Chapter 1 – Introduction to Laegna Base‑4 Frequentials

Welcome to the Laegna Playground. This chapter introduces the Laegna Base‑4 IOAE system — a number system designed not only for computation, but for intuition, frequency‑based thinking, and child‑friendly learning. It is the foundation for the interactive counters and 3D visualizations that appear later.

The Task (for users who want to repeat the whole construction)

The Laegna system is built from a simple challenge:

“Can we create a base‑4 number system that behaves like a frequency ladder, aligns with binary exponentiation, and uses two kinds of zero?”

This chapter explains the rules so that you can reconstruct the system yourself. Later chapters show the math, code, and 3D visualization.

1. Why Base‑4?

Base‑4 is the smallest base that naturally expresses:

binary × binary structures
2‑dimensional frequency grids
octave‑like doubling (2, 4, 8, 16…)
clean symmetry around a center

In Laegna, base‑4 is not written as 0, 1, 2, 3. Instead, it uses IOAE — four letters that represent four “frequency tones”.

2. The Digits: I, O, A, E

These four symbols correspond to the four non‑zero values:

Digit	Value
I	1
O	2
A	3
E	4

The unusual part is that Laegna does not start from 0. Instead, it introduces two special zeroes.

3. The Two Zeroes: U and V

Laegna uses:

U — the unsigned zero (the “empty frequency”)
V — the signed zero (the “centered frequency”)

These two zeroes behave differently:

U appears at the beginning of each digit‑length cycle.
V appears at the center of each range, where the system “balances”.

This is why Laegna is called a frequential number system: each digit length forms a frequency band with a beginning (U) and a center (V).

4. R‑Levels: The Four Starting Points

Laegna numbers are grouped by R, the number of digits:

R = 1 → start at U
R = 2 → start at UU
R = 3 → start at UUU
R = 4 → start at UUUU

Each R‑level contains exactly 4ᴿ values. This mirrors binary exponentiation: 4ᴿ = (2²)ᴿ = 2^(2R).

This is why Laegna aligns naturally with:

Fourier decomposition
binary grids
2D wave interference
octave‑like doubling

5. The Counting Pattern

Each R‑level begins with U…U (R times). Then the system cycles through:

I (T=1, S=–2)
O (T=2, S=–1)
V (T=0, S=0) — the signed zero
A (T=3, S=+1)
E (T=4, S=+2)

The pair (T, S) is the “frequency coordinate” of each digit:

T = tonal height (0–4)
S = signed shift (–2 to +2)

When T = 0 or S = 0, the digit is displayed longer, with special visual effects (birth/death of a frequency).

6. Why This Matters

Laegna numbers are not just symbols — they are frequency coordinates. They map cleanly to:

binary exponentiation
2D Fourier tiles
octave‑like doubling
Hilbert‑space simplifications

This is why the system is used for:

teaching children to count
visualizing multidimensional waves
building intuitive math intuition
constructing the Laegna 3D hologram numbers

7. What Comes Next

In Chapter 2, you will see:

the formal math
the code generators (Python, Julia, Prolog, Spapython)
the OA (octavic) system
the 3D applet that animates the counters

For now, this chapter gives you the conceptual foundation: a base‑4 system with two zeroes, frequency coordinates, and octave‑aligned ranges.

Chapter 2 – Formal Math, Code, and the OA System

This chapter presents the formal mathematical structure of the Laegna Base‑4 IOAE system, including the two zeroes U and V, the (T, S) coordinate system, and the related OA (Octavic) base‑2 system. It also includes code generators so users can reproduce the system.

1. The Formal Digit Mapping

Laegna uses four non‑zero digits:

Digit	Value	T	S
I	1	1	-2
O	2	2	-1
V	0 (signed)	0	0
A	3	3	+1
E	4	4	+2

The unsigned zero U is not part of the digit cycle. It appears only at the start of each R‑level.

2. The Two Zeroes in LaTeX

The unsigned zero:

\[ U = \text{“empty frequency”}, \quad T(U) = 0,\quad S(U) = 0 \]

The signed zero:

\[ V = \text{“centered frequency”}, \quad T(V) = 0,\quad S(V) = 0 \]

They share the same coordinate but differ in role:

U marks the beginning of a range.
V marks the center of a range.

3. R‑Levels and Frequency Bands

Each R‑digit number spans:

\[ 4^R = 2^{2R} \]

This is the key reason Laegna aligns with:

binary exponentiation
2D Fourier tiles
octave‑like doubling

The starting point of each band is:

\[ \underbrace{UU\ldots U}_{R\text{ times}} \]

4. Digit Coordinates (T, S)

Each digit is mapped to a 2‑dimensional coordinate:

\[ d \mapsto (T(d), S(d)) \]

where:

T is tonal height (0–4)
S is signed shift (–2 to +2)

For a multi‑digit number:

\[ N = d_1 d_2 \ldots d_R \]

the full coordinate is:

\[ (T_N, S_N) = \left( \sum_{i=1}^R T(d_i)\,4^{R-i},\; \sum_{i=1}^R S(d_i)\,4^{R-i} \right) \]

This is the “frequency hologram” representation.

5. The OA (Octavic) System

OA is the base‑2 sibling of Laegna. It uses only two digits:

O → +1
A → –1

The two zeroes are:

U → unsigned zero
V → signed zero

OA divides each range into 8 equal parts (octavic symmetry).

This is why OA is used for:

binary waveforms
±1 signal encoding
Hilbert‑space simplification

6. Code Generators

Below are code blocks for generating Laegna numbers. Use the tabs above to switch languages.

# Python – Laegna generator
digits = ["I","O","V","A","E"]

def laegna_digits(R):
    if R == 0:
        return []
    if R == 1:
        return ["U"] + digits
    base = laegna_digits(R-1)
    return ["U"*R] + [d + x for d in digits for x in base]
        

# Julia – Laegna generator
digits = ["I","O","V","A","E"]

function laegna_digits(R)
    R == 1 && return ["U", digits...]
    prev = laegna_digits(R-1)
    return vcat(["U"^R], [d * x for d in digits for x in prev])
end
        

% Prolog – Laegna generator
digit("I"). digit("O"). digit("V"). digit("A"). digit("E").

laegna(1, ["U" | D]) :- findall(X, digit(X), D).

laegna(R, [U | Out]) :-
    R > 1,
    atom_concat_list(["U"], R, U),
    R1 is R - 1,
    laegna(R1, Prev),
    findall(DX, (digit(D), member(X, Prev), atom_concat(D, X, DX)), Out).
        

# Spapython – flat, simple
digits = ["I","O","V","A","E"]

def laegna(R):
    if R == 1:
        return ["U"] + digits
    prev = laegna(R-1)
    return ["U"*R] + [d + x for d in digits for x in prev]
        

7. Counter Placeholders

[Laegna Counter Placeholder]

[OA Counter Placeholder]

These counters will be animated in the 3D applet in the next stage.

8. What Comes Next

The next step is to integrate the 3D visualization:

digit holograms
frequency transitions
birth/death animations for U and V
range‑level octave shifts

The placeholders in the fullscreen applet are already prepared. The next phase will insert the rendering logic.