
and then you blend him together by
assigning weights for each of the
variations you want and adding
it all together so you can
have like half of a spherical one
quarter of a linear one and
another quarter
of the cosine one.
You can also make it more than one, but
then it kind of looks weird.
So then you can also
then they also added post transforms with
this, which is you take another one of
these and do it again and a final
one.
Yeah, they they really like that
transfer as they do it everywhere, which
is kind of cool.
So when I said
the things that Mackerras gave earlier,
so if you only do the Keres
game that's normal described, you get a
black and white picture, some
pixels are hit, some aren't.
That's not really pretty.
You can improve that by counting
the number of times you hit a particular
pixel and you have a grayscale image,
which is better.
It's a lot better if you take longer
to do that because you have lots of spots
that get hit. A lot of times, lots of
randomness in there.
So you want to smooth out some
of these smaller things.
So small differences, but you
still want to show like some of the
bigger ones. So it kind of worked
for them.
And if you like a column that
the way they do that is like every
function you have because you still have
more than one function.
Every function gets an index on a color
map and you mix them all together
and then you get much prettier pictures
and then you add some
vibrancy algorithm.
Always good.
Now, as
I said this, everything is strictly in
2D, so
it looks pretty, but it's not.
So can we do that in 3-D?
Well, we can't do if the can't because we
don't have enough memory for that,
but we can do it.
We can do it with a discrete version.
So,
yeah, it's been some fun time on this on
this program stuff,
so it doesn't crash,
OK, so
we want more possession.
We want this to be bigger.
It's already kind of looks cool.
It's already in 40.
So let's stick with her three decades
from now.
You still get the effect of it turning
inside out and everything, so it's kind
of a random one, just the first best seat
I had.
You notice that it's too deep because I
just took a plane and subdivided it.
A lot of times.
And then you run the
transforms, you can sort of extend some
of them, like I've seen the cosine one.
And you can just add a number of cosine.
You've seen this very good one.
You have a formula for Steer's narrative.
And so all this.
Well, and we can do
that.
We can't do it. Do that in 40 also,
which means our stuff can turn it inside
out. The cool part of an effective flame
is actually the way the colors work,
which turns out is really, really easy to
implement in a in a pixel shader.
If you're willing to do a logarithm and a
pixel shader
sounds worse than it is, it actually
works pretty well in real time.
So this is rendered in real time.
It's not the prettiest said ever,
but there's some room for improvement
here. So
and it's kind of nice to see that,
that it's still pretty damn smooth
so I can do some random colors.
OK, let's take this one.
So like the yellow and the
purple and the blue, which
is sort of visible, there are all
different functions and they get mixed
together in the pixel shader, which works
surprisingly well
and can do all sorts
of pretty well and.
Four years on, it's actually pretty nice
because it highlights
some of the properties.
This one is pretty good.
So depending on from where we look,
our pics of Qader with your original
Ibram has this kind of property of
amplifying geometry, which is pretty
cool. And we can again do this
in any kind of mention we want.
So we have this cool thing and
we can make it turn inside out.
Well, sorta.
Right.
So you get this nebula
and by doing a 40, 40
insideout thing, you can
make some pretty amazing pictures
if you find that kind of seeing
everything.
So this thing,
this piece of code is all
open source.
It's all all available on on GitHub.
You can play with it every once in your
browser, although unfortunately,
it turns out that Web T.L.
doesn't support
doesn't seem to support
this floating point texture's.
We just need to cound ones
up, unfortunately, Vavasour clan.
So it doesn't really work, which is kind
of sad, but we can,
but it still works in principle and you
can play it with most of the most of the
other things. Um, so yeah.
Uh, that's a program if you're
interested. There's a block series
explaining most of the formulas more in
more detail because it just wouldn't cut
it.
Uh, this is my, uh, Twitter,
if you're interested.
Here's your original paper about the fact
of Flamini written by Scott Race and
Every Corexit, which is a really amazing
read. And they put a lot of cool stuff in
there. So I always give it a read.
And as I said, something
much more French
we've all seen I'll see one.
And that's the end of that one.
So if anyone has any questions.
OK, thank you, Magnum, for this really
interesting talk. I really, really liked
fractals. I'm sure everybody here does
as well.
Before we start with questions, if you
absolutely have to leave, please be
quiet. That involves no talking.
If you can, please just stay until the
end of questions.
So are there any
questions from the Internet signal
angel?
Can you wave at me if you have any
questions?
Is there a question?
Yes, OK.
Ask away.
All the angels or so,
could you please turn on the second
largest microphone?
Maybe start. Let's start with a question
from microphone number three until
the signal is just set up.
Hey, thank you for your interest in, um,
I was wondering, in the production,
uh, transformation that you explain, you
assume that the last
dimension, it's one.
So it's going to be depth
and then you you you
assume that the rest are going to
collapse.
Uh, my question is, what happens if you
don't assume that or if there is
anything that changes?
Because in the in the
projection, you're kind of assuming
how the prediction is done to 3-D.
Right.
If I understand correctly that that is
true, I haven't actually explored what
happens if we do a different kind of 3D
transform in in this case.
So,
I mean, this perceptually could be
different.
It could be.
I've just tried to take the
prospective projection as you see the and
extend it to to 14, 14, 15
higher. So I'm an expert if it gets
different, but it would definitely be
amazing to try that out.
So I totally agree.
Yeah.
OK, single angel, do you have a
microphone.
Yes I have.
OK, please ask away standard user
from ISC asks are these visualization
applicable to the 3D virtual
reality experience and
will that give us the ability to better
understand one of the
fractals and project?
Since I'm I'm sure
that if we have um,
if we have something, a virtual reality
answer that can that can
do things in 3D right in front of us,
that would completely,
completely improve the experience and it
would definitely help us explore behere
dimensions better.
So I have again
tried it.
But if someone would like to try that
with VR, dear, the first
test it.
OK, let's go back to microphone number
three, please.
Yeah, just a quick question.
I noticed that even for the
multidimensional cases you used,
apparently to the primitives as the
primitive sea.
Would it be possible to use the high
dimensional primitives?
It would definitely be possible.
So, as you said, what I did
was I still use tries
and I kind of cheated by basically making
them see fruit as in because if you had
a 40, if you're looking
at a 3D object with 40, you can see
all the all facets simultaneously.
So it would definitely
be possible to to take
some action from it is it's just that
it's a lot easier to reason about if you
have something like it to do primitive
and you're trying to wrap your head
around it.
OK, are there are there
any more questions from the Internet,
Angel? Yes, there is.
OK, I'll ask away.
Um, Twilight Sparkle for my
he asks, do you think a game in 40
plus would be possible?
And do you think it would be playable?
So there actually are
games in 40 and higher.
I remember at least a game called 40
Mays, and there's a couple of adaptations
of the Rubik's Cube to at least 40,
50. And I think I've seen 70
with Rubik's Cube in particular.
I have to admit that I can solve a pretty
one.
So
I would be amazed to see anyone solve a
forty one. But it's possible
and it can be done.
So actually when one when I was thinking
about doing this stuff, the
transformations kind of look really cool
and it would be kind of cool to use it as
like a transformation effect.
Like you have your scene and you have no
see and then you can do a 40 swap
over, which I think would be really cool.
But I'm not sure, like,
I would love to see more games like this.
And there are some, but and.
Yeah.
OK, are there more intimate questions?
Not so far. OK,
one more microphone number three.
Thank you for your talk.
I have one question regarding
using it, not just those factors,
but just inserting 40 or
indeed datasets
from real data.
And does it work?
And how would
maybe look like and what trick
you used to get the transparency
effect for this data center?
So, I mean, in the demo,
I, um, I
mean, I basically just use a transfer
to the surface that that part was
easy. But I could
see this use in the higher dimensions
to visualize data sets.
So, I mean, when you're looking
at a lot of a lot of stuff where you
where you analyze stuff into into
different categories, you always end
up with more on free.
So if you would like to draw a graph, it
would probably be very
easy to make a plan or one in three
dimensions.
But I've not seen this done yet.
I would love to see this done.
But unfortunately, it is kind of hard to
wrap your head around a 40 rotation part.
So I'm not sure
if because we've got so used to make
stuff easier to see.
So I'm not sure if it will make it easier
to see if you're not trying to see me,
but it would probably be cool.
OK, we have time up for about
one or two more questions.
Let's try microphone number two for a
change, please.
Thank you for the talk.
I have some sort of more theoretical
question I was wondering.
You told us that there are forty seven
types of variations and I've never heard
of variations.
So what actually do they have some
sort of properties? It's kind of a very
small class.
Um, so they're just they're
just basically defined at
the end of that particular paper
with the severo here.
So this particular paper
so very specifically defined a
fairly common property they really have
is that they're sort of esthetically
pleasing on their own and
that they're all tuti so.
So you can extend some of them.
You can extend numbers, but they're
specifically defined in the appendix of
that paper.
OK, thank you very much.
OK, one last question.
Yes, not a question.
Just a quick info
on the 40 games question.
So there is a game that made the rounds
on the Internet this year called, and
I have no clue where the word comes from,
so I can't pronounce it mega churi.
Am I a k
u r e?
You can look it up and
what you can do in that game is like you
have a 3D environment, you look at it in
the 3D sense and then like you
get to a wall and in your dimension
that you're currently in, you can't walk
around the wall and then you can shift
to the fourth dimension or into
another version of your 3D world
along the fourth axis.
And then you can go around
the wall.
And in gameplay terms,
it's really just that you shift to
another dimension. I mean, a lot of games
do that. But the funny thing is that
that game actually implements it with the
correct math. So it looks really, really
fancy.
I remember seeing that I haven't
tried it. I wanted to try it.
So, I
mean, it makes sense
that it works because, well, when you do
40 projects, you really end up with the
free world. So it totally makes sense
that you could use this to
turn parts of the world around.
So I've seen it look very cool,
but I've tried it, but
probably. Well,
uh, let's thank the speaker
again, please.