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Los mundos de Mandelbrot y Julia
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Contenido
Introduction
History: worlds nobody saw for a hundred years
Lesson 1. The 128 × 128 screen: seven bits instead of six
1.1. The idea: one more bit
1.2. The idea: reset from outside
1.3. Building
1.4. Check
Lesson 1 paste (Cmd+I)
Lesson 2. Fractions: the point nobody sees
2.1. The idea: a fixed point
2.2. The idea: add with the same adder
2.3. Building
2.4. Check
Lesson 2 paste (Cmd+I)
Lesson 3. The multiplier: long multiplication, all at once
3.1. The idea: long multiplication in binary
3.2. The idea: the sign separately
3.3. The idea: where the point went
3.4. Building
3.5. Check
Lesson 3 paste (Cmd+I)
Lesson 4. The Barnsley fern: a leaf from random jumps
4.1. The idea: four rules
4.2. The idea: chance — 7 shifts per clock
4.3. The idea: coefficients from a ROM
4.4. The idea: a screen without multiplication
4.5. Building
4.6. Check
Lesson 4 paste (Cmd+I)
Lesson 5. The Mandelbrot set: z = z² + c
5.1. The idea: does it run away or not
5.2. The idea: "escaped" in 14 bits
5.3. The idea: dot by dot
5.4. The price of a picture in clocks
5.5. Building
5.6. Check
Lesson 5 paste (Cmd+I)
Lesson 6. Julia worlds: a switch changes the universe
6.1. The idea: the same formula the other way round
6.2. The idea: a multiplexer
6.3. The idea: a different window
6.4. Building
6.5. Check
6.6. Julia worlds
Lesson 6 paste (Cmd+I)
Summary
All the parts
Instructions in pictures
Lesson 1 in pictures
← Back to the lesson: Lesson 1. The 128 × 128 screen: seven bits instead of six
1
The 128 × 128 screen
seven bits: CNT7 and the raster scan
kh
+03
64 × 64
128 × 128
VCC
1×
GND
1×
IN
3×
OUT
8×
XOR
7×
-1
AND
21×
-1
DFF
7×
-1
Hz
1×
0
1×
NOT
2×
-1
OR
5×
-1
NOR
1×
-1
1×
∞
Ctrl
+
G
CNT7
© 2026 Daniil Khanin · khanin.info
4
CANVAS
CNT7
1
VCC
1×
GND
1×
2×
VCC
VCC
GND
GND
2
IN
3×
OUT
8×
CLK
EN
nRST
Q0
Q1
Q2
Q3
Q4
Q5
Q6
CO
3
XOR
1×
-1
AND
2×
-1
9×
XOR
I0
Q0
I1
EN
O0
t0
AND
I0
t0
I1
nRST
O0
d0
AND
I0
EN
I1
Q0
O0
c1
© 2026 Daniil Khanin · khanin.info
5
CNT7 …
4
DFF
1×
-1
XOR
1×
-1
AND
1×
-1
9×
DFF
D
d0
CLK
CLK
Q
Q0
nQ
XOR
I0
Q1
I1
c1
O0
t1
AND
I0
t1
I1
nRST
O0
d1
5
AND
1×
-1
DFF
1×
-1
XOR
1×
-1
9×
AND
I0
c1
I1
Q1
O0
c2
DFF
D
d1
CLK
CLK
Q
Q1
nQ
XOR
I0
Q2
I1
c2
O0
t2
© 2026 Daniil Khanin · khanin.info
6
CNT7 …
6
AND
2×
-1
DFF
1×
-1
9×
AND
I0
t2
I1
nRST
O0
d2
AND
I0
c2
I1
Q2
O0
c3
DFF
D
d2
CLK
CLK
Q
Q2
nQ
7
XOR
1×
-1
AND
2×
-1
9×
XOR
I0
Q3
I1
c3
O0
t3
AND
I0
t3
I1
nRST
O0
d3
AND
I0
c3
I1
Q3
O0
c4
8
DFF
1×
-1
XOR
1×
-1
AND
1×
-1
9×
DFF
D
d3
CLK
CLK
Q
Q3
nQ
XOR
I0
Q4
I1
c4
O0
t4
AND
I0
t4
I1
nRST
O0
d4
© 2026 Daniil Khanin · khanin.info
7
CNT7 …
9
AND
1×
-1
DFF
1×
-1
XOR
1×
-1
9×
AND
I0
c4
I1
Q4
O0
c5
DFF
D
d4
CLK
CLK
Q
Q4
nQ
XOR
I0
Q5
I1
c5
O0
t5
10
AND
2×
-1
DFF
1×
-1
9×
AND
I0
t5
I1
nRST
O0
d5
AND
I0
c5
I1
Q5
O0
c6
DFF
D
d5
CLK
CLK
Q
Q5
nQ
11
XOR
1×
-1
AND
2×
-1
9×
XOR
I0
Q6
I1
c6
O0
t6
AND
I0
t6
I1
nRST
O0
d6
AND
I0
c6
I1
Q6
O0
CO
© 2026 Daniil Khanin · khanin.info
8
CNT7 …
CANVAS
12
DFF
1×
-1
3×
DFF
D
d6
CLK
CLK
Q
Q6
nQ
13
Ctrl
+
G
Library
…
Chip N
2×
Library
…
CNT7
Enter
CNT7
CLK
EN
nRST
Q0
Q1
Q2
Q3
Q4
Q5
Q6
CO
14
Dot display → ⚙: Layout Dots, 128 × 128, Port width 1 dot
15
Hz
1×
0
1×
NOT
1×
-1
4×
60Hz
CLK
0
RST
NOT
I0
RST
O0
nRST
© 2026 Daniil Khanin · khanin.info
9
CANVAS …
16
NOT
1×
-1
2×
NOT
I0
CLK
O0
nCLK
17
CNT7
1×
11×
CNT7
CLK
CLK
EN
VCC
nRST
nRST
Q0
x0
Q1
x1
Q2
x2
Q3
x3
Q4
x4
Q5
x5
Q6
x6
CO
cx
18
CNT7
1×
11×
CNT7
CLK
CLK
EN
cx
nRST
nRST
Q0
y0
Q1
y1
Q2
y2
Q3
y3
Q4
y4
Q5
y5
Q6
y6
CO
cy
19
AND
3×
-1
9×
AND
I0
x0
I1
y0
O0
a0
AND
I0
x1
I1
y1
O0
a1
AND
I0
x2
I1
y2
O0
a2
© 2026 Daniil Khanin · khanin.info
10
CANVAS …
20
AND
3×
-1
9×
AND
I0
x3
I1
y3
O0
a3
AND
I0
x4
I1
y4
O0
a4
AND
I0
x5
I1
y5
O0
a5
21
AND
1×
-1
OR
2×
-1
9×
AND
I0
x6
I1
y6
O0
a6
OR
I0
a0
I1
a1
O0
o01
OR
I0
a2
I1
a3
O0
o23
22
OR
3×
-1
9×
OR
I0
a4
I1
a5
O0
o45
OR
I0
o01
I1
o23
O0
o03
OR
I0
o45
I1
a6
O0
o46
© 2026 Daniil Khanin · khanin.info
11
CANVAS …
23
NOR
1×
-1
3×
NOR
I0
o03
I1
o46
O0
pix
24
1×
18×
Dot display
X0–X6
x0…x6
Y0–Y6
y0…y6
D0
pix
WE
VCC
CLK
nCLK
CLR
RST
2187
© 2026 Daniil Khanin · khanin.info
12
← Back to the lesson
Lesson 1. The 128 × 128 screen: seven bits instead of six