muzik wrote:Has any research been done into zero-sided polygons?
Me, a while ago, wrote:I'm currently trying to figure out how a universe in which Newton's second law is F=mj would behave. So far, I've determined that particles travel in parabolas (possibly degenerate) when not acted upon by any force, objects falling under a constant force with zero horizontal acceleration descend cubic curves, and that colliding objects attracted by a force rebound either superelastically or inelastic with a probability of 1 depending on how the force varies with distance — no kind of energy is in any way conserved, at least if my intuition is correct. I'm wondering what kind of work has been done on this problem.
from __future__ import print_function
from time import sleep
import operator
ARRAY_SIZE = (100, 50)
XRES_OVER_YRES = 0.5
class V:
def __init__(self, *args):
if len(args) == 1 and hasattr(args[0], '__iter__'):
self._t = tuple(map(float, args[0]))
self._d = len(self._t)
else:
self._t = tuple(map(float, args))
self._d = len(args)
def __str__(self):
return "<" + str(self._t)[1:-1] + ">"
def __repr__(self):
return "V" + str(self._t)
def __eq__(self, other):
return self._t == other._t
def __ne__(self, other):
return self._t != other._t
__hash__ = None
def __len__(self):
return self._d
def __getitem__(self, ind):
return self._t[ind]
def __setitem__(self, ind, val):
foo = self._t[ind] #To raise the correct exceptions
l = list(self._t)
l[ind] = val
self._t = tuple(l)
def __iter__(self):
return (i for i in self._t)
def __add__(self, other):
return V(map(operator.add, self._t, other._t))
def __sub__(self, other):
return V(map(operator.sub, self._t, other._t))
def __mul__(self, other):
if type(other) == type(self):
return sum(map(operator.mul, self._t, other._t))
else:
return V(map(operator.mul, self._t, (other,)*self._d))
def __div__(self, other):
return V(map(operator.div, self._t, (other,)*self._d))
def __truediv__(self, other):
return V(map(operator.truediv, self._t, (other,)*self._d))
def __mod__(self, other):
return V(map(operator.mod, self._t, (other,)*self._d))
def __neg__(self):
return V(map(operator.neg, self._t))
def __concat__(self, other):
return V(self._t+other._t)
def __abs__(self):
return sum(map(operator.mul, self._t, self._t))**0.5
class Obj:
def __init__(self, x, v, a, disp=None):
self.x = x
self.v = v
self.a = a
self.visible = disp is not None
if self.visible:
self.disp = disp
def tick(self, dt, n):
for i in xrange(n):
self.v += self.a*dt
self.x += self.v*dt
if self.visible:
self.disp.objsToDisp.append(self)
def jerk(self, j, dt):
self.a += j*dt
def getSymbol(self):
return "?"
class Disp:
def __init__(self, x_ul, xres):
self.objsToDisp = []
self.x = x_ul
self.xres = xres
self.yres = xres/XRES_OVER_YRES
def display(self):
disp = [[" " for i in xrange(ARRAY_SIZE[0])] for j in xrange(ARRAY_SIZE[1])]
for obj in self.objsToDisp:
x_obj = obj.x
s = obj.getSymbol()
disp[int(x_obj[1]/self.yres)][int(x_obj[0]/self.xres)] = s
print("\n".join(["".join(i) for i in disp]), end="\r"+"\x1b[A"*(len(disp)-1))
self.objsToDisp = []
d = Disp(V(0,0),1)
q = Obj(V(0,25),V(0,-3),V(0,0),d)
z = Obj(V(15,25),V(0,0),V(0,0),d)
t = 0.1
for i in xrange(int(15/t)):
q.jerk((z.x-q.x)/abs(z.x-q.x)**3*60, t)
q.tick(t, not not i)
z.tick(t, not not i)
d.display()
sleep(t)
gameoflifemaniac wrote:calcyman wrote:gameoflifemaniac wrote:Is it possible to emulate a normal computer in a quantum computer?
Yes, there's a standard trick for this. Given a function f : {0, 1}^n --> {0, 1}^m expressed as a circuit C of classical logic gates, you can replace the gates in the circuit with reversible equivalents to yield a reversible circuit C' (which may have lots of ancillary '0' inputs and some arbitrary messy outputs):
[n input bits][k ancillary '0' bits] --> [m output bits][n+k-m unwanted garbage bits]
Suppose we have another m ancillary '0' bits at this stage. Then we can CNOT the output bits with these ancillary bits to produce another copy of the output, like so:
[n input bits][k ancillary '0' bits][m extra '0' bits] --> [m output bits][n+k-m unwanted garbage bits][m output bits]
Now apply C' in reverse to the first n+k bits to clean up the mess we created:
[n input bits][k ancillary '0' bits][m extra '0' bits] --> [n input bits][k ancillary '0' bits][m output bits]
The upshot of this is that the ancillary '0' bits can be reused in a future computation. This combined circuit, ignoring the k ancillary '0' bits, actually computes the reversible function:
f : {0, 1}^(n+m) --> {0, 1}^(n+m)
(x, y) --> (x, f(x) XOR y)
Now, every reversible classical circuit is a quantum circuit (permutation matrices are unitary matrices), so this can be built out of quantum gates.
Adam, you are smart as hell.
x = 122, y = 107, rule = B3/S23
102bo$102bo$101b2o$36bo64bo$36bo63bo$37bo62bo$37bo61bo$37bo61bo$38bo
59bo$38bo58bo$38bo58bo$39bo56bo$39bo55bo$40bo54bo$40bo53bo$40bo52bo$
41bo51bo$41bo50bo$42bo48bo$42bo47bo$43bo45bo$43bo44bo$44bo42bo$44bo42b
o$45bo40bo$45bo39bo$45bo37b2o$45bo36bo$46bo34bo$46bo33bo$46bo33bo$47bo
31bo$47bo30bo$48bo28bo$48bo28bo$49bo26bo$49bo25bo$50bo23bo$50bo23bo$
50bo22bo$51bo20bo$51bo19bo$52bo17bo$52bo16bo$53bo14bo$53bo14bo$54bo12b
o$54bo11bo$55bo9bo$55bo8bo$55bo7bo$56bo5bo$56bo3b2o$56bo2bo$8o49b2o$8b
21o28bo$29b19o8bobo$48b74o$54bo3bo$53bo4bo$52bo5bo$51bo7bo$50bo8bo$49b
o9bo$48bo10bo$47bo11bo$46bo13bo$46bo13bo$45bo14bo$44bo16bo$43bo17bo$
43bo17bo$42bo19bo$41bo20bo$40bo22bo$40bo22bo$39bo23bo$38bo24bo$37bo26b
o$36bo27bo$35bo28bo$34bo29bo$33bo31bo$33bo31bo$32bo33bo$31bo34bo$30bo
35bo$30bo36bo$29bo37bo$28bo38bo$27bo40bo$26bo41bo$25bo42bo$24bo43bo$
23bo44bo$23bo45bo$22bo46bo$22bo46bo$20b2o47bo$19bo49bo$18bo50bo$17bo
51bo$16bo$14b2o$12b2o$10b2o$8b2o!
x = 63, y = 86, rule = B3/S23
28bo$28bo$28bo$28bo$28bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$
29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$
29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$29bo$
29bo$29bo$29bo$28b2o$29bo$28b2o$28bobo$27bo2bo$26bo4bo$26bo5bo$25bo7bo
$24bo9bo$24bo10bo$23bo11bo$23bo12bo$22bo14b2o$21bo17bo$21bo18bo$20bo
20b2o$19bo23b2o$18bo26bo$18bo27bo$17bo29b2o$16bo32bo$15bo34bo$15bo35bo
$14bo37bo$13bo39bo$12bo41bo$12bo42bo$11bo44bo$10bo46bo$10bo47bo$9bo49b
o$8bo51bo$8bo52bo$7bo54bo$6bo$5bo$5bo$4bo$3bo$2bo$2o!
gameoflifemaniac wrote:How are bits converted into music?
calcyman wrote:In a .wav file, you simply specify the amplitude of the sound wave (using 16-bit fixed-precision reals) every 1/44100th of a second. By Nyquist's sampling theorem, this is sufficient to fully recover any frequencies up to 22050 Hz, which is the upper range of human hearing. Then, you just pass it through a digital-to-analogue converter and into a loudspeaker, which therefore vibrates according to that sound wave.
muzik wrote:Why do the small stellated dodecahedron and great dodecahedron fit onto a sphere so well if they have a Euler characteristic of -6 - wouldn't it make more sense for them to tesselate a quadruple torus instead?
Macbi wrote: Since its faces intersect each other, the small stellated dodecahedron can't really be considered a valid polyhedron embedded in 3D space.
muzik wrote:Macbi wrote: Since its faces intersect each other, the small stellated dodecahedron can't really be considered a valid polyhedron embedded in 3D space.
By this logic wouldn't that invalidate every star polygon in 2D space?
muzik wrote:How do i calculate the Euler characteristics of regular hyperbolic polygonal tilings, and is it possible to tessellate a certain order torus with these?
gameoflifemaniac wrote:There exists a formula for the reduced quartic equation, but is there a formula for the full quartic equation?
Macbi wrote:gameoflifemaniac wrote:There exists a formula for the reduced quartic equation, but is there a formula for the full quartic equation?
Yes. It's easy to express the coefficients of the reduced quartic in terms of the original quartic, and then you can just substitute them in. But it's hilariously awful https://suhaimiramly.files.wordpress.co ... uartic.jpg
gameoflifemaniac wrote:Macbi wrote:gameoflifemaniac wrote:There exists a formula for the reduced quartic equation, but is there a formula for the full quartic equation?
Yes. It's easy to express the coefficients of the reduced quartic in terms of the original quartic, and then you can just substitute them in. But it's hilariously awful https://suhaimiramly.files.wordpress.co ... uartic.jpg
No. This is the formula for the reduced quartic equation, x^4+ax^3+bx^2+cx+d. But not for ax^4+bx^3+cx^2+dx+e.
And I knew about this formula, but I didn't see a formula for the full quartic equation.
x = -b/(4*a) - Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^
2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)]/2 -
Sqrt[b^2/(2*a^2) - (4*c)/(3*a) - (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e +
Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)) -
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^
(1/3)/(3*2^(1/3)*a) - (-(b^3/a^3) + (4*b*c)/a^2 - (8*d)/a)/(4*Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e -
72*a*c*e)^2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)])]/2
x = -b/(4*a) - Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^
2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)]/2 +
Sqrt[b^2/(2*a^2) - (4*c)/(3*a) - (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e +
Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)) -
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^
(1/3)/(3*2^(1/3)*a) - (-(b^3/a^3) + (4*b*c)/a^2 - (8*d)/a)/(4*Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e -
72*a*c*e)^2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)])]/2
x = -b/(4*a) + Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^
2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)]/2 -
Sqrt[b^2/(2*a^2) - (4*c)/(3*a) - (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e +
Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)) -
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^
(1/3)/(3*2^(1/3)*a) + (-(b^3/a^3) + (4*b*c)/a^2 - (8*d)/a)/(4*Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e -
72*a*c*e)^2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)])]/2
x = -b/(4*a) + Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^
2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)]/2 +
Sqrt[b^2/(2*a^2) - (4*c)/(3*a) - (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e +
Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)) -
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^
(1/3)/(3*2^(1/3)*a) + (-(b^3/a^3) + (4*b*c)/a^2 - (8*d)/a)/(4*Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e -
72*a*c*e)^2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)])]/2
calcyman wrote:The four solutions, according to Mathematica, are:Code: Select allx = -b/(4*a) - Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^
2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)]/2 -
Sqrt[b^2/(2*a^2) - (4*c)/(3*a) - (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e +
Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)) -
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^
(1/3)/(3*2^(1/3)*a) - (-(b^3/a^3) + (4*b*c)/a^2 - (8*d)/a)/(4*Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e -
72*a*c*e)^2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)])]/2
Code: Select allx = -b/(4*a) - Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^
2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)]/2 +
Sqrt[b^2/(2*a^2) - (4*c)/(3*a) - (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e +
Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)) -
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^
(1/3)/(3*2^(1/3)*a) - (-(b^3/a^3) + (4*b*c)/a^2 - (8*d)/a)/(4*Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e -
72*a*c*e)^2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)])]/2
Code: Select allx = -b/(4*a) + Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^
2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)]/2 -
Sqrt[b^2/(2*a^2) - (4*c)/(3*a) - (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e +
Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)) -
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^
(1/3)/(3*2^(1/3)*a) + (-(b^3/a^3) + (4*b*c)/a^2 - (8*d)/a)/(4*Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e -
72*a*c*e)^2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)])]/2Code: Select allx = -b/(4*a) + Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^
2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)]/2 +
Sqrt[b^2/(2*a^2) - (4*c)/(3*a) - (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e +
Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)) -
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^
(1/3)/(3*2^(1/3)*a) + (-(b^3/a^3) + (4*b*c)/a^2 - (8*d)/a)/(4*Sqrt[b^2/(4*a^2) - (2*c)/(3*a) + (2^(1/3)*(c^2 - 3*b*d + 12*a*e))/
(3*a*(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e -
72*a*c*e)^2])^(1/3)) + (2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e + Sqrt[-4*(c^2 - 3*b*d + 12*a*e)^3 +
(2*c^3 - 9*b*c*d + 27*a*d^2 + 27*b^2*e - 72*a*c*e)^2])^(1/3)/(3*2^(1/3)*a)])]/2
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