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The Fractal Machine
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Contents
Introduction
History: patterns that repeat themselves
Lesson 0. Gates from transistors: for those who start from scratch
0.1. The idea: a transistor is a switch
0.2. Building
0.3. Check
Lesson 0 paste (Cmd+I)
Lesson 1. Memory: the latch and the D flip-flop
1.1. The idea: a loop that holds
1.2. Building
1.3. Check
Lesson 1 paste (Cmd+I)
Lesson 2. The counter: six flip-flops count to 63
2.1. The idea: add 1 in a column
2.2. Building
2.3. Check
Lesson 2 paste (Cmd+I)
Lesson 3. The raster scan: the screen dot by dot
3.1. The idea: the beam runs along the rows
3.2. Building
Try the patterns
3.3. Check
Lesson 3 paste (Cmd+I)
Lesson 4. The Sierpinski triangle: a fractal from ten gates
4.1. The idea: not a single common bit
4.2. Building
4.3. Check
Lesson 4 paste (Cmd+I)
Lesson 5. The chaos game: a triangle from chance
5.1. The idea: a half-way step
5.2. The idea: a step is a shift
5.3. Building
5.4. Check
Lesson 5 paste (Cmd+I)
Lesson 6 (bonus). Munching Squares: a living pattern from 1962
6.1. The idea: (x XOR y) < t
6.2. Building
6.3. Check
Lesson 6 paste (Cmd+I)
Summary
All the parts
Instructions in pictures
Lesson 3 in pictures
← Back to the lesson: Lesson 3. The raster scan: the screen dot by dot
3
Raster scan
counters X and Y draw the 64 × 64 screen
kh
HB-03
x 0 … 63
y 0 … 63
Hz
1×
0
1×
NOT
1×
0
CNT6
2×
2
XOR
1×
0
1×
∞
© 2026 Daniil Khanin · khanin.info · HB-03
23
CANVAS
1
Delete
2Hz
CLK
0
EN
0
RST
Q0
Q1
Q2
Q3
Q4
Q5
2
Delete
CO
6b
0
Q
3
Delete
CNT6
CLK
CLK
EN
EN
RST
RST
Q0
Q0
Q1
Q1
Q2
Q2
Q3
Q3
Q4
Q4
Q5
Q5
CO
CO
© 2026 Daniil Khanin · khanin.info · HB-03
24
CANVAS …
4
Dot display → ⚙: Layout Dots, 64 × 64, Port width 1 dot
5
Hz
1×
0
1×
NOT
1×
0
4×
60Hz
CLK
0
RST
NOT
I0
CLK
O0
nCLK
6
CNT6
1×
2
10×
CNT6
CLK
CLK
EN
VCC
RST
RST
Q0
x0
Q1
x1
Q2
x2
Q3
x3
Q4
x4
Q5
x5
CO
cx
7
CNT6
1×
2
10×
CNT6
CLK
CLK
EN
cx
RST
RST
Q0
y0
Q1
y1
Q2
y2
Q3
y3
Q4
y4
Q5
y5
CO
cy
8
XOR
1×
0
3×
XOR
I0
x3
I1
y3
O0
pix
© 2026 Daniil Khanin · khanin.info · HB-03
25
CANVAS …
9
1×
16×
Dot display
X0
x0
X1
x1
X2
x2
X3
x3
X4
x4
X5
x5
Y0
y0
Y1
y1
Y2
y2
Y3
y3
Y4
y4
Y5
y5
D0
pix
WE
VCC
CLK
nCLK
CLR
GND
60Hz → 4kHz
© 2026 Daniil Khanin · khanin.info · HB-03
26
← Back to the lesson
Lesson 3. The raster scan: the screen dot by dot