Les équations quadratiques maudites
"this weird shape is the solution of a
cursed quadratic equation where a is a
triangle B is a circle and C is a
spinning star and we can use different
shapes to get different solutions this
one is
satisfying this one is stressful and
this one is
boring there are way too many options to
put them all in the video so I made a
website where you can switch things up
yourself but before you do that let me
explain what's going on what does it
mean to put shapes in this equation
instead of just regular
numbers each shape is a collection of
points and each point is a complex
number the complex numbers are
two-dimensional so we can use them to
make shapes in my previous video I
explored applying a function to a shape
we take each point of the shape as an
input to the function and graph the
output the results are pretty
cool and doing this made me wonder what
if we combine different shapes like a
triangle times a
circle this presents a problem two
different shapes have two different sets
of points given this point on the
triangle which point of the circle do we
multiply with I decided to match points
using their angles so this point on the
triangle combines with this point on the
circle since they are the same angle
from the
xaxis so now that we can match the
points we can multiply shapes or add
them or subtract them there's a lot we
could do I think the quadratic formula
is a wonderful way to Showcase this it
has addition subtraction multiplication
division and even a square root but it's
kind of a pain to write so we'll instead
write the equation that it solves all
right let's do it let's say a is A
square B is a circle and C is a triangle
and the result
is that's actually kind of
disappointing uh what if a is a
hexagon or if C is a
pentagon H what if B is a triangle this
is not very fun why do they all look the
same to understand why let's make all
three shapes a triangle the result is
just two points what about all a square
all a hexagon it's always just those two
points since all the shapes are the same
we could divide out that shape so that A
B and C are all one input those into the
quadratic formula and we get2 plus or
minus < tk32 I and that's these two
points returning viewers May recognize
these as the eisenstein integers Omega
and Omega
squar three copies of any shape will
just give these points but what if
they're different
shapes well since we're using points on
the same angle the points will be close
together so those Solutions are close to
Omega and Omega
squ I'd like something more interesting
so let's split up those input points
we'll make the circle negative the
circle's point will have the negative
angle of the other
points okay this is a little better
let's switch around the
shapes and what if we make the triangle
-2 okay what about a star well at any
given angle there are two points which
one do we pick to draw the star we have
to make two laps around the origin so we
can say that this point is 45s Pi
radians since it's on the first lap
and this is 14 fths Pi radians since
it's on the second lap now each point
has a unique angle so we know which to
pick to match one: one with the circle
we just take half of the angle so the
first lap matches with the first half of
the circle and the second with the
second now we can get all three input
points away from each other using a star
a negative circle and a
pentagon hey this one actually looks
nice
and so does this let's rotate that
star
cool this one's interesting and what if
we add the points of the triangle wait
there's six the triangle only has three
points where the other three come
from it's because of the square root
each number has two Roots so each set of
points gives two solutions the solution
shape is actually a combination of a
positive shape and a negative shape but
it's not always clear which is which if
we vary the radius of the circle we see
the shapes split and this is why I
animate just single points instead of
trying to connect them when I first
started writing the code yes okay very
interesting more importantly changing
the radius looks cool let's do that
again and add the
points very
nice now now this pentagon is regular
which is fine but what if it were a
flower or if we made it wiggle or
smooth let's add wiggle to the 11 gone
these new shape types are really fun but
I don't always know what to use so I
added a random button to switch things
around sometimes the result is amazing
but sometimes it's not so I also added a
Hearts button to cycle through some of
my favorites though I won't spoil them
all in this video you'll have to check
them out
yourself and if you find an equation
that you really like you can copy the
settings and then paste them here later
to get right back where you were and you
should also paste them in the comments I
want to see the cool patterns you find
and I'll add my favorite ones to this
heart
section that's all I have to say so
thanks for watching and go play around
with the site
"
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