A sequence where each value in the sequence is defined by the previous value.
A system that enacts rules on a set of variables to produce a state.
The study of dynamical systems defined by complex functions.
TODO
$fk+1 := f ˆ f^k$
Given
\begin{talign*} \onslide<3->{f^0(x) & = x \} \onslide<4->{f^1(x) & = x + 1 \} \onslide<5->{f^2(x) & = (x + 1) + 1 \} \onslide<6->{f^3(x) & = \left((x + 1) + 1\right) + 1 \} \onslide<6->{\vdots} \end{talign*}
Let
\begin{talign*} (a + b\symbf{i}) + (x + y\symbf{i}) \onslide<4->{& = (a + x) + (b + y)\symbf{i}} \end{talign*}
Let
\begin{talign*} (a + b\symbf{i}) ∗ (x + y\symbf{i}) \onslide<7->{& = ax + ay\symbf{i} + bx\symbf{i} + by\symbf{i}^2 \} \onslide<8->{& = (ax - by) + (ay + bx)\symbf{i}} \end{talign*}
$f^0(z) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \symbf{i}$ $f^1(z) \only<2>{= \left( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \symbf{i} \right)^2 = \left( \frac{1}{\sqrt{2}} \right)^2 - \left(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}\right)^2 \symbf{i}} = \symbf{i}$ $f^2(z) = -1$ $f^3(z) = 1$ $f^4(z) = f^5(z) = f^6(z) = 1$
$f^0(z) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \symbf{i}$ $f^1(z) \only<2>{= -\frac{1}{10} + \left(1 - \frac{1}{10}\right)\symbf{i}} = -0.1 + 0.9\symbf{i}$ $f^2(z) = -0.9-0.28\symbf{i}$ $f^3(z) = 0.6316+0.404\symbf{i}$ $f^4(z) ≈ 0.13570256+0.4103328\symbf{i}$ $f^5(z) ≈ -0.24995782+0.01136642\symbf{i}$ $f^6(z) ≈ -0.03765028-0.10568225\symbf{i}$
$f^0(z) = 0.5$ $f^1(z) = 0.05$ $f^2(z) = -0.1975$ $f^3(z) = -0.16099375$ $f^4(z) ≈ -0.1740810125$
$f^0(z) = 0.5 - 0.5 \symbf{i}$ $f^1(z) = -0.2 - 0.1 \symbf{i}$ $f^2(z) = -0.17 + 0.44 \symbf{i}$ $f^3(z) = -0.3647 + 0.2504 \symbf{i}$ $f^4(z) = -0.12969407 + 0.21735824 \symbf{i}$
N = 128
B = 4
c = complex(-0.2, 0.4)
def iterate(z):
for n in range(N):
z = z*z + c
if abs(z) > B*B: break
return n
Defined by iterative function in complex space
$f_c (z) = z^2 - 0.675 - 0.112\symbf{i}$ $K_c = \left\{ z_0 ∈ \symbb{C}: \abs{f^k_c \left(z_0 \right)} > B \text{ as } k → ∞\right\}$
- @@b:<2->@@
$\symbf{i}^2 = \symbf{ijk}$ \begin{talign*} \symbf{i}-1 \symbf{i}^2 & = \symbf{i}-1 \symbf{ijk}
\symbf{i} & = \symbf{jk} \end{talign*} - @@b:<3->@@
$\symbf{k}^2 = \symbf{ijk}$ \begin{talign*} \symbf{k}^2 \symbf{k}-1 & = \symbf{ijk} \symbf{k}-1
\symbf{k} & = \symbf{ij} \end{talign*} - @@b:<3->@@
$\symbf{j} = \symbf{ki}$
- @@b:<4->@@
$\symbf{i} = \symbf{jk}$ \begin{talign*} \symbf{ji} & = \symbf{jjk}
\symbf{ji} & = \symbf{j}^2 \symbf{k} \ \symbf{ji} & = -\symbf{k} \ -\symbf{k} & = \symbf{ji} \end{talign*} - @@b:<5->@@
$-\symbf{i} = \symbf{kj}$ - @@b:<5->@@
$-\symbf{j} = \symbf{ik}$
\begin{align*} p ∗ q \only<1-2>{& = dw + dx\symbf{i} + dy\symbf{j} + dz\symbf{k} \} \only<1-2>{& + aw\symbf{i} + ax\symbf{i}^2 + ay\symbf{ij} + az\symbf{ik} \} \only<1-2>{& + bw\symbf{j} + bx\symbf{ji} + by\symbf{j}^2 + bz\symbf{jk} \} \only<1-2>{& + cw\symbf{k} + cx\symbf{ki} + cy\symbf{kj} + cz\symbf{k}^2 \} \only<2-3>{& = dw - ax - by - cz \} \only<4>{& = dw - (ax + by + cz) \} \only<5->{& = dw - \begin{asvector} a\b\c \end{asvector} ⋅ \begin{asvector} x\y\z \end{asvector} \} \only<2-5>{& + dx\symbf{i} + aw\symbf{i} + bz\symbf{i} - cy\symbf{i} \} \only<2-5>{& + dy\symbf{j} - az\symbf{j} + bw\symbf{j} + cx\symbf{j} \} \only<2-5>{& + dz\symbf{k} + ay\symbf{k} - bx\symbf{k} + cw\symbf{k} \} \onslide<6->{& + \begin{avector}} \onslide<6->{dx + aw + bz - cy \} \onslide<6->{dy - az + bw + cx \} \onslide<6->{dz + ay - bx + cw} \onslide<6->{\end{avector}} \onslide<6->{⋅ \begin{avector} \symbf{i} \ \symbf{j} \ \symbf{k} \end{avector} \} \onslide<7->{& = dw - \begin{asvector} a\b\c \end{asvector} ⋅ \begin{asvector} x\y\z \end{asvector} + \left(d \begin{asvector} x\y\z \end{asvector} \only<8->{+ w \begin{asvector} a\b\c \end{asvector}} \only<9->{+ \begin{asvector} a\b\c \end{asvector} × \begin{asvector} x\y\z \end{asvector}} \only<7-8>{\cdots} \right) ⋅ \begin{asvector} \symbf{i}\\symbf{j}\\symbf{k} \end{asvector}} \end{align*}
def q_mult(p, q):
r = Quat(
p.w*q.w – p.x*q.x – p.y*q.y - p.z*q.z,
p.w*q.x + p.x*q.w + p.y*q.z - p.z*q.y,
p.w*q.y – p.x*q.z + p.y*q.w + p.y*q.x,
p.w*q.z + p.x*q.y – p.y*q.x + p.z*q.w
)
return r
def q_square(q):
r = Quat(
q.w*q.w – q.x*q.x – q.y*q.y - q.z*q.z,
2*q.w*q.x,
2*q.w*q.y,
2*q.w*q.z
)
return r
def q_add(p, q):
r = Quat(
p.w + q.w,
p.x + q.x,
p.y + q.y,
p.z + q.z
)
return r
def q_abs(q):
return q.w*q.w +
q.x*q.x +
q.y*q.y +
q.z*q.z
N = 12
B = 16
q = Quat(-0.2, 0.4, -0.4, -0.4)
def iterate(z):
for n in range(N):
z = q_add(q_square(z), q)
if q_abs(z) > B*B: break
return n
$f^0(z) = 0.5 - 0.5\symbf{i} + 0.5\symbf{j} - 0.5\symbf{k}$ $f^1(z) = -0.2 - 0.875\symbf{i} - 0.175\symbf{j} - 0.612\symbf{k}$ $f^2(z) = -0.831 - 0.025\symbf{i} - 0.605\symbf{j} + 0.133\symbf{k}$ $f^3(z) ≈ 0.6066 - 0.333\symbf{i} + 0.330\symbf{j} - 0.333\symbf{k}$ $f^4(z) ≈ 0.336 - 0.779\symbf{i} - 0.274\symbf{j} - 0.515\symbf{k}$ $f^5(z) ≈ -0.535 - 0.899\symbf{i} - 0.860\symbf{j} - 0.458\symbf{k}$
Let
- Defined by iterative function in 4D Quaternion space